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32 lines
1.5 KiB
ReStructuredText
32 lines
1.5 KiB
ReStructuredText
**elefun.h** -- support for evaluation of elementary functions
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===============================================================================
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The exponential function
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--------------------------------------------------------------------------------
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.. function:: long elefun_exp_taylor_bound(long mag, long prec)
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Returns *n* such that
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`\left|\sum_{k=n}^{\infty} x^k / k!\right| \le 2^{-\mathrm{prec}}`,
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assuming `|x| \le 2^{\mathrm{mag}} \le 1/4`.
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.. function:: void elefun_exp_fixed_taylor_horner_precomp(fmpz_t y, fmpz_t h, const fmpz_t x, long n, long p)
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Letting `\epsilon = 2^{-p}`, computes `(y, h)` such that
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`|y \epsilon - \sum_{k=0}^{n-1} (x \epsilon)^k / k!| \le h \epsilon`
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using Horner's rule and a precomputed table of
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inverse factorials. We assume that *n* and *p* are small enough
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and that `|x \epsilon| < 1`.
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The coefficients `c_k \approx 1/k!` are precomputed using truncating divisions,
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with error at most `2^{-q}` for some `q \ge p`. Rounding to `p` bits is
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then done using truncation. Since repeated truncation preserves
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correctness of rounding, the coefficients have error at most `\epsilon`.
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We now evaluate
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`s_0 = c_{n-1}, s_1 = R(x \epsilon s_0) + c_{n-2}, \ldots s_{n-1} = R(x \epsilon s_{n-2}) + c_0`.
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where `R(t)` denotes fixed-point truncation which adds an error
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of at most `\epsilon`. The error of `s_{i+1}` exceeds that of that
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of `s_i` by at most `2\epsilon`. Using induction, we obtain
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the bound `h = 2n-1`.
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