From 19a8b9a300f2f1f0012aff0f2b70b09430f50d9e Mon Sep 17 00:00:00 2001 From: Wilco Dijkstra Date: Tue, 3 Apr 2018 16:24:29 +0100 Subject: [PATCH 1/7] sin/cos slow paths: avoid slow paths for small inputs This series of patches removes the slow patchs from sin, cos and sincos. Besides greatly simplifying the implementation, the new version is also much faster for inputs up to PI (41% faster) and for large inputs needing range reduction (27% faster). ULP is ~0.55 with no errors found after testing 1.6 billion inputs across most of the range with mpsin and mpcos. The number of incorrectly rounded results (ie. ULP >0.5) is at most ~2750 per million inputs between 0.125 and 0.5, the average is ~850 per million between 0 and PI. Tested on AArch64 and x86_64 with no regressions. The first patch removes the slow paths for the cases where the input is small and doesn't require range reduction. Update ULP tables for sin, cos and sincos on AArch64 and x86_64. * sysdeps/aarch64/libm-test-ulps: Update ULP for sin, cos, sincos. * sysdeps/ieee754/dbl-64/s_sin.c (__sin): Remove slow paths for small inputs. (__cos): Likewise. * sysdeps/x86_64/fpu/libm-test-ulps: Update ULP for sin, cos, sincos. --- sysdeps/ieee754/dbl-64/s_sin.c | 40 ++++++++++++++-------------------------- 1 file changed, 14 insertions(+), 26 deletions(-) (limited to 'sysdeps/ieee754/dbl-64/s_sin.c') diff --git a/sysdeps/ieee754/dbl-64/s_sin.c b/sysdeps/ieee754/dbl-64/s_sin.c index 8c589cb..0c16b72 100644 --- a/sysdeps/ieee754/dbl-64/s_sin.c +++ b/sysdeps/ieee754/dbl-64/s_sin.c @@ -448,7 +448,7 @@ SECTION #endif __sin (double x) { - double xx, res, t, cor; + double xx, t, cor; mynumber u; int4 k, m; double retval = 0; @@ -471,26 +471,22 @@ __sin (double x) xx = x * x; /* Taylor series. */ t = POLYNOMIAL (xx) * (xx * x); - res = x + t; - cor = (x - res) + t; - retval = (res == res + 1.07 * cor) ? res : slow (x); + /* Max ULP of x + t is 0.535. */ + retval = x + t; } /* else if (k < 0x3fd00000) */ /*---------------------------- 0.25<|x|< 0.855469---------------------- */ else if (k < 0x3feb6000) { - res = do_sin (x, 0, &cor); - retval = (res == res + 1.096 * cor) ? res : slow1 (x); - retval = __copysign (retval, x); + /* Max ULP is 0.548. */ + retval = __copysign (do_sin (x, 0, &cor), x); } /* else if (k < 0x3feb6000) */ /*----------------------- 0.855469 <|x|<2.426265 ----------------------*/ else if (k < 0x400368fd) { - t = hp0 - fabs (x); - res = do_cos (t, hp1, &cor); - retval = (res == res + 1.020 * cor) ? res : slow2 (x); - retval = __copysign (retval, x); + /* Max ULP is 0.51. */ + retval = __copysign (do_cos (t, hp1, &cor), x); } /* else if (k < 0x400368fd) */ #ifndef IN_SINCOS @@ -541,7 +537,7 @@ SECTION #endif __cos (double x) { - double y, xx, res, cor, a, da; + double y, xx, cor, a, da; mynumber u; int4 k, m; @@ -561,8 +557,8 @@ __cos (double x) else if (k < 0x3feb6000) { /* 2^-27 < |x| < 0.855469 */ - res = do_cos (x, 0, &cor); - retval = (res == res + 1.020 * cor) ? res : cslow2 (x); + /* Max ULP is 0.51. */ + retval = do_cos (x, 0, &cor); } /* else if (k < 0x3feb6000) */ else if (k < 0x400368fd) @@ -571,20 +567,12 @@ __cos (double x) a = y + hp1; da = (y - a) + hp1; xx = a * a; + /* Max ULP is 0.501 if xx < 0.01588 or 0.518 otherwise. + Range reduction uses 106 bits here which is sufficient. */ if (xx < 0.01588) - { - res = TAYLOR_SIN (xx, a, da, cor); - cor = 1.02 * cor + __copysign (1.0e-31, cor); - retval = (res == res + cor) ? res : sloww (a, da, x, true); - } + retval = TAYLOR_SIN (xx, a, da, cor); else - { - res = do_sin (a, da, &cor); - cor = 1.035 * cor + __copysign (1.0e-31, cor); - retval = ((res == res + cor) ? __copysign (res, a) - : sloww1 (a, da, x, true)); - } - + retval = __copysign (do_sin (a, da, &cor), a); } /* else if (k < 0x400368fd) */ -- cgit v1.1