Positive * pos_inf should be pos_inf, not 0 +- inf. Same for other combinations of signs. This addresses github issue #408.

arb/addmul.c

@@ -24,6 +24,13 @@ arb_addmul_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
         if (inexact)
             arf_mag_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec);
     }
+    else if (arf_is_inf(y) && arb_is_nonzero(x))
+    {
+        if (arf_sgn(arb_midref(x)) > 0)
+            arb_add_arf(z, z, y, prec);
+        else
+            arb_sub_arf(z, z, y, prec);
+    }
     else if (ARB_IS_LAGOM(x) && ARF_IS_LAGOM(y) && ARB_IS_LAGOM(z))
     {
         mag_fast_init_set_arf(ym, y);
@@ -60,6 +67,20 @@ arb_addmul(arb_t z, const arb_t x, const arb_t y, slong prec)
     {
         arb_addmul_arf(z, y, arb_midref(x), prec);
     }
+    else if (arf_is_inf(arb_midref(x)) && mag_is_finite(arb_radref(x)) && arb_is_nonzero(y))
+    {
+        if (arf_sgn(arb_midref(y)) > 0)
+            arb_add_arf(z, z, arb_midref(x), prec);
+        else
+            arb_sub_arf(z, z, arb_midref(x), prec);
+    }
+    else if (arf_is_inf(arb_midref(y)) && mag_is_finite(arb_radref(y)) && arb_is_nonzero(x))
+    {
+        if (arf_sgn(arb_midref(x)) > 0)
+            arb_add_arf(z, z, arb_midref(y), prec);
+        else
+            arb_sub_arf(z, z, arb_midref(y), prec);
+    }
     else if (ARB_IS_LAGOM(x) && ARB_IS_LAGOM(y) && ARB_IS_LAGOM(z))
     {
         mag_fast_init_set_arf(xm, arb_midref(x));

arb/fma.c

@@ -26,6 +26,13 @@ arb_fma_arf(arb_t res, const arb_t x, const arf_t y, const arb_t z, slong prec)
         else
             mag_set(arb_radref(res), arb_radref(z));
     }
+    else if (arf_is_inf(y) && arb_is_nonzero(x))
+    {
+        if (arf_sgn(arb_midref(x)) > 0)
+            arb_add_arf(res, z, y, prec);
+        else
+            arb_sub_arf(res, z, y, prec);
+    }
     else if (ARB_IS_LAGOM(res) && ARB_IS_LAGOM(x) && ARF_IS_LAGOM(y) && ARB_IS_LAGOM(z))
     {
         mag_t tm;
@@ -73,6 +80,20 @@ arb_fma(arb_t res, const arb_t x, const arb_t y, const arb_t z, slong prec)
     {
         arb_fma_arf(res, y, arb_midref(x), z, prec);
     }
+    else if (arf_is_inf(arb_midref(x)) && mag_is_finite(arb_radref(x)) && arb_is_nonzero(y))
+    {
+        if (arf_sgn(arb_midref(y)) > 0)
+            arb_add_arf(res, z, arb_midref(x), prec);
+        else
+            arb_sub_arf(res, z, arb_midref(x), prec);
+    }
+    else if (arf_is_inf(arb_midref(y)) && mag_is_finite(arb_radref(y)) && arb_is_nonzero(x))
+    {
+        if (arf_sgn(arb_midref(x)) > 0)
+            arb_add_arf(res, z, arb_midref(y), prec);
+        else
+            arb_sub_arf(res, z, arb_midref(y), prec);
+    }
     else if (ARB_IS_LAGOM(res) && ARB_IS_LAGOM(x) && ARB_IS_LAGOM(y) && ARB_IS_LAGOM(z))
     {
         mag_fast_init_set_arf(xm, arb_midref(x));

