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-rw-r--r--sysdeps/ieee754/dbl-64/s_sin.c904
1 files changed, 102 insertions, 802 deletions
diff --git a/sysdeps/ieee754/dbl-64/s_sin.c b/sysdeps/ieee754/dbl-64/s_sin.c
index ca2532fb63..b369ac9f5b 100644
--- a/sysdeps/ieee754/dbl-64/s_sin.c
+++ b/sysdeps/ieee754/dbl-64/s_sin.c
@@ -1,7 +1,7 @@
/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
- * Copyright (C) 2001-2016 Free Software Foundation, Inc.
+ * Copyright (C) 2001-2018 Free Software Foundation, Inc.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published by
@@ -22,22 +22,11 @@
/* */
/* FUNCTIONS: usin */
/* ucos */
-/* slow */
-/* slow1 */
-/* slow2 */
-/* sloww */
-/* sloww1 */
-/* sloww2 */
-/* bsloww */
-/* bsloww1 */
-/* bsloww2 */
-/* cslow2 */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
-/* branred.c sincos32.c dosincos.c mpa.c */
-/* sincos.tbl */
+/* branred.c sincos.tbl */
/* */
-/* An ultimate sin and routine. Given an IEEE double machine number x */
-/* it computes the correctly rounded (to nearest) value of sin(x) or cos(x) */
+/* An ultimate sin and cos routine. Given an IEEE double machine number x */
+/* it computes sin(x) or cos(x) with ~0.55 ULP. */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/* */
@@ -52,6 +41,8 @@
#include "MathLib.h"
#include <math.h>
#include <math_private.h>
+#include <math-underflow.h>
+#include <libm-alias-double.h>
#include <fenv.h>
/* Helper macros to compute sin of the input values. */
@@ -65,35 +56,11 @@
a - a^3/3! + a^5/5! - a^7/7! + a^9/9! + (1 - a^2) * da / 2
The constants s1, s2, s3, etc. are pre-computed values of 1/3!, 1/5! and so
- on. The result is returned to LHS and correction in COR. */
-#define TAYLOR_SIN(xx, a, da, cor) \
+ on. The result is returned to LHS. */
+#define TAYLOR_SIN(xx, a, da) \
({ \
double t = ((POLYNOMIAL (xx) * (a) - 0.5 * (da)) * (xx) + (da)); \
double res = (a) + t; \
- (cor) = ((a) - res) + t; \
- res; \
-})
-
-/* This is again a variation of the Taylor series expansion with the term
- x^3/3! expanded into the following for better accuracy:
-
- bb * x ^ 3 + 3 * aa * x * x1 * x2 + aa * x1 ^ 3 + aa * x2 ^ 3
-
- The correction term is dx and bb + aa = -1/3!
- */
-#define TAYLOR_SLOW(x0, dx, cor) \
-({ \
- static const double th2_36 = 206158430208.0; /* 1.5*2**37 */ \
- double xx = (x0) * (x0); \
- double x1 = ((x0) + th2_36) - th2_36; \
- double y = aa * x1 * x1 * x1; \
- double r = (x0) + y; \
- double x2 = ((x0) - x1) + (dx); \
- double t = (((POLYNOMIAL2 (xx) + bb) * xx + 3.0 * aa * x1 * x2) \
- * (x0) + aa * x2 * x2 * x2 + (dx)); \
- t = (((x0) - r) + y) + t; \
- double res = r + t; \
- (cor) = (r - res) + t; \
res; \
})
@@ -123,156 +90,69 @@ static const double
cs4 = -4.16666666666664434524222570944589E-02,
cs6 = 1.38888874007937613028114285595617E-03;
-static const double t22 = 0x1.8p22;
-
-void __dubsin (double x, double dx, double w[]);
-void __docos (double x, double dx, double w[]);
-double __mpsin (double x, double dx, bool reduce_range);
-double __mpcos (double x, double dx, bool reduce_range);
-static double slow (double x);
-static double slow1 (double x);
-static double slow2 (double x);
-static double sloww (double x, double dx, double orig, int n);
-static double sloww1 (double x, double dx, double orig, int m, int n);
