From d9469deb14ba6f55bd8af1652951ab306a8f63bd Mon Sep 17 00:00:00 2001 From: Wilco Dijkstra Date: Tue, 3 Apr 2018 16:33:13 +0100 Subject: [PATCH 3/7] sin/cos slow paths: remove slow paths from small range reduction This patch improves the accuracy of the range reduction. When the input is large (2^27) and very close to a multiple of PI/2, using 110 bits of PI is not enough. Improve range reduction accuracy to 136 bits. As a result the special checks for results close to zero can be removed. The ULP of the polynomials is at worst 0.55ULP, so there is no reason for the slow functions, and they can be removed. * sysdeps/ieee754/dbl-64/s_sin.c (reduce_sincos_1): Rename to reduce_sincos, improve accuracy to 136 bits. (do_sincos_1): Rename to do_sincos, remove fallbacks to slow functions. (__sin): Use improved reduction and simplified do_sincos calculation. (__cos): Likewise. * sysdeps/ieee754/dbl-64/s_sincos.c (__sincos): Likewise. --- sysdeps/ieee754/dbl-64/s_sin.c | 94 ++++++++++++++++++--------------------- sysdeps/ieee754/dbl-64/s_sincos.c | 6 +-- 2 files changed, 47 insertions(+), 53 deletions(-) (limited to 'sysdeps/ieee754/dbl-64') diff --git a/sysdeps/ieee754/dbl-64/s_sin.c b/sysdeps/ieee754/dbl-64/s_sin.c index c86fb9f..b8c366a 100644 --- a/sysdeps/ieee754/dbl-64/s_sin.c +++ b/sysdeps/ieee754/dbl-64/s_sin.c @@ -295,9 +295,13 @@ reduce_and_compute (double x, bool shift_quadrant) return retval; } +/* Reduce range of x to within PI/2 with abs (x) < 105414350. The high part + is written to *a, the low part to *da. Range reduction is accurate to 136 + bits so that when x is large and *a very close to zero, all 53 bits of *a + are correct. */ static inline int4 __always_inline -reduce_sincos_1 (double x, double *a, double *da) +reduce_sincos (double x, double *a, double *da) { mynumber v; @@ -306,62 +310,45 @@ reduce_sincos_1 (double x, double *a, double *da) v.x = t; double y = (x - xn * mp1) - xn * mp2; int4 n = v.i[LOW_HALF] & 3; - double db = xn * mp3; - double b = y - db; - db = (y - b) - db; + + double b, db, t1, t2; + t1 = xn * pp3; + t2 = y - t1; + db = (y - t2) - t1; + + t1 = xn * pp4; + b = t2 - t1; + db += (t2 - b) - t1; *a = b; *da = db; - return n; } -/* Compute sin (A + DA). cos can be computed by passing SHIFT_QUADRANT as - true, which results in shifting the quadrant N clockwise. */ +/* Compute sin or cos (A + DA) for the given quadrant N. */ static double __always_inline -do_sincos_1 (double a, double da, double x, int4 n, bool shift_quadrant) +do_sincos (double a, double da, int4 n) { - double xx, retval, res, cor; - double eps = fabs (x) * 1.2e-30; + double retval, cor; - int k1 = (n + shift_quadrant) & 3; - switch (k1) - { /* quarter of unit circle */ - case 2: - a = -a; - da = -da; - /* Fall through. */ - case 0: - xx = a * a; + if (n & 1) + /* Max ULP is 0.513. */ + retval = do_cos (a, da, &cor); + else + { + double xx = a * a; + /* Max ULP is 0.501 if xx < 0.01588, otherwise ULP is 0.518. */ if (xx < 0.01588) - { - /* Taylor series. */ - res = TAYLOR_SIN (xx, a, da, cor); - cor = 1.02 * cor + __copysign (eps, cor); - retval = (res == res + cor) ? res : sloww (a, da, x, shift_quadrant); - } + retval = TAYLOR_SIN (xx, a, da, cor); else - { - res = do_sin (a, da, &cor); - cor = 1.035 * cor + __copysign (eps, cor); - retval = ((res == res + cor) ? __copysign (res, a) - : sloww1 (a, da, x, shift_quadrant)); - } - break; - - case 1: - case 3: - res = do_cos (a, da, &cor); - cor = 1.025 * cor + __copysign (eps, cor); - retval = ((res == res + cor) ? ((n & 2) ? -res : res) - : sloww2 (a, da, x, n)); - break; + retval = __copysign (do_sin (a, da, &cor), a); } - return retval; + return (n & 2) ? -retval : retval; } + /*******************************************************************/ /* An ultimate sin routine. Given an IEEE double machine number x */ /* it computes the correctly rounded (to nearest) value of sin(x) */ @@ -374,13 +361,18 @@ SECTION #endif __sin (double x) { - double xx, t, cor; +#ifndef IN_SINCOS + double xx, t, a, da, cor; mynumber u; - int4 k, m; + int4 k, m, n; double retval = 0; -#ifndef IN_SINCOS SET_RESTORE_ROUND_53BIT (FE_TONEAREST); +#else + double xx, t, cor; + mynumber u; + int4 k, m; + double retval = 0; #endif u.x = x; @@ -419,9 +411,8 @@ __sin (double x) /*-------------------------- 2.426265<|x|< 105414350 ----------------------*/ else if (k < 0x419921FB) { - double a, da; - int4 n = reduce_sincos_1 (x, &a, &da); - retval = do_sincos_1 (a, da, x, n, false); + n = reduce_sincos (x, &a, &da); + retval = do_sincos (a, da, n); } /* else if (k < 0x419921FB ) */ /* --------------------105414350 <|x| <2^1024------------------------------*/ @@ -456,7 +447,11 @@ __cos (double x) { double y, xx, cor, a, da; mynumber u; +#ifndef IN_SINCOS + int4 k, m, n; +#else int4 k, m; +#endif double retval = 0; @@ -496,9 +491,8 @@ __cos (double x) #ifndef IN_SINCOS else if (k < 0x419921FB) { /* 2.426265<|x|< 105414350 */ - double a, da; - int4 n = reduce_sincos_1 (x, &a, &da); - retval = do_sincos_1 (a, da, x, n, true); + n = reduce_sincos (x, &a, &da); + retval = do_sincos (a, da, n + 1); } /* else if (k < 0x419921FB ) */ /* 105414350 <|x| <2^1024 */ diff --git a/sysdeps/ieee754/dbl-64/s_sincos.c b/sysdeps/ieee754/dbl-64/s_sincos.c index a9af8ce..4f032d2 100644 --- a/sysdeps/ieee754/dbl-64/s_sincos.c +++ b/sysdeps/ieee754/dbl-64/s_sincos.c @@ -79,10 +79,10 @@ __sincos (double x, double *sinx, double *cosx) if (k < 0x419921FB) { double a, da; - int4 n = reduce_sincos_1 (x, &a, &da); + int4 n = reduce_sincos (x, &a, &da); - *sinx = do_sincos_1 (a, da, x, n, false); - *cosx = do_sincos_1 (a, da, x, n, true); + *sinx = do_sincos (a, da, n); + *cosx = do_sincos (a, da, n + 1); return; } -- cgit v1.1