arb/mul.c

@@ -26,6 +26,15 @@ arb_mul_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
         else
             mag_zero(arb_radref(z));
     }
+    else if (arf_is_inf(y) && arb_is_nonzero(x))
+    {
+        mag_zero(arb_radref(z));
+
+        if (arf_sgn(arb_midref(x)) * arf_sgn(y) > 0)
+            arf_pos_inf(arb_midref(z));
+        else
+            arf_neg_inf(arb_midref(z));
+    }
     else if (ARB_IS_LAGOM(x) && ARF_IS_LAGOM(y) && ARB_IS_LAGOM(z))
     {
         mag_fast_init_set_arf(ym, y);
@@ -74,6 +83,16 @@ arb_mul(arb_t z, const arb_t x, const arb_t y, slong prec)
     {
         arb_mul_arf(z, x, arb_midref(y), prec);
     }
+    else if (arf_is_inf(arb_midref(x)) && mag_is_finite(arb_radref(x)) && arb_is_nonzero(y) ||
+             arf_is_inf(arb_midref(y)) && mag_is_finite(arb_radref(y)) && arb_is_nonzero(x))
+    {
+        mag_zero(arb_radref(z));
+
+        if (arf_sgn(arb_midref(x)) * arf_sgn(arb_midref(y)) > 0)
+            arf_pos_inf(arb_midref(z));
+        else
+            arf_neg_inf(arb_midref(z));
+    }
     else if (ARB_IS_LAGOM(x) && ARB_IS_LAGOM(y) && ARB_IS_LAGOM(z))
     {
         mag_fast_init_set_arf(xm, arb_midref(x));

arb/submul.c

@@ -24,6 +24,13 @@ arb_submul_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
         if (inexact)
             arf_mag_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec);
     }
+    else if (arf_is_inf(y) && arb_is_nonzero(x))
+    {
+        if (arf_sgn(arb_midref(x)) > 0)
+            arb_sub_arf(z, z, y, prec);
+        else
+            arb_add_arf(z, z, y, prec);
+    }
     else if (ARB_IS_LAGOM(x) && ARF_IS_LAGOM(y) && ARB_IS_LAGOM(z))
     {
         mag_fast_init_set_arf(ym, y);
@@ -60,6 +67,20 @@ arb_submul(arb_t z, const arb_t x, const arb_t y, slong prec)
     {
         arb_submul_arf(z, y, arb_midref(x), prec);
     }
+    else if (arf_is_inf(arb_midref(x)) && mag_is_finite(arb_radref(x)) && arb_is_nonzero(y))
+    {
+        if (arf_sgn(arb_midref(y)) > 0)
+            arb_sub_arf(z, z, arb_midref(x), prec);
+        else
+            arb_add_arf(z, z, arb_midref(x), prec);
+    }
+    else if (arf_is_inf(arb_midref(y)) && mag_is_finite(arb_radref(y)) && arb_is_nonzero(x))
+    {
+        if (arf_sgn(arb_midref(x)) > 0)
+            arb_sub_arf(z, z, arb_midref(y), prec);
+        else
+            arb_add_arf(z, z, arb_midref(y), prec);
+    }
     else if (ARB_IS_LAGOM(x) && ARB_IS_LAGOM(y) && ARB_IS_LAGOM(z))
     {
         mag_fast_init_set_arf(xm, arb_midref(x));

arb/test/t-addmul.c

@@ -39,6 +39,19 @@ mag_close(const mag_t am, const mag_t bm)
     return res1 && res2;
 }
 