-static double sloww2 (double x, double dx, double orig, int n);
-static double bsloww (double x, double dx, double orig, int n);
-static double bsloww1 (double x, double dx, double orig, int n);
-static double bsloww2 (double x, double dx, double orig, int n);
int __branred (double x, double *a, double *aa);
-static double cslow2 (double x);
-/* Given a number partitioned into U and X such that U is an index into the
- sin/cos table, this macro computes the cosine of the number by combining
- the sin and cos of X (as computed by a variation of the Taylor series) with
- the values looked up from the sin/cos table to get the result in RES and a
- correction value in COR. */
-static double
-do_cos (mynumber u, double x, double *corp)
+/* Given a number partitioned into X and DX, this function computes the cosine
+ of the number by combining the sin and cos of X (as computed by a variation
+ of the Taylor series) with the values looked up from the sin/cos table to
+ get the result. */
+static inline double
+__always_inline
+do_cos (double x, double dx)
{
- double xx, s, sn, ssn, c, cs, ccs, res, cor;
+ mynumber u;
+
+ if (x < 0)
+ dx = -dx;
+
+ u.x = big + fabs (x);
+ x = fabs (x) - (u.x - big) + dx;
+
+ double xx, s, sn, ssn, c, cs, ccs, cor;
xx = x * x;
s = x + x * xx * (sn3 + xx * sn5);
c = xx * (cs2 + xx * (cs4 + xx * cs6));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
cor = (ccs - s * ssn - cs * c) - sn * s;
- res = cs + cor;
- cor = (cs - res) + cor;
- *corp = cor;
- return res;
+ return cs + cor;
}
-/* A more precise variant of DO_COS where the number is partitioned into U, X
- and DX. EPS is the adjustment to the correction COR. */
-static double
-do_cos_slow (mynumber u, double x, double dx, double eps, double *corp)
+/* Given a number partitioned into X and DX, this function computes the sine of
+ the number by combining the sin and cos of X (as computed by a variation of
+ the Taylor series) with the values looked up from the sin/cos table to get
+ the result. */
+static inline double
+__always_inline
+do_sin (double x, double dx)
{
- double xx, y, x1, x2, e1, e2, res, cor;
- double s, sn, ssn, c, cs, ccs;
- xx = x * x;
- s = x * xx * (sn3 + xx * sn5);
- c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
- SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
- x1 = (x + t22) - t22;
- x2 = (x - x1) + dx;
- e1 = (sn + t22) - t22;
- e2 = (sn - e1) + ssn;
- cor = (ccs - cs * c - e1 * x2 - e2 * x) - sn * s;
- y = cs - e1 * x1;
- cor = cor + ((cs - y) - e1 * x1);
- res = y + cor;
- cor = (y - res) + cor;
- if (cor > 0)
- cor = 1.0005 * cor + eps;
- else
- cor = 1.0005 * cor - eps;
- *corp = cor;
- return res;
-}
+ double xold = x;
+ /* Max ULP is 0.501 if |x| < 0.126, otherwise ULP is 0.518. */
+ if (fabs (x) < 0.126)
+ return TAYLOR_SIN (x * x, x, dx);
-/* Given a number partitioned into U and X and DX such that U is an index into
- the sin/cos table, this macro computes the sine of the number by combining
- the sin and cos of X (as computed by a variation of the Taylor series) with
- the values looked up from the sin/cos table to get the result in RES and a
- correction value in COR. */
-static double
-do_sin (mynumber u, double x, double dx, double *corp)
-{
- double xx, s, sn, ssn, c, cs, ccs, cor, res;
+ mynumber u;
+
+ if (x <= 0)
+ dx = -dx;
+ u.x = big + fabs (x);
