arb/doc/source/arb.rst

.. _arb:

**arb.h** -- real numbers represented as floating-point balls
===============================================================================

An :type:`arb_t` represents a ball over the real numbers,
that is, an interval `[m \pm r] \equiv [m-r, m+r]` where the midpoint `m` and the
radius `r` are (extended) real numbers and `r` is nonnegative (possibly infinite).
The result of an (approximate) operation done on :type:`arb_t` variables
is a ball which contains the result of the (mathematically exact) operation
applied to any choice of points in the input balls.
In general, the output ball is not the smallest possible.

The precision parameter passed to each function roughly indicates the
precision to which calculations on the midpoint are carried out
(operations on the radius are always done using a fixed, small
precision.)

For arithmetic operations, the precision parameter currently simply
specifies the precision of the corresponding :type:`arf_t` operation.
In the future, the arithmetic might be made faster by incorporating
sloppy rounding (typically equivalent to a loss of 1-2 bits of effective
working precision) when the result is known to be inexact (while still
propagating errors rigorously, of course).
Arithmetic operations done on exact input with exactly
representable output are always guaranteed to produce exact output.

For more complex operations, the precision parameter indicates a minimum
working precision (algorithms might allocate extra internal precision to
attempt to produce an output accurate to the requested number of bits,
especially when the required precision can be estimated easily, but this
is not generally required).

If the precision is increased and the inputs either are exact or are
computed with increased accuracy as well, the output should
converge proportionally, absent any bugs.
The general intended strategy for using ball arithmetic is to add a few
guard bits, and then repeat the calculation as necessary with an
exponentially increasing number of guard bits (Ziv's strategy) until the
result is exact
enough for one's purposes (typically the first attempt will be successful).

The following balls with an infinite or NaN component are permitted,
and may be returned as output from functions.

* The ball `[+\infty \pm c]`, where `c` is finite, represents the point at positive infinity. Such a ball can always be replaced by `[+\infty \pm 0]` while preserving mathematical correctness (this is currently not done automatically by the library).
* The ball `[-\infty \pm c]`, where `c` is finite, represents the point at negative infinity. Such a ball can always be replaced by `[-\infty \pm 0]` while preserving mathematical correctness (this is currently not done automatically by the library).
* The ball `[c \pm \infty]`, where `c` is finite or infinite, represents the whole extended real line `[-\infty,+\infty]`. Such a ball can always be replaced by `[0 \pm \infty]` while preserving mathematical correctness (this is currently not done automatically by the library). Note that there is no way to represent a half-infinite interval such as `[0,\infty]`.
* The ball `[\operatorname{NaN} \pm c]`, where `c` is finite or infinite, represents an indeterminate value (the value could be any extended real number, or it could represent a function being evaluated outside its domain of definition, for example where the result would be complex). Such an indeterminate ball can always be replaced by `[\operatorname{NaN} \pm \infty]` while preserving mathematical correctness (this is currently not done automatically by the library).

The :type:`arb_t` type is almost identical semantically to
the legacy :type:`fmprb_t` type, but uses a more efficient
internal representation.
Whereas the midpoint and radius of an :type:`fmprb_t` both have the
same type, the :type:`arb_t` type uses an :type:`arf_t` for the midpoint
and a :type:`mag_t` for the radius.  Code designed to manipulate the
radius of an :type:`fmprb_t` directly can be ported to the :type:`arb_t` type
by writing the radius to a temporary :type:`arf_t` variable, manipulating
that variable, and then converting back to the :type:`mag_t` radius.
Alternatively, :type:`mag_t` methods can be used directly where available.

Types, macros and constants
-------------------------------------------------------------------------------

.. type:: arb_struct

.. type:: arb_t

    An :type:`arb_struct` consists of an :type:`arf_struct` (the midpoint) and
    a :type:`mag_struct` (the radius).
    An :type:`arb_t` is defined as an array of length one of type
    :type:`arb_struct`, permitting an :type:`arb_t` to be passed by
    reference.

.. type:: arb_ptr

   Alias for ``arb_struct *``, used for vectors of numbers.

.. type:: arb_srcptr

   Alias for ``const arb_struct *``, used for vectors of numbers
   when passed as constant input to functions.

.. macro:: arb_midref(x)

    Macro returning a pointer to the midpoint of *x* as an :type:`arf_t`.

.. macro:: arb_radref(x)

    Macro returning a pointer to the radius of *x* as a :type:`mag_t`.

Memory management
-------------------------------------------------------------------------------

.. function:: void arb_init(arb_t x)

    Initializes the variable *x* for use. Its midpoint and radius are both
    set to zero.

.. function:: void arb_clear(arb_t x)

    Clears the variable *x*, freeing or recycling its allocated memory.

.. function:: arb_ptr _arb_vec_init(long n)

    Returns a pointer to an array of *n* initialized :type:`arb_struct`
    entries.

.. function:: void _arb_vec_clear(arb_ptr v, long n)

    Clears an array of *n* initialized :type:`arb_struct` entries.

.. function:: void arb_swap(arb_t x, arb_t y)

    Swaps *x* and *y* efficiently.

Assignment and rounding
-------------------------------------------------------------------------------

.. function:: void arb_set_fmprb(arb_t y, const fmprb_t x)

.. function:: void arb_get_fmprb(fmprb_t y, const arb_t x)

.. function:: void arb_set(arb_t y, const arb_t x)

.. function:: void arb_set_arf(arb_t y, const arf_t x)

.. function:: void arb_set_si(arb_t y, long x)

.. function:: void arb_set_ui(arb_t y, ulong x)

.. function:: void arb_set_d(arb_t y, double x)

.. function:: void arb_set_fmpz(arb_t y, const fmpz_t x)

    Sets *y* to the value of *x* without rounding.

.. function:: void arb_set_fmpz_2exp(arb_t y, const fmpz_t x, const fmpz_t e)

    Sets *y* to `x \cdot 2^e`.

.. function:: void arb_set_round(arb_t y, const arb_t x, long prec)

.. function:: void arb_set_round_fmpz(arb_t y, const fmpz_t x, long prec)

    Sets *y* to the value of *x*, rounded to *prec* bits.

.. function:: void arb_set_round_fmpz_2exp(arb_t y, const fmpz_t x, const fmpz_t e, long prec)

    Sets *y* to `x \cdot 2^e`, rounded to *prec* bits.

.. function:: void arb_set_fmpq(arb_t y, const fmpq_t x, long prec)

    Sets *y* to the rational number *x*, rounded to *prec* bits.

.. function:: int arb_set_str(arb_t res, const char * inp, long prec)

    Sets *res* to the value specified by the human-readable string *inp*.
    The input may be a decimal floating-point literal,
    such as "25", "0.001", "7e+141" or "-31.4159e-1", and may also consist
    of two such literals separated by the symbol "+/-" and optionally
    enclosed in brackets, e.g. "[3.25 +/- 0.0001]", or simply
    "[+/- 10]" with an implicit zero midpoint.
    The output is rounded to *prec* bits, and if the binary-to-decimal
    conversion is inexact, the resulting error is added to the radius.

    The symbols "inf" and "nan" are recognized (a nan midpoint results in an
    indeterminate interval, with infinite radius).

    Returns 0 if successful and nonzero if unsuccessful. If unsuccessful,
    the result is set to an indeterminate interval.

.. function:: char * arb_get_str(const arb_t x, long n, ulong flags)

    Returns a nice human-readable representation of *x*, with at most *n*
    digits of the midpoint printed.

    With default flags, the output can be parsed back with :func:`arb_set_str`,
    and this is guaranteed to produce an interval containing the original
    interval *x*.

    By default, the output is rounded so that the value given for the
    midpoint is correct up to 1 ulp (unit in the last decimal place).

    If *ARB_STR_MORE* is added to *flags*, more (possibly incorrect)
    digits may be printed.

    If *ARB_STR_NO_RADIUS* is added to *flags*, the radius is not
    included in the output if at least 1 digit of the midpoint
    can be printed.