+int
+arb_equal_mid_close_mag(const arb_t a, const arb_t b)
+{
+    return arf_equal(arb_midref(a), arb_midref(b)) &&
+        (mag_close(arb_radref(a), arb_radref(b)) ||
+            /* If a's and b's centres are infinite but their radii are finite, the radii don't need
+               to be close: they represent signed infinity regardless. If their centres are NaN,
+               then we should ignore their radii. */
+            (arf_is_inf(arb_midref(a)) && arf_is_inf(arb_midref(b)) &&
+                mag_is_finite(arb_radref(a)) && mag_is_finite(arb_radref(b))) ||
+            (arf_is_nan(arb_midref(a)) && arf_is_nan(arb_midref(b))));
+}
+
 void
 arb_addmul_naive(arb_t z, const arb_t x, const arb_t y, slong prec)
 {
@@ -214,8 +227,7 @@ int main()
                 arb_addmul(z, x, y, prec);
                 arb_addmul_naive(v, x, y, prec);
 
-                if (!arf_equal(arb_midref(z), arb_midref(v))
+                if (!arb_equal_mid_close_mag(z, v))
-                    || !mag_close(arb_radref(z), arb_radref(v)))
                 {
                     flint_printf("FAIL!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -232,8 +244,7 @@ int main()
                 arb_addmul(z, x, y, prec);
                 arb_addmul(v, x, x, prec);
 
-                if (!arf_equal(arb_midref(z), arb_midref(v))
+                if (!arb_equal_mid_close_mag(z, v))
-                    || !mag_close(arb_radref(z), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 1)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -248,8 +259,7 @@ int main()
                 arb_addmul(v, x, x, prec);
                 arb_addmul(x, x, x, prec);
 
-                if (!arf_equal(arb_midref(x), arb_midref(v))
+                if (!arb_equal_mid_close_mag(x, v))
-                    || !mag_close(arb_radref(x), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 2)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -263,8 +273,7 @@ int main()
                 arb_addmul(v, x, y, prec);
                 arb_addmul(x, x, y, prec);
 
-                if (!arf_equal(arb_midref(x), arb_midref(v))
+                if (!arb_equal_mid_close_mag(x, v))
-                    || !mag_close(arb_radref(x), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 3)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -279,8 +288,7 @@ int main()
                 arb_addmul(v, x, y, prec);
                 arb_addmul(x, y, x, prec);
 
-                if (!arf_equal(arb_midref(x), arb_midref(v))
+                if (!arb_equal_mid_close_mag(x, v))
-                    || !mag_close(arb_radref(x), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 4)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");

arb/test/t-mul.c

@@ -39,6 +39,19 @@ mag_close(const mag_t am, const mag_t bm)
     return res1 && res2;
 }
 
+int
+arb_equal_mid_close_mag(const arb_t a, const arb_t b)
+{
+    return arf_equal(arb_midref(a), arb_midref(b)) &&
+        (mag_close(arb_radref(a), arb_radref(b)) ||
+            /* If a's and b's centres are infinite but their radii are finite, the radii don't need
+               to be close: they represent signed infinity regardless. If their centres are NaN,
+               then we should ignore their radii. */
+            (arf_is_inf(arb_midref(a)) && arf_is_inf(arb_midref(b)) &&
+                mag_is_finite(arb_radref(a)) && mag_is_finite(arb_radref(b))) ||
+            (arf_is_nan(arb_midref(a)) && arf_is_nan(arb_midref(b))));
+}
+
 void
 arb_mul_naive(arb_t z, const arb_t x, const arb_t y, slong prec)
 {
@@ -73,15 +86,18 @@ arb_mul_naive(arb_t z, const arb_t x, const arb_t y, slong prec)
         fmpz_clear(e);
     }
 
-    /* propagated error */
+    /* propagated error - note that (signed infinity) * nonzero should be a signed
-    if (!arb_is_exact(x))
+       infinity, meaning the error should *not* propagate */
+    if (!arb_is_exact(x) && !(
+            arb_is_nonzero(x) && arf_is_inf(arb_midref(y)) && mag_is_finite(arb_radref(y))))
     {
         arf_set_mag(t, arb_radref(x));
         arf_abs(u, arb_midref(y));
         arf_addmul(zr, t, u, MAG_BITS, ARF_RND_UP);
     }
 