+ x = fabs (x) - (u.x - big);
+
+ double xx, s, sn, ssn, c, cs, ccs, cor;
xx = x * x;
s = x + (dx + x * xx * (sn3 + xx * sn5));
c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
cor = (ssn + s * ccs - sn * c) + cs * s;
- res = sn + cor;
- cor = (sn - res) + cor;
- *corp = cor;
- return res;
-}
-
-/* A more precise variant of res = do_sin where the number is partitioned into U, X
- and DX. EPS is the adjustment to the correction COR. */
-static double
-do_sin_slow (mynumber u, double x, double dx, double eps, double *corp)
-{
- double xx, y, x1, x2, c1, c2, res, cor;
- double s, sn, ssn, c, cs, ccs;
- xx = x * x;
- s = x * xx * (sn3 + xx * sn5);
- c = xx * (cs2 + xx * (cs4 + xx * cs6));
- SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
- x1 = (x + t22) - t22;
- x2 = (x - x1) + dx;
- c1 = (cs + t22) - t22;
- c2 = (cs - c1) + ccs;
- cor = (ssn + s * ccs + cs * s + c2 * x + c1 * x2 - sn * x * dx) - sn * c;
- y = sn + c1 * x1;
- cor = cor + ((sn - y) + c1 * x1);
- res = y + cor;
- cor = (y - res) + cor;
- if (cor > 0)
- cor = 1.0005 * cor + eps;
- else
- cor = 1.0005 * cor - eps;
- *corp = cor;
- return res;
-}
-
-/* Reduce range of X and compute sin of a + da. K is the amount by which to
- rotate the quadrants. This allows us to use the same routine to compute cos
- by simply rotating the quadrants by 1. */
-static inline double
-__always_inline
-reduce_and_compute (double x, unsigned int k)
-{
- double retval = 0, a, da;
- unsigned int n = __branred (x, &a, &da);
- k = (n + k) % 4;
- switch (k)
- {
- case 0:
- if (a * a < 0.01588)
- retval = bsloww (a, da, x, n);
- else
- retval = bsloww1 (a, da, x, n);
- break;
- case 2:
- if (a * a < 0.01588)
- retval = bsloww (-a, -da, x, n);
- else
- retval = bsloww1 (-a, -da, x, n);
- break;
-
- case 1:
- case 3:
- retval = bsloww2 (a, da, x, n);
- break;
- }
- return retval;
+ return __copysign (sn + cor, xold);
}
+/* 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;
@@ -281,198 +161,54 @@ 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;
-
- *a = b;
- *da = db;
-
- return n;
-}
-
-/* Compute sin (A + DA). cos can be computed by shifting the quadrant N
- clockwise. */
-static double
-__always_inline
-do_sincos_1 (double a, double da, double x, int4 n, int4 k)
-{
- double xx, retval, res, cor, y;
- mynumber u;
- int m;
- double eps = fabs (x) * 1.2e-30;
-
- int k1 = (n + k) & 3;
- switch (k1)
- { /* quarter of unit circle */
- case 2:
- a = -a;
- da = -da;
- case 0:
- xx = a * a;
- if (xx < 0.01588)
- {
- /* Taylor series. */
- res = TAYLOR_SIN (xx, a, da, cor);
- cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
- retval = (res == res + cor) ? res : sloww (a, da, x, k);
- }
- else
- {
- if (a > 0)
- m = 1;
- else
- {
- m = 0;
- a = -a;
- da = -da;
- }
- u.x = big + a;
- y = a - (u.x - big);
- res = do_sin (u, y, da, &cor);
- cor = (cor > 0) ? 1.035 * cor + eps : 1.035 * cor - eps;
- retval = ((res == res + cor) ? ((m) ? res : -res)
- : sloww1 (a, da, x, m, k));
- }
- break;
-
- case 1:
- case 3:
- if (a < 0)
- {
- a = -a;
- da = -da;
- }
- u.x = big + a;
- y = a - (u.x - big) + da;
- res = do_cos (u, y, &cor);
- cor = (cor > 0) ? 1.025 * cor + eps : 1.025 * cor - eps;
- retval = ((res == res + cor) ? ((k1 & 2) ? -res : res)
- : sloww2 (a, da, x, n));
- break;
- }
-
- return retval;
-}
-static inline int4
-__always_inline