    By adding a multiple *m* of *ARB_STR_CONDENSE* to *flags*, strings
    of more than three times *m* consecutive digits are condensed, only
    printing the leading and trailing *m* digits along with
    brackets indicating the number of digits omitted
    (useful when computing values to extremely high precision).

Assignment of special values
-------------------------------------------------------------------------------

.. function:: void arb_zero(arb_t x)

    Sets *x* to zero.

.. function:: void arb_one(arb_t f)

    Sets *x* to the exact integer 1.

.. function:: void arb_pos_inf(arb_t x)

    Sets *x* to positive infinity, with a zero radius.

.. function:: void arb_neg_inf(arb_t x)

    Sets *x* to negative infinity, with a zero radius.

.. function:: void arb_zero_pm_inf(arb_t x)

    Sets *x* to `[0 \pm \infty]`, representing the whole extended real line.

.. function:: void arb_indeterminate(arb_t x)

    Sets *x* to `[\operatorname{NaN} \pm \infty]`, representing
    an indeterminate result.

Input and output
-------------------------------------------------------------------------------

.. function:: void arb_print(const arb_t x)

    Prints the internal representation of *x*.

.. function:: void arb_printd(const arb_t x, long digits)

    Prints *x* in decimal. The printed value of the radius is not adjusted
    to compensate for the fact that the binary-to-decimal conversion
    of both the midpoint and the radius introduces additional error.

.. function:: void arb_printn(const arb_t x, long digits, ulong flags)

    Prints a nice decimal representation of *x*.
    By default, the output is guaranteed to be correct to within one unit
    in the last digit. An error bound is also printed explicitly.
    See :func:`arb_get_str` for details.

Random number generation
-------------------------------------------------------------------------------

.. function:: void arb_randtest(arb_t x, flint_rand_t state, long prec, long mag_bits)

    Generates a random ball. The midpoint and radius will both be finite.

.. function:: void arb_randtest_exact(arb_t x, flint_rand_t state, long prec, long mag_bits)

    Generates a random number with zero radius.

.. function:: void arb_randtest_precise(arb_t x, flint_rand_t state, long prec, long mag_bits)

    Generates a random number with radius around `2^{-\text{prec}}`
    the magnitude of the midpoint.

.. function:: void arb_randtest_wide(arb_t x, flint_rand_t state, long prec, long mag_bits)

    Generates a random number with midpoint and radius chosen independently,
    possibly giving a very large interval.

.. function:: void arb_randtest_special(arb_t x, flint_rand_t state, long prec, long mag_bits)

    Generates a random interval, possibly having NaN or an infinity
    as the midpoint and possibly having an infinite radius.

.. function:: void arb_get_rand_fmpq(fmpq_t q, flint_rand_t state, const arb_t x, long bits)

    Sets *q* to a random rational number from the interval represented by *x*.
    A denominator is chosen by multiplying the binary denominator of *x*
    by a random integer up to *bits* bits.

    The outcome is undefined if the midpoint or radius of *x* is non-finite,
    or if the exponent of the midpoint or radius is so large or small
    that representing the endpoints as exact rational numbers would
    cause overflows.

Radius and interval operations
-------------------------------------------------------------------------------

.. function:: void arb_get_mid_arb(arb_t m, const arb_t x)

    Sets *m* to the midpoint of *x*.

.. function:: void arb_get_rad_arb(arb_t r, const arb_t x)

    Sets *m* to the radius of *x*.

.. function:: void arb_add_error_arf(arb_t x, const arf_t err)

.. function:: void arb_add_error_mag(arb_t x, const mag_t err)

.. function:: void arb_add_error(arb_t x, const arb_t err)

    Adds the absolute value of *err* to the radius of *x* (the operation
    is done in-place).

.. function:: void arb_add_error_2exp_si(arb_t x, long e)

.. function:: void arb_add_error_2exp_fmpz(arb_t x, const fmpz_t e)

    Adds `2^e` to the radius of *x*.

.. function:: void arb_union(arb_t z, const arb_t x, const arb_t y, long prec)

    Sets *z* to a ball containing both *x* and *y*.

.. function:: void arb_get_abs_ubound_arf(arf_t u, const arb_t x, long prec)

    Sets *u* to the upper bound for the absolute value of *x*,
    rounded up to *prec* bits. If *x* contains NaN, the result is NaN.

.. function:: void arb_get_abs_lbound_arf(arf_t u, const arb_t x, long prec)

    Sets *u* to the lower bound for the absolute value of *x*,
    rounded down to *prec* bits. If *x* contains NaN, the result is NaN.

.. function:: void arb_get_mag(mag_t z, const arb_t x)

    Sets *z* to an upper bound for the absolute value of *x*. If *x* contains
    NaN, the result is positive infinity.

.. function:: void arb_get_mag_lower(mag_t z, const arb_t x)

    Sets *z* to a lower bound for the absolute value of *x*. If *x* contains
    NaN, the result is zero.

.. function:: void arb_get_mag_lower_nonnegative(mag_t z, const arb_t x)

    Sets *z* to a lower bound for the signed value of *x*, or zero
    if *x* overlaps with the negative half-axis. If *x* contains NaN,
    the result is zero.

.. function:: void arb_get_interval_fmpz_2exp(fmpz_t a, fmpz_t b, fmpz_t exp, const arb_t x)

    Computes the exact interval represented by *x*, in the form of an integer
    interval multiplied by a power of two, i.e. `x = [a, b] \times 2^{\text{exp}}`.

    The outcome is undefined if the midpoint or radius of *x* is non-finite,
    or if the difference in magnitude between the midpoint and radius
    is so large that representing the endpoints exactly would cause overflows.

.. function:: void arb_set_interval_arf(arb_t x, const arf_t a, const arf_t b, long prec)

.. function:: void arb_set_interval_mpfr(arb_t x, const mpfr_t a, const mpfr_t b, long prec)

    Sets *x* to a ball containing the interval `[a, b]`. We
    require that `a \le b`.

.. function:: void arb_get_interval_arf(arf_t a, arf_t b, const arb_t x, long prec)

.. function:: void arb_get_interval_mpfr(mpfr_t a, mpfr_t b, const arb_t x)

    Constructs an interval `[a, b]` containing the ball *x*. The MPFR version
    uses the precision of the output variables.

.. function:: long arb_rel_error_bits(const arb_t x)

    Returns the effective relative error of *x* measured in bits, defined as
    the difference between the position of the top bit in the radius
    and the top bit in the midpoint, plus one.
    The result is clamped between plus/minus *ARF_PREC_EXACT*.

.. function:: long arb_rel_accuracy_bits(const arb_t x)

    Returns the effective relative accuracy of *x* measured in bits,
    equal to the negative of the return value from :func:`arb_rel_error_bits`.

.. function:: long arb_bits(const arb_t x)

    Returns the number of bits needed to represent the absolute value
    of the mantissa of the midpoint of *x*, i.e. the minimum precision
    sufficient to represent *x* exactly. Returns 0 if the midpoint
    of *x* is a special value.

.. function:: void arb_trim(arb_t y, const arb_t x)

    Sets *y* to a trimmed copy of *x*: rounds *x* to a number of bits
    equal to the accuracy of *x* (as indicated by its radius),
    plus a few guard bits. The resulting ball is guaranteed to
    contain *x*, but is more economical if *x* has
    less than full accuracy.

.. function:: int arb_get_unique_fmpz(fmpz_t z, const arb_t x)

    If *x* contains a unique integer, sets *z* to that value and returns
    nonzero. Otherwise (if *x* represents no integers or more than one integer),
    returns zero.

.. function:: void arb_floor(arb_t y, const arb_t x, long prec)

.. function:: void arb_ceil(arb_t y, const arb_t x, long prec)

    Sets *y* to a ball containing `\lfloor x \rfloor` and `\lceil x \rceil`
    respectively, with the midpoint of *y* rounded to at most *prec* bits.