-    if (!arb_is_exact(y))
+    if (!arb_is_exact(y) && !(
+            arb_is_nonzero(y) && arf_is_inf(arb_midref(x)) && mag_is_finite(arb_radref(x))))
     {
         arf_set_mag(t, arb_radref(y));
         arf_abs(u, arb_midref(x));
@@ -265,8 +281,7 @@ int main()
                 arb_mul(z, x, y, prec);
                 arb_mul_naive(v, x, y, prec);
 
-                if (!arf_equal(arb_midref(z), arb_midref(v))
+                if (!arb_equal_mid_close_mag(z, v))
-                    || !mag_close(arb_radref(z), arb_radref(v)))
                 {
                     flint_printf("FAIL!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -282,8 +297,7 @@ int main()
                 arb_mul(z, x, y, prec);
                 arb_mul(v, x, x, prec);
 
-                if (!arf_equal(arb_midref(z), arb_midref(v))
+                if (!arb_equal_mid_close_mag(z, v))
-                    || !mag_close(arb_radref(z), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 1)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");

arb/test/t-submul.c

@@ -39,6 +39,19 @@ mag_close(const mag_t am, const mag_t bm)
     return res1 && res2;
 }
 
+int
+arb_equal_mid_close_mag(const arb_t a, const arb_t b)
+{
+    return arf_equal(arb_midref(a), arb_midref(b)) &&
+        (mag_close(arb_radref(a), arb_radref(b)) ||
+            /* If a's and b's centres are infinite but their radii are finite, the radii don't need
+               to be close: they represent signed infinity regardless. If their centres are NaN,
+               then we should ignore their radii. */
+            (arf_is_inf(arb_midref(a)) && arf_is_inf(arb_midref(b)) &&
+                mag_is_finite(arb_radref(a)) && mag_is_finite(arb_radref(b))) ||
+            (arf_is_nan(arb_midref(a)) && arf_is_nan(arb_midref(b))));
+}
+
 void
 arb_submul_naive(arb_t z, const arb_t x, const arb_t y, slong prec)
 {
@@ -214,8 +227,7 @@ int main()
                 arb_submul(z, x, y, prec);
                 arb_submul_naive(v, x, y, prec);
 
-                if (!arf_equal(arb_midref(z), arb_midref(v))
+                if (!arb_equal_mid_close_mag(z, v))
-                    || !mag_close(arb_radref(z), arb_radref(v)))
                 {
                     flint_printf("FAIL!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -232,8 +244,7 @@ int main()
                 arb_submul(z, x, y, prec);
                 arb_submul(v, x, x, prec);
 
-                if (!arf_equal(arb_midref(z), arb_midref(v))
+                if (!arb_equal_mid_close_mag(z, v))
-                    || !mag_close(arb_radref(z), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 1)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -248,8 +259,7 @@ int main()
                 arb_submul(v, x, x, prec);
                 arb_submul(x, x, x, prec);
 
-                if (!arf_equal(arb_midref(x), arb_midref(v))
+                if (!arb_equal_mid_close_mag(x, v))
-                    || !mag_close(arb_radref(x), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 2)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -263,8 +273,7 @@ int main()
                 arb_submul(v, x, y, prec);
                 arb_submul(x, x, y, prec);
 
-                if (!arf_equal(arb_midref(x), arb_midref(v))
+                if (!arb_equal_mid_close_mag(x, v))
-                    || !mag_close(arb_radref(x), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 3)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");
@@ -279,8 +288,7 @@ int main()
                 arb_submul(v, x, y, prec);
                 arb_submul(x, y, x, prec);
 
-                if (!arf_equal(arb_midref(x), arb_midref(v))
+                if (!arb_equal_mid_close_mag(x, v))
-                    || !mag_close(arb_radref(x), arb_radref(v)))
                 {
                     flint_printf("FAIL (aliasing 4)!\n");
                     flint_printf("x = "); arb_print(x); flint_printf("\n\n");