-reduce_sincos_2 (double x, double *a, double *da)
-{
- mynumber v;
+ double b, db, t1, t2;
+ t1 = xn * pp3;
+ t2 = y - t1;
+ db = (y - t2) - t1;
- double t = (x * hpinv + toint);
- double xn = t - toint;
- v.x = t;
- double xn1 = (xn + 8.0e22) - 8.0e22;
- double xn2 = xn - xn1;
- double y = ((((x - xn1 * mp1) - xn1 * mp2) - xn2 * mp1) - xn2 * mp2);
- int4 n = v.i[LOW_HALF] & 3;
- double db = xn1 * pp3;
- t = y - db;
- db = (y - t) - db;
- db = (db - xn2 * pp3) - xn * pp4;
- double b = t + db;
- db = (t - b) + db;
+ 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 shifting the quadrant N
- clockwise. */
+/* Compute sin or cos (A + DA) for the given quadrant N. */
static double
__always_inline
-do_sincos_2 (double a, double da, double x, int4 n, int4 k)
+do_sincos (double a, double da, int4 n)
{
- double res, retval, cor, xx;
- mynumber u;
-
- double eps = 1.0e-24;
-
- k = (n + k) & 3;
-
- switch (k)
- {
- case 2:
- a = -a;
- da = -da;
- /* Fall through. */
- case 0:
- xx = a * a;
- if (xx < 0.01588)
- {
- /* Taylor series. */
- res = TAYLOR_SIN (xx, a, da, cor);
- cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
- retval = (res == res + cor) ? res : bsloww (a, da, x, n);
- }
- else
- {
- double t, db, y;
- int m;
- if (a > 0)
- {
- m = 1;
- t = a;
- db = da;
- }
- else
- {
- m = 0;
- t = -a;
- db = -da;
- }
- u.x = big + t;
- y = t - (u.x - big);
- res = do_sin (u, y, db, &cor);
- cor = (cor > 0) ? 1.035 * cor + eps : 1.035 * cor - eps;
- retval = ((res == res + cor) ? ((m) ? res : -res)
- : bsloww1 (a, da, x, n));
- }
- break;
+ double retval;
- case 1:
- case 3:
- if (a < 0)
- {
- a = -a;
- da = -da;
- }
- u.x = big + a;
- double y = a - (u.x - big) + da;
- res = do_cos (u, y, &cor);
- cor = (cor > 0) ? 1.025 * cor + eps : 1.025 * cor - eps;
- retval = ((res == res + cor) ? ((n & 2) ? -res : res)
- : bsloww2 (a, da, x, n));
- break;
- }
+ if (n & 1)
+ /* Max ULP is 0.513. */
+ retval = do_cos (a, da);
+ else
+ /* Max ULP is 0.501 if xx < 0.01588, otherwise ULP is 0.518. */
+ retval = do_sin (a, da);
- 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) */
/*******************************************************************/
-#ifdef IN_SINCOS
-static double
-#else
+#ifndef IN_SINCOS
double
SECTION
-#endif
__sin (double x)
{
- double xx, res, t, cor, y, s, c, sn, ssn, cs, ccs;
+ double t, a, da;
mynumber u;
- int4 k, m;
+ int4 k, m, n;
double retval = 0;
-#ifndef IN_SINCOS
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
-#endif
u.x = x;
m = u.i[HIGH_HALF];
@@ -482,77 +218,34 @@ __sin (double x)
math_check_force_underflow (x);
retval = x;
}
- /*---------------------------- 2^-26 < |x|< 0.25 ----------------------*/
- else if (k < 0x3fd00000)
- {
- 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);
- } /* else if (k < 0x3fd00000) */
-/*---------------------------- 0.25<|x|< 0.855469---------------------- */
+/*--------------------------- 2^-26<|x|< 0.855469---------------------- */
else if (k < 0x3feb6000)
{
- u.x = (m > 0) ? big + x : big - x;
- y = (m > 0) ? x - (u.x - big) : x + (u.x - big);
- xx = y * y;
- s = y + y * xx * (sn3 + xx * sn5);
- c = xx * (cs2 + xx * (cs4 + xx * cs6));
- SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
- if (m <= 0)
- {
- sn = -sn;
- ssn = -ssn;
- }
- cor = (ssn + s * ccs - sn * c) + cs * s;
- res = sn