.. function:: void arb_get_fmpz_mid_rad_10exp(fmpz_t mid, fmpz_t rad, fmpz_t exp, const arb_t x, long n)

    Assuming that *x* is finite and not exactly zero, computes integers *mid*,
    *rad*, *exp* such that `x \in [m-r, m+r] \times 10^e` and such that the
    larger out of *mid* and *rad* has at least *n* digits plus a few guard
    digits. If *x* is infinite or exactly zero, the outputs are all set
    to zero.

Comparisons
-------------------------------------------------------------------------------

.. function:: int arb_is_zero(const arb_t x)

    Returns nonzero iff the midpoint and radius of *x* are both zero.

.. function:: int arb_is_nonzero(const arb_t x)

    Returns nonzero iff zero is not contained in the interval represented
    by *x*.

.. function:: int arb_is_one(const arb_t f)

    Returns nonzero iff *x* is exactly 1.

.. function:: int arb_is_finite(const arb_t x)

    Returns nonzero iff the midpoint and radius of *x* are both finite
    floating-point numbers, i.e. not infinities or NaN.

.. function:: int arb_is_exact(const arb_t x)

    Returns nonzero iff the radius of *x* is zero.

.. function:: int arb_is_int(const arb_t x)

    Returns nonzero iff *x* is an exact integer.

.. function:: int arb_equal(const arb_t x, const arb_t y)

    Returns nonzero iff *x* and *y* are equal as balls, i.e. have both the
    same midpoint and radius.

    Note that this is not the same thing as testing whether both
    *x* and *y* certainly represent the same real number, unless
    either *x* or *y* is exact (and neither contains NaN).
    To test whether both operands *might* represent the same mathematical
    quantity, use :func:`arb_overlaps` or :func:`arb_contains`,
    depending on the circumstance.

.. function:: int arb_is_positive(const arb_t x)

.. function:: int arb_is_nonnegative(const arb_t x)

.. function:: int arb_is_negative(const arb_t x)

.. function:: int arb_is_nonpositive(const arb_t x)

    Returns nonzero iff all points *p* in the interval represented by *x*
    satisfy, respectively, `p > 0`, `p \ge 0`, `p < 0`, `p \le 0`.
    If *x* contains NaN, returns zero.

.. function:: int arb_overlaps(const arb_t x, const arb_t y)

    Returns nonzero iff *x* and *y* have some point in common.
    If either *x* or *y* contains NaN, this function always returns nonzero
    (as a NaN could be anything, it could in particular contain any
    number that is included in the other operand).

.. function:: int arb_contains_arf(const arb_t x, const arf_t y)

.. function:: int arb_contains_fmpq(const arb_t x, const fmpq_t y)

.. function:: int arb_contains_fmpz(const arb_t x, const fmpz_t y)

.. function:: int arb_contains_si(const arb_t x, long y)

.. function:: int arb_contains_mpfr(const arb_t x, const mpfr_t y)

.. function:: int arb_contains(const arb_t x, const arb_t y)

    Returns nonzero iff the given number (or ball) *y* is contained in
    the interval represented by *x*.

    If *x* is contains NaN, this function always returns nonzero (as it
    could represent anything, and in particular could represent all
    the points included in *y*).
    If *y* contains NaN and *x* does not, it always returns zero.

.. function:: int arb_contains_zero(const arb_t x)

.. function:: int arb_contains_negative(const arb_t x)

.. function:: int arb_contains_nonpositive(const arb_t x)

.. function:: int arb_contains_positive(const arb_t x)

.. function:: int arb_contains_nonnegative(const arb_t x)

    Returns nonzero iff there is any point *p* in the interval represented
    by *x* satisfying, respectively, `p = 0`, `p < 0`, `p \le 0`, `p > 0`, `p \ge 0`.
    If *x* contains NaN, returns nonzero.

.. function:: int arb_eq(const arb_t x, const arb_t y)

.. function:: int arb_ne(const arb_t x, const arb_t y)

.. function:: int arb_lt(const arb_t x, const arb_t y)

.. function:: int arb_le(const arb_t x, const arb_t y)

.. function:: int arb_gt(const arb_t x, const arb_t y)

.. function:: int arb_ge(const arb_t x, const arb_t y)

    Respectively performs the comparison `x = y`, `x \ne y`,
    `x < y`, `x \le y`, `x > y`, `x \ge y` in a mathematically meaningful way.
    If the comparison `t \, (\operatorname{op}) \, u` holds for all
    `t \in x` and all `u \in y`, returns 1.
    Otherwise, returns 0.

    The balls *x* and *y* are viewed as subintervals of the extended real line.
    Note that balls that are formally different can compare as equal
    under this definition: for example, `[-\infty \pm 3] = [-\infty \pm 0]`.
    Also `[-\infty] \le [\infty \pm \infty]`.

    The output is always 0 if either input has NaN as midpoint.

Arithmetic
-------------------------------------------------------------------------------

.. function:: void arb_neg(arb_t y, const arb_t x)

.. function:: void arb_neg_round(arb_t y, const arb_t x, long prec)

    Sets *y* to the negation of *x*.

.. function:: void arb_abs(arb_t y, const arb_t x)

    Sets *y* to the absolute value of *x*. No attempt is made to improve the
    interval represented by *x* if it contains zero.

.. function:: void arb_add(arb_t z, const arb_t x, const arb_t y, long prec)

.. function:: void arb_add_arf(arb_t z, const arb_t x, const arf_t y, long prec)

.. function:: void arb_add_ui(arb_t z, const arb_t x, ulong y, long prec)

.. function:: void arb_add_si(arb_t z, const arb_t x, long y, long prec)

.. function:: void arb_add_fmpz(arb_t z, const arb_t x, const fmpz_t y, long prec)

    Sets `z = x + y`, rounded to *prec* bits. The precision can be
    *ARF_PREC_EXACT* provided that the result fits in memory.

.. function:: void arb_add_fmpz_2exp(arb_t z, const arb_t x, const fmpz_t m, const fmpz_t e, long prec)

    Sets `z = x + m \cdot 2^e`, rounded to *prec* bits. The precision can be
    *ARF_PREC_EXACT* provided that the result fits in memory.

.. function:: void arb_sub(arb_t z, const arb_t x, const arb_t y, long prec)

.. function:: void arb_sub_arf(arb_t z, const arb_t x, const arf_t y, long prec)

.. function:: void arb_sub_ui(arb_t z, const arb_t x, ulong y, long prec)

.. function:: void arb_sub_si(arb_t z, const arb_t x, long y, long prec)

.. function:: void arb_sub_fmpz(arb_t z, const arb_t x, const fmpz_t y, long prec)

    Sets `z = x - y`, rounded to *prec* bits. The precision can be
    *ARF_PREC_EXACT* provided that the result fits in memory.

.. function:: void arb_mul(arb_t z, const arb_t x, const arb_t y, long prec)

.. function:: void arb_mul_arf(arb_t z, const arb_t x, const arf_t y, long prec)

.. function:: void arb_mul_si(arb_t z, const arb_t x, long y, long prec)

.. function:: void arb_mul_ui(arb_t z, const arb_t x, ulong y, long prec)

.. function:: void arb_mul_fmpz(arb_t z, const arb_t x, const fmpz_t y, long prec)

    Sets `z = x \cdot y`, rounded to *prec* bits. The precision can be
    *ARF_PREC_EXACT* provided that the result fits in memory.

.. function:: void arb_mul_2exp_si(arb_t y, const arb_t x, long e)

.. function:: void arb_mul_2exp_fmpz(arb_t y, const arb_t x, const fmpz_t e)

    Sets *y* to *x* multiplied by `2^e`.

.. function:: void arb_addmul(arb_t z, const arb_t x, const arb_t y, long prec)

.. function:: void arb_addmul_arf(arb_t z, const arb_t x, const arf_t y, long prec)

.. function:: void arb_addmul_si(arb_t z, const arb_t x, long y, long prec)

.. function:: void arb_addmul_ui(arb_t z, const arb_t x, ulong y, long prec)

.. function:: void arb_addmul_fmpz(arb_t z, const arb_t x, const fmpz_t y, long prec)

    Sets `z = z + x \cdot y`, rounded to prec bits. The precision can be
    *ARF_PREC_EXACT* provided that the result fits in memory.

.. function:: void arb_submul(arb_t z, const arb_t x, const arb_t y, long prec)

.. function:: void arb_submul_arf(arb_t z, const arb_t x, const arf_t y, long prec)

.. function:: void arb_submul_si(arb_t z, const arb_t x, long y, long prec)

.. function:: void arb_submul_ui(arb_t z, const arb_t x, ulong y, long prec)

.. function:: void arb_submul_fmpz(arb_t z, const arb_t x, const fmpz_t y, long prec)

    Sets `z = z - x \cdot y`, rounded to prec bits. The precision can be
    *ARF_PREC_EXACT* provided that the result fits in memory.

.. function:: void arb_inv(arb_t y, const arb_t x, long prec)

    Sets *z* to `1 / x`.

.. function:: void arb_div(arb_t z, const arb_t x, const arb_t y, long prec)

.. function:: void arb_div_arf(arb_t z, const arb_t x, const arf_t y, long prec)

.. function:: void arb_div_si(arb_t z, const arb_t x, long y, long prec)

.. function:: void arb_div_ui(arb_t z, const arb_t x, ulong y, long prec)

.. function:: void arb_div_fmpz(arb_t z, const arb_t x, const fmpz_t y, long prec)

.. function:: void arb_fmpz_div_fmpz(arb_t z, const fmpz_t x, const fmpz_t y, long prec)

.. function:: void arb_ui_div(arb_t z, ulong x, const arb_t y, long prec)

    Sets `z = x / y`, rounded to *prec* bits. If *y* contains zero, *z* is
    set to `0 \pm \infty`. Otherwise, error propagation uses the rule

    .. math ::
        \left| \frac{x}{y} - \frac{x+\xi_1 a}{y+\xi_2 b} \right| =
        \left|\frac{x \xi_2 b - y \xi_1 a}{y (y+\xi_2 b)}\right| \le
        \frac{|xb|+|ya|}{|y| (|y|-b)}

    where `-1 \le \xi_1, \xi_2 \le 1`, and
    where the triangle inequality has been applied to the numerator and
    the reverse triangle inequality has been applied to the denominator.

.. function:: void arb_div_2expm1_ui(arb_t z, const arb_t x, ulong n, long prec)

    Sets `z = x / (2^n - 1)`, rounded to *prec* bits.

Powers and roots
-------------------------------------------------------------------------------

.. function:: void arb_sqrt(arb_t z, const arb_t x, long prec)

.. function:: void arb_sqrt_arf(arb_t z, const arf_t x, long prec)

.. function:: void arb_sqrt_fmpz(arb_t z, const fmpz_t x, long prec)

.. function:: void arb_sqrt_ui(arb_t z, ulong x, long prec)

    Sets *z* to the square root of *x*, rounded to *prec* bits.

    If `x = m \pm x` where `m \ge r \ge 0`, the propagated error is bounded by
    `\sqrt{m} - \sqrt{m-r} = \sqrt{m} (1 - \sqrt{1 - r/m}) \le \sqrt{m} (r/m + (r/m)^2)/2`.

.. function:: void arb_sqrtpos(arb_t z, const arb_t x, long prec)

    Sets *z* to the square root of *x*, assuming that *x* represents a
    nonnegative number (i.e. discarding any negative numbers in the input
    interval).

.. function:: void arb_hypot(arb_t z, const arb_t x, const arb_t y, long prec)

    Sets *z* to `\sqrt{x^2 + y^2}`.

.. function:: void arb_rsqrt(arb_t z, const arb_t x, long prec)

.. function:: void arb_rsqrt_ui(arb_t z, ulong x, long prec)

    Sets *z* to the reciprocal square root of *x*, rounded to *prec* bits.
    At high precision, this is faster than computing a square root.

.. function:: void arb_sqrt1pm1(arb_t z, const arb_t x, long prec)

    Sets `z = \sqrt{1+x}-1`, computed accurately when `x \approx 0`.

.. function:: void arb_root(arb_t z, const arb_t x, ulong k, long prec)

    Sets *z* to the *k*-th root of *x*, rounded to *prec* bits.
    This function selects between different algorithms. For large *k*,
    it evaluates `\exp(\log(x)/k)`. For small *k*, it uses :func:`arf_root`
    at the midpoint and computes a propagated error bound as follows:
    if input interval is `[m-r, m+r]` with `r \le m`, the error is largest at
    `m-r` where it satisfies

    .. math ::

        m^{1/k} - (m-r)^{1/k} = m^{1/k} [1 - (1-r/m)^{1/k}]

        = m^{1/k} [1 - \exp(\log(1-r/m)/k)]

        \le m^{1/k} \min(1, -\log(1-r/m)/k)

        = m^{1/k} \min(1, \log(1+r/(m-r))/k).

    This is evaluated using :func:`mag_log1p`.

.. function:: void arb_pow_fmpz_binexp(arb_t y, const arb_t b, const fmpz_t e, long prec)

.. function:: void arb_pow_fmpz(arb_t y, const arb_t b, const fmpz_t e, long prec)

.. function:: void arb_pow_ui(arb_t y, const arb_t b, ulong e, long prec)

.. function:: void arb_ui_pow_ui(arb_t y, ulong b, ulong e, long prec)

.. function:: void arb_si_pow_ui(arb_t y, long b, ulong e, long prec)

    Sets `y = b^e` using binary exponentiation (with an initial division
    if `e < 0`). Provided that *b* and *e*
    are small enough and the exponent is positive, the exact power can be
    computed by setting the precision to *ARF_PREC_EXACT*.

    Note that these functions can get slow if the exponent is
    extremely large (in such cases :func:`arb_pow` may be superior).

.. function:: void arb_pow_fmpq(arb_t y, const arb_t x, const fmpq_t a, long prec)

    Sets `y = b^e`, computed as `y = (b^{1/q})^p` if the denominator of
    `e = p/q` is small, and generally as `y = \exp(e \log b)`.

    Note that this function can get slow if the exponent is
    extremely large (in such cases :func:`arb_pow` may be superior).

.. function:: void arb_pow(arb_t z, const arb_t x, const arb_t y, long prec)

    Sets `z = x^y`, computed using binary exponentiation if `y` if
    a small exact integer, as `z = (x^{1/2})^{2y}` if `y` is a small exact
    half-integer, and generally as `z = \exp(y \log x)`.

Exponentials and logarithms
-------------------------------------------------------------------------------

.. function:: void arb_log_ui(arb_t z, ulong x, long prec)

.. function:: void arb_log_fmpz(arb_t z, const fmpz_t x, long prec)

.. function:: void arb_log_arf(arb_t z, const arf_t x, long prec)

.. function:: void arb_log(arb_t z, const arb_t x, long prec)

    Sets `z = \log(x)`.

    At low to medium precision (up to about 4096 bits), :func:`arb_log_arf`
    uses table-based argument reduction and fast Taylor series evaluation
    via :func:`_arb_atan_taylor_rs`. At high precision, it falls back to MPFR.
    The function :func:`arb_log` simply calls :func:`arb_log_arf` with
    the midpoint as input, and separately adds the propagated error.

.. function:: void arb_log_ui_from_prev(arb_t log_k1, ulong k1, arb_t log_k0, ulong k0, long prec)

    Computes `\log(k_1)`, given `\log(k_0)` where `k_0 < k_1`.
    At high precision, this function uses the formula
    `\log(k_1) = \log(k_0) + 2 \operatorname{atanh}((k_1-k_0)/(k_1+k_0))`,
    evaluating the inverse hyperbolic tangent using binary splitting
    (for best efficiency, `k_0` should be large and `k_1 - k_0` should
    be small). Otherwise, it ignores `\log(k_0)` and evaluates the logarithm
    the usual way.

.. function:: void arb_log1p(arb_t z, const arb_t x, long prec)

    Sets `z = \log(1+x)`, computed accurately when `x \approx 0`.

.. function:: void arb_exp(arb_t z, const arb_t x, long prec)

    Sets `z = \exp(x)`. Error propagation is done using the following rule:
    assuming `x = m \pm r`, the error is largest at `m + r`, and we have
    `\exp(m+r) - \exp(m) = \exp(m) (\exp(r)-1) \le r \exp(m+r)`.

.. function:: void arb_expm1(arb_t z, const arb_t x, long prec)

    Sets `z = \exp(x)-1`, computed accurately when `x \approx 0`.

.. function:: void arb_exp_invexp(arb_t z, arb_t w, const arb_t x, long prec)

    Sets `z = \exp(x)` and `w = \exp(-x)`. The second exponential is computed
    from the first using a division, but propagated error bounds are
    computed separately.

Trigonometric functions
-------------------------------------------------------------------------------

.. function:: void arb_sin(arb_t s, const arb_t x, long prec)

.. function:: void arb_cos(arb_t c, const arb_t x, long prec)

.. function:: void arb_sin_cos(arb_t s, arb_t c, const arb_t x, long prec)

    Sets `s = \sin(x)`, `c = \cos(x)`. Error propagation uses the rule
    `|\sin(m \pm r) - \sin(m)| \le \min(r,2)`.

.. function:: void arb_sin_pi(arb_t s, const arb_t x, long prec)

.. function:: void arb_cos_pi(arb_t c, const arb_t x, long prec)

.. function:: void arb_sin_cos_pi(arb_t s, arb_t c, const arb_t x, long prec)

    Sets `s = \sin(\pi x)`, `c = \cos(\pi x)`.

.. function:: void arb_tan(arb_t y, const arb_t x, long prec)

    Sets `y = \tan(x) = \sin(x) / \cos(y)`.

.. function:: void arb_cot(arb_t y, const arb_t x, long prec)

    Sets `y = \cot(x) = \cos(x) / \sin(y)`.

.. function:: void arb_sin_cos_pi_fmpq(arb_t s, arb_t c, const fmpq_t x, long prec)

.. function:: void arb_sin_pi_fmpq(arb_t s, const fmpq_t x, long prec)

.. function:: void arb_cos_pi_fmpq(arb_t c, const fmpq_t x, long prec)

    Sets `s = \sin(\pi x)`, `c = \cos(\pi x)` where `x` is a rational
    number (whose numerator and denominator are assumed to be reduced).
    We first use trigonometric symmetries to reduce the argument to the
    octant `[0, 1/4]`. Then we either multiply by a numerical approximation
    of `\pi` and evaluate the trigonometric function the usual way,
    or we use algebraic methods, depending on which is estimated to be faster.
    Since the argument has been reduced to the first octant, the
    first of these two methods gives full accuracy even if the original
    argument is close to some root other the origin.

.. function:: void arb_tan_pi(arb_t y, const arb_t x, long prec)

    Sets `y = \tan(\pi x)`.

.. function:: void arb_cot_pi(arb_t y, const arb_t x, long prec)

    Sets `y = \cot(\pi x)`.

Inverse trigonometric functions
-------------------------------------------------------------------------------

.. function:: void arb_atan_arf(arb_t z, const arf_t x, long prec)

.. function:: void arb_atan(arb_t z, const arb_t x, long prec)

    Sets `z = \operatorname{atan}(x)`.

    At low to medium precision (up to about 4096 bits), :func:`arb_atan_arf`
    uses table-based argument reduction and fast Taylor series evaluation
    via :func:`_arb_atan_taylor_rs`. At high precision, it falls back to MPFR.
    The function :func:`arb_atan` simply calls :func:`arb_atan_arf` with
    the midpoint as input, and separately adds the propagated error.

    The function :func:`arb_atan_arf` uses lookup tables if
    possible, and otherwise falls back to :func:`arb_atan_arf_bb`.

.. function:: void arb_atan2(arb_t z, const arb_t b, const arb_t a, long prec)

    Sets *r* to an the argument (phase) of the complex number
    `a + bi`, with the branch cut discontinuity on `(-\infty,0]`.
    We define `\operatorname{atan2}(0,0) = 0`, and for `a < 0`,
    `\operatorname{atan2}(0,a) = \pi`.

.. function:: void arb_asin(arb_t z, const arb_t x, long prec)

    Sets `z = \operatorname{asin}(x) = \operatorname{atan}(x / \sqrt{1-x^2})`.
    If `x` is not contained in the domain `[-1,1]`, the result is an
    indeterminate interval.

.. function:: void arb_acos(arb_t z, const arb_t x, long prec)

    Sets `z = \operatorname{acos}(x) = \pi/2 - \operatorname{asin}(x)`.
    If `x` is not contained in the domain `[-1,1]`, the result is an
    indeterminate interval.

Hyperbolic functions
-------------------------------------------------------------------------------

.. function:: void arb_sinh(arb_t s, const arb_t x, long prec)

.. function:: void arb_cosh(arb_t c, const arb_t x, long prec)

.. function:: void arb_sinh_cosh(arb_t s, arb_t c, const arb_t x, long prec)

    Sets `s = \sinh(x)`, `c = \cosh(x)`. If the midpoint of `x` is close
    to zero and the hyperbolic sine is to be computed,
    evaluates `(e^{2x}\pm1) / (2e^x)` via :func:`arb_expm1`
    to avoid loss of accuracy. Otherwise evaluates `(e^x \pm e^{-x}) / 2`.

.. function:: void arb_tanh(arb_t y, const arb_t x, long prec)

    Sets `y = \tanh(x) = \sinh(x) / \cosh(x)`, evaluated
    via :func:`arb_expm1` as `\tanh(x) = (e^{2x} - 1) / (e^{2x} + 1)`
    if `|x|` is small, and as
    `\tanh(\pm x) = 1 - 2 e^{\mp 2x} / (1 + e^{\mp 2x})`
    if `|x|` is large.

.. function:: void arb_coth(arb_t y, const arb_t x, long prec)

    Sets `y = \coth(x) = \cosh(x) / \sinh(x)`, evaluated using
    the same strategy as :func:`arb_tanh`.

Inverse hyperbolic functions
-------------------------------------------------------------------------------

.. function:: void arb_atanh(arb_t z, const arb_t x, long prec)

    Sets `z = \operatorname{atanh}(x)`.

.. function:: void arb_asinh(arb_t z, const arb_t x, long prec)

    Sets `z = \operatorname{asinh}(x)`.

.. function:: void arb_acosh(arb_t z, const arb_t x, long prec)

    Sets `z = \operatorname{acosh}(x)`.
    If `x < 1`, the result is an indeterminate interval.


Constants
-------------------------------------------------------------------------------

The following functions cache the computed values to speed up repeated
calls at the same or lower precision.
For further implementation details, see :ref:`algorithms_constants`.

.. function:: void arb_const_pi(arb_t z, long prec)

    Computes `\pi`.

.. function:: void arb_const_sqrt_pi(arb_t z, long prec)

    Computes `\sqrt{\pi}`.

.. function:: void arb_const_log_sqrt2pi(arb_t z, long prec)

    Computes `\log \sqrt{2 \pi}`.

.. function:: void arb_const_log2(arb_t z, long prec)

    Computes `\log(2)`.

.. function:: void arb_const_log10(arb_t z, long prec)

    Computes `\log(10)`.

.. function:: void arb_const_euler(arb_t z, long prec)

    Computes Euler's constant `\gamma = \lim_{k \rightarrow \infty} (H_k - \log k)`
    where `H_k = 1 + 1/2 + \ldots + 1/k`.

.. function:: void arb_const_catalan(arb_t z, long prec)

    Computes Catalan's constant `C = \sum_{n=0}^{\infty} (-1)^n / (2n+1)^2`.

.. function:: void arb_const_e(arb_t z, long prec)

    Computes `e = \exp(1)`.

.. function:: void arb_const_khinchin(arb_t z, long prec)

    Computes Khinchin's constant `K_0`.

.. function:: void arb_const_glaisher(arb_t z, long prec)

    Computes the Glaisher-Kinkelin constant `A = \exp(1/12 - \zeta'(-1))`.

.. function:: void arb_const_apery(arb_t z, long prec)

    Computes Apery's constant `\zeta(3)`.

Gamma function and factorials
-------------------------------------------------------------------------------

.. function:: void arb_rising_ui_bs(arb_t z, const arb_t x, ulong n, long prec)

.. function:: void arb_rising_ui_rs(arb_t z, const arb_t x, ulong n, ulong step, long prec)

.. function:: void arb_rising_ui_rec(arb_t z, const arb_t x, ulong n, long prec)

.. function:: void arb_rising_ui(arb_t z, const arb_t x, ulong n, long prec)

.. function:: void arb_rising(arb_t z, const arb_t x, const arb_t n, long prec)

    Computes the rising factorial `z = x (x+1) (x+2) \cdots (x+n-1)`.

    The *bs* version uses binary splitting. The *rs* version uses rectangular
    splitting. The *rec* version uses either *bs* or *rs* depending
    on the input. The default version uses the gamma function unless
    *n* is a small integer.

    The *rs* version takes an optional *step* parameter for tuning
    purposes (to use the default step length, pass zero).

.. function:: void arb_rising_fmpq_ui(arb_t z, const fmpq_t x, ulong n, long prec)

    Computes the rising factorial `z = x (x+1) (x+2) \cdots (x+n-1)` using
    binary splitting. If the denominator or numerator of *x* is large
    compared to *prec*, it is more efficient to convert *x* to an approximation
    and use :func:`arb_rising_ui`.

.. function :: void arb_rising2_ui_bs(arb_t u, arb_t v, const arb_t x, ulong n, long prec)

.. function :: void arb_rising2_ui_rs(arb_t u, arb_t v, const arb_t x, ulong n, ulong step, long prec)

.. function :: void arb_rising2_ui(arb_t u, arb_t v, const arb_t x, ulong n, long prec)

    Letting `u(x) = x (x+1) (x+2) \cdots (x+n-1)`, simultaneously compute
    `u(x)` and `v(x) = u'(x)`, respectively using binary splitting,
    rectangular splitting (with optional nonzero step length *step*
    to override the default choice), and an automatic algorithm choice.

.. function:: void arb_fac_ui(arb_t z, ulong n, long prec)

    Computes the factorial `z = n!` via the gamma function.

.. function:: void arb_fac2_ui(arb_t z, ulong n, long prec)

    Computes the double factorial `z = n!!` via the gamma function.

.. function:: void arb_bin_ui(arb_t z, const arb_t n, ulong k, long prec)

.. function:: void arb_bin_uiui(arb_t z, ulong n, ulong k, long prec)

    Computes the binomial coefficient `z = {n \choose k}`, via the
    rising factorial as `{n \choose k} = (n-k+1)_k / k!`.

.. function:: void arb_gamma(arb_t z, const arb_t x, long prec)

.. function:: void arb_gamma_fmpq(arb_t z, const fmpq_t x, long prec)

.. function:: void arb_gamma_fmpz(arb_t z, const fmpz_t x, long prec)

    Computes the gamma function `z = \Gamma(x)`.

.. function:: void arb_lgamma(arb_t z, const arb_t x, long prec)

    Computes the logarithmic gamma function `z = \log \Gamma(x)`.
    The complex branch structure is assumed, so if `x \le 0`, the
    result is an indeterminate interval.

.. function:: void arb_rgamma(arb_t z, const arb_t x, long prec)

    Computes the reciprocal gamma function `z = 1/\Gamma(x)`,
    avoiding division by zero at the poles of the gamma function.

.. function:: void arb_digamma(arb_t y, const arb_t x, long prec)

    Computes the digamma function `z = \psi(x) = (\log \Gamma(x))' = \Gamma'(x) / \Gamma(x)`.


Zeta function
-------------------------------------------------------------------------------

.. function:: void arb_zeta_ui_vec_borwein(arb_ptr z, ulong start, long num, ulong step, long prec)

    Evaluates `\zeta(s)` at `\mathrm{num}` consecutive integers *s* beginning
    with *start* and proceeding in increments of *step*.
    Uses Borwein's formula ([Bor2000]_, [GS2003]_),
    implemented to support fast multi-evaluation
    (but also works well for a single *s*).

    Requires `\mathrm{start} \ge 2`. For efficiency, the largest *s*
    should be at most about as
    large as *prec*. Arguments approaching *LONG_MAX* will cause
    overflows.
    One should therefore only use this function for *s* up to about *prec*, and
    then switch to the Euler product.

    The algorithm for single *s* is basically identical to the one used in MPFR
    (see [MPFR2012]_ for a detailed description).
    In particular, we evaluate the sum backwards to avoid storing more than one
    `d_k` coefficient, and use integer arithmetic throughout since it
    is convenient and the terms turn out to be slightly larger than
    `2^\mathrm{prec}`.
    The only numerical error in the main loop comes from the division by `k^s`,
    which adds less than 1 unit of error per term.
    For fast multi-evaluation, we repeatedly divide by `k^{\mathrm{step}}`.
    Each division reduces the input error and adds at most 1 unit of
    additional rounding error, so by induction, the error per term
    is always smaller than 2 units.

.. function:: void arb_zeta_ui_asymp(arb_t x, ulong s, long prec)

    Assuming `s \ge 2`, approximates `\zeta(s)` by `1 + 2^{-s}` along with
    a correct error bound. We use the following bounds: for `s > b`,
    `\zeta(s) - 1 < 2^{-b}`, and generally,
    `\zeta(s) - (1 + 2^{-s}) < 2^{2-\lfloor 3 s/2 \rfloor}`.

.. function:: void arb_zeta_ui_euler_product(arb_t z, ulong s, long prec)

    Computes `\zeta(s)` using the Euler product. This is fast only if *s*
    is large compared to the precision.

    Writing `P(a,b) = \prod_{a \le p \le b} (1 - p^{-s})`, we have
    `1/\zeta(s) = P(a,M) P(M+1,\infty)`.

    To bound the error caused by truncating
    the product at `M`, we write `P(M+1,\infty) = 1 - \epsilon(s,M)`.
    Since `0 < P(a,M) \le 1`, the absolute error for `\zeta(s)` is
    bounded by `\epsilon(s,M)`.

    According to the analysis in [Fil1992]_, it holds for all `s \ge 6` and `M \ge 1`
    that `1/P(M+1,\infty) - 1 \le f(s,M) \equiv 2 M^{1-s} / (s/2 - 1)`.
    Thus, we have `1/(1-\epsilon(s,M)) - 1 \le f(s,M)`, and expanding
    the geometric series allows us to conclude that
    `\epsilon(M) \le f(s,M)`.

.. function:: void arb_zeta_ui_bernoulli(arb_t x, ulong s, long prec)

    Computes `\zeta(s)` for even *s* via the corresponding Bernoulli number.

.. function:: void arb_zeta_ui_borwein_bsplit(arb_t x, ulong s, long prec)

    Computes `\zeta(s)` for arbitrary `s \ge 2` using a binary splitting
    implementation of Borwein's algorithm. This has quasilinear complexity
    with respect to the precision (assuming that `s` is fixed).

.. function:: void arb_zeta_ui_vec(arb_ptr x, ulong start, long num, long prec)

.. function:: void arb_zeta_ui_vec_even(arb_ptr x, ulong start, long num, long prec)

.. function:: void arb_zeta_ui_vec_odd(arb_ptr x, ulong start, long num, long prec)

    Computes `\zeta(s)` at *num* consecutive integers (respectively *num*
    even or *num* odd integers) beginning with `s = \mathrm{start} \ge 2`,
    automatically choosing an appropriate algorithm.

.. function:: void arb_zeta_ui(arb_t x, ulong s, long prec)

    Computes `\zeta(s)` for nonnegative integer `s \ne 1`, automatically
    choosing an appropriate algorithm. This function is
    intended for numerical evaluation of isolated zeta values; for
    multi-evaluation, the vector versions are more efficient.

.. function:: void arb_zeta(arb_t z, const arb_t s, long prec)

    Sets *z* to the value of the Riemann zeta function `\zeta(s)`.

    For computing derivatives with respect to `s`,
    use :func:`arb_poly_zeta_series`.

.. function:: void arb_hurwitz_zeta(arb_t z, const arb_t s, const arb_t a, long prec)

    Sets *z* to the value of the Hurwitz zeta function `\zeta(s,a)`.

    For computing derivatives with respect to `s`,
    use :func:`arb_poly_zeta_series`.

Bernoulli numbers
-------------------------------------------------------------------------------

.. function:: void arb_bernoulli_ui(arb_t b, ulong n, long prec)

    Sets `b` to the numerical value of the Bernoulli number `B_n` accurate
    to *prec* bits, computed by a division of the exact fraction if `B_n` is in
    the global cache or the exact numerator roughly is larger than
    *prec* bits, and using :func:`arb_bernoulli_ui_zeta`
    otherwise. This function reads `B_n` from the global cache
    if the number is already cached, but does not automatically extend
    the cache by itself.

.. function:: void arb_bernoulli_ui_zeta(arb_t b, ulong n, long prec)

    Sets `b` to the numerical value of `B_n` accurate to *prec* bits,
    computed using the formula `B_{2n} = (-1)^{n+1} 2 (2n)! \zeta(2n) / (2 \pi)^n`.

    To avoid potential infinite recursion, we explicitly call the
    Euler product implementation of the zeta function.
    We therefore assume that the precision is small
    enough and `n` large enough for the Euler product to converge
    rapidly (otherwise this function will effectively hang).

.. function:: void arb_power_sum_vec(arb_ptr res, const arb_t a, const arb_t b, long len, long prec)

    For *n* from 0 to *len* - 1, sets entry *n* in the output vector *res* to

    .. math ::

        S_n(a,b) = \frac{1}{n+1}\left(B_{n+1}(b) - B_{n+1}(a)\right)

    where `B_n(x)` is a Bernoulli polynomial. If *a* and *b* are integers
    and `b \ge a`, this is equivalent to

    .. math ::

        S_n(a,b) = \sum_{k=a}^{b-1} k^n.

    The computation uses the generating function for Bernoulli polynomials.

Polylogarithms
-------------------------------------------------------------------------------

.. function:: void arb_polylog(arb_t w, const arb_t s, const arb_t z, long prec)

.. function:: void arb_polylog_si(arb_t w, long s, const arb_t z, long prec)

    Sets *w* to the polylogarithm `\operatorname{Li}_s(z)`.

Other special functions
-------------------------------------------------------------------------------

.. function:: void arb_fib_fmpz(arb_t z, const fmpz_t n, long prec)

.. function:: void arb_fib_ui(arb_t z, ulong n, long prec)

    Computes the Fibonacci number `F_n`. Uses the binary squaring
    algorithm described in [Tak2000]_.
    Provided that *n* is small enough, an exact Fibonacci number can be
    computed by setting the precision to *ARF_PREC_EXACT*.

.. function:: void arb_agm(arb_t z, const arb_t x, const arb_t y, long prec)

    Sets *z* to the arithmetic-geometric mean of *x* and *y*.

.. function:: void arb_chebyshev_t_ui(arb_t a, ulong n, const arb_t x, long prec)

.. function:: void arb_chebyshev_u_ui(arb_t a, ulong n, const arb_t x, long prec)

    Evaluates the Chebyshev polynomial of the first kind `a = T_n(x)`
    or the Chebyshev polynomial of the second kind `a = U_n(x)`.

.. function:: void arb_chebyshev_t2_ui(arb_t a, arb_t b, ulong n, const arb_t x, long prec)

.. function:: void arb_chebyshev_u2_ui(arb_t a, arb_t b, ulong n, const arb_t x, long prec)

    Simultaneously evaluates `a = T_n(x), b = T_{n-1}(x)` or
    `a = U_n(x), b = U_{n-1}(x)`.
    Aliasing between *a*, *b* and *x* is not permitted.

.. function:: void arb_bell_sum_bsplit(arb_t res, const fmpz_t n, const fmpz_t a, const fmpz_t b, const fmpz_t mmag, long prec)

.. function:: void arb_bell_sum_taylor(arb_t res, const fmpz_t n, const fmpz_t a, const fmpz_t b, const fmpz_t mmag, long prec)

    Helper functions for Bell numbers, evaluating the sum
    `\sum_{k=a}^{b-1} k^n / k!`. If *mmag* is non-NULL, it may be used
    to indicate that the target error tolerance should be
    `2^{mmag - prec}`.

.. function:: void arb_bell_fmpz(arb_t res, const fmpz_t n, long prec)

.. function:: void arb_bell_ui(arb_t res, ulong n, long prec)

    Sets *res* to the Bell number `B_n`. If the number is too large to
    fit exactly in *prec* bits, a numerical approximation is computed
    efficiently.

Internals for computing elementary functions
-------------------------------------------------------------------------------

.. function:: void _arb_atan_taylor_naive(mp_ptr y, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N, int alternating)

.. function:: void _arb_atan_taylor_rs(mp_ptr y, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N, int alternating)

    Computes an approximation of `y = \sum_{k=0}^{N-1} x^{2k+1} / (2k+1)`
    (if *alternating* is 0) or `y = \sum_{k=0}^{N-1} (-1)^k x^{2k+1} / (2k+1)`
    (if *alternating* is 1). Used internally for computing arctangents
    and logarithms. The *naive* version uses the forward recurrence, and the
    *rs* version uses a division-avoiding rectangular splitting scheme.

    Requires `N \le 255`, `0 \le x \le 1/16`, and *xn* positive.
    The input *x* and output *y* are fixed-point numbers with *xn* fractional
    limbs. A bound for the ulp error is written to *error*.

.. function:: void _arb_exp_taylor_naive(mp_ptr y, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N)

.. function:: void _arb_exp_taylor_rs(mp_ptr y, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N)

    Computes an approximation of `y = \sum_{k=0}^{N-1} x^k / k!`. Used internally
    for computing exponentials. The *naive* version uses the forward recurrence,
    and the *rs* version uses a division-avoiding rectangular splitting scheme.

    Requires `N \le 287`, `0 \le x \le 1/16`, and *xn* positive.
    The input *x* is a fixed-point number with *xn* fractional
    limbs, and the output *y* is a fixed-point number with *xn* fractional
    limbs plus one extra limb for the integer part of the result.

    A bound for the ulp error is written to *error*.

.. function:: void _arb_sin_cos_taylor_naive(mp_ptr ysin, mp_ptr ycos, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N)

.. function:: void _arb_sin_cos_taylor_rs(mp_ptr ysin, mp_ptr ycos, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N, int sinonly, int alternating)

    Computes approximations of `y_s = \sum_{k=0}^{N-1} (-1)^k x^{2k+1} / (2k+1)!`
    and `y_c = \sum_{k=0}^{N-1} (-1)^k x^{2k} / (2k)!`.
    Used internally for computing sines and cosines. The *naive* version uses
    the forward recurrence, and the *rs* version uses a division-avoiding
    rectangular splitting scheme.

    Requires `N \le 143`, `0 \le x \le 1/16`, and *xn* positive.
    The input *x* and outputs *ysin*, *ycos* are fixed-point numbers with
    *xn* fractional limbs. A bound for the ulp error is written to *error*.

    If *sinonly* is 1, only the sine is computed; if *sinonly* is 0
    both the sine and cosine are computed.
    To compute sin and cos, *alternating* should be 1. If *alternating* is 0,
    the hyperbolic sine is computed (this is currently only intended to
    be used together with *sinonly*).

.. function:: int _arb_get_mpn_fixed_mod_log2(mp_ptr w, fmpz_t q, mp_limb_t * error, const arf_t x, mp_size_t wn)

    Attempts to write `w = x - q \log(2)` with `0 \le w < \log(2)`, where *w*
    is a fixed-point number with *wn* limbs and ulp error *error*.
    Returns success.

.. function:: int _arb_get_mpn_fixed_mod_pi4(mp_ptr w, fmpz_t q, int * octant, mp_limb_t * error, const arf_t x, mp_size_t wn)

    Attempts to write `w = |x| - q \pi/4` with `0 \le w < \pi/4`, where *w*
    is a fixed-point number with *wn* limbs and ulp error *error*.
    Returns success.

    The value of *q* mod 8 is written to *octant*. The output variable *q*
    can be NULL, in which case the full value of *q* is not stored.

.. function:: long _arb_exp_taylor_bound(long mag, long prec)

    Returns *n* such that
    `\left|\sum_{k=n}^{\infty} x^k / k!\right| \le 2^{-\mathrm{prec}}`,
    assuming `|x| \le 2^{\mathrm{mag}} \le 1/4`.

.. function:: void arb_exp_arf_bb(arb_t z, const arf_t x, long prec, int m1)

    Computes the exponential function using the bit-burst algorithm.
    If *m1* is nonzero, the exponential function minus one is computed
    accurately.

    Aborts if *x* is extremely small or large (where another algorithm
    should be used).

    For large *x*, repeated halving is used. In fact, we always
    do argument reduction until `|x|` is smaller than about `2^{-d}`
    where `d \approx 16` to speed up convergence. If `|x| \approx 2^m`,
    we thus need about `m+d` squarings.

    Computing `\log(2)` costs roughly 100-200 multiplications, so is not
    usually worth the effort at very high precision. However, this function
    could be improved by using `\log(2)` based reduction at precision low
    enough that the value can be assumed to be cached.

.. function:: void _arb_exp_sum_bs_simple(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp, const fmpz_t x, mp_bitcnt_t r, long N)

.. function:: void _arb_exp_sum_bs_powtab(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp, const fmpz_t x, mp_bitcnt_t r, long N)

    Computes *T*, *Q* and *Qexp* such that
    `T / (Q 2^{\text{Qexp}}) = \sum_{k=1}^N (x/2^r)^k/k!` using binary splitting.
    Note that the sum is taken to *N* inclusive and omits the constant term.

    The *powtab* version precomputes a table of powers of *x*,
    resulting in slightly higher memory usage but better speed. For best
    efficiency, *N* should have many trailing zero bits.

.. function:: void _arb_atan_sum_bs_simple(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp, const fmpz_t x, mp_bitcnt_t r, long N)

.. function:: void _arb_atan_sum_bs_powtab(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp, const fmpz_t x, mp_bitcnt_t r, long N)

    Computes *T*, *Q* and *Qexp* such that
    `T / (Q 2^{\text{Qexp}}) = \sum_{k=1}^N (-1)^k (x/2^r)^{2k} / (2k+1)`
    using binary splitting.
    Note that the sum is taken to *N* inclusive, omits the linear term,
    and requires a final multiplication by `(x/2^r)` to give the
    true series for atan.

    The *powtab* version precomputes a table of powers of *x*,
    resulting in slightly higher memory usage but better speed. For best
    efficiency, *N* should have many trailing zero bits.

.. function:: void arb_atan_arf_bb(arb_t z, const arf_t x, long prec)

    Computes the arctangent of *x*.
    Initially, the argument-halving formula

    .. math ::

        \operatorname{atan}(x) = 2 \operatorname{atan}\left(\frac{x}{1+\sqrt{1+x^2}}\right)

    is applied up to 8 times to get a small argument.
    Then a version of the bit-burst algorithm is used.
    The functional equation

    .. math ::

        \operatorname{atan}(x) = \operatorname{atan}(p/q) +
            \operatorname{atan}(w),
            \quad w = \frac{qx-p}{px+q},
            \quad p = \lfloor qx \rfloor

    is applied repeatedly instead of integrating a differential
    equation for the arctangent, as this appears to be more efficient.

Vector functions
-------------------------------------------------------------------------------

.. function:: void _arb_vec_zero(arb_ptr vec, long n)

    Sets all entries in *vec* to zero.

.. function:: int _arb_vec_is_zero(arb_srcptr vec, long len)

    Returns nonzero iff all entries in *x* are zero.

.. function:: int _arb_vec_is_finite(arb_srcptr x, long len)

    Returns nonzero iff all entries in *x* certainly are finite.

.. function:: void _arb_vec_set(arb_ptr res, arb_srcptr vec, long len)

    Sets *res* to a copy of *vec*.

.. function:: void _arb_vec_set_round(arb_ptr res, arb_srcptr vec, long len, long prec)

    Sets *res* to a copy of *vec*, rounding each entry to *prec* bits.

.. function:: void _arb_vec_swap(arb_ptr vec1, arb_ptr vec2, long len)

    Swaps the entries of *vec1* and *vec2*.

.. function:: void _arb_vec_neg(arb_ptr B, arb_srcptr A, long n)

.. function:: void _arb_vec_sub(arb_ptr C, arb_srcptr A, arb_srcptr B, long n, long prec)

.. function:: void _arb_vec_add(arb_ptr C, arb_srcptr A, arb_srcptr B, long n, long prec)

.. function:: void _arb_vec_scalar_mul(arb_ptr res, arb_srcptr vec, long len, const arb_t c, long prec)

.. function:: void _arb_vec_scalar_div(arb_ptr res, arb_srcptr vec, long len, const arb_t c, long prec)

.. function:: void _arb_vec_scalar_mul_fmpz(arb_ptr res, arb_srcptr vec, long len, const fmpz_t c, long prec)

.. function:: void _arb_vec_scalar_mul_2exp_si(arb_ptr res, arb_srcptr src, long len, long c)

.. function:: void _arb_vec_scalar_addmul(arb_ptr res, arb_srcptr vec, long len, const arb_t c, long prec)

   Performs the respective scalar operation elementwise.

.. function:: void _arb_vec_dot(arb_t res, arb_srcptr vec1, arb_srcptr vec2, long len2, long prec)

    Sets *res* to the dot product of *vec1* and *vec2*.

.. function:: void _arb_vec_norm(arb_t res, arb_srcptr vec, long len, long prec)

    Sets *res* to the dot product of *vec* with itself.

.. function:: void _arb_vec_get_mag(mag_t bound, arb_srcptr vec, long len, long prec)

    Sets *bound* to an upper bound for the entries in *vec*.

.. function:: long _arb_vec_bits(arb_srcptr x, long len)

    Returns the maximum of :func:`arb_bits` for all entries in *vec*.

.. function:: void _arb_vec_set_powers(arb_ptr xs, const arb_t x, long len, long prec)

    Sets *xs* to the powers `1, x, x^2, \ldots, x^{len-1}`.

.. function:: void _arb_vec_add_error_arf_vec(arb_ptr res, arf_srcptr err, long len)

.. function:: void _arb_vec_add_error_mag_vec(arb_ptr res, mag_srcptr err, long len)

    Adds the magnitude of each entry in *err* to the radius of the
    corresponding entry in *res*.

.. function:: void _arb_vec_indeterminate(arb_ptr vec, long len)

    Applies :func:`arb_indeterminate` elementwise.

.. function:: void _arb_vec_trim(arb_ptr res, arb_srcptr vec, long len)

    Applies :func:`arb_trim` elementwise.

.. function:: int _arb_vec_get_unique_fmpz_vec(fmpz * res,  arb_srcptr vec, long len)

    Calls :func:`arb_get_unique_fmpz` elementwise and returns nonzero if
    all entries can be rounded uniquely to integers. If any entry in *vec*
    cannot be rounded uniquely to an integer, returns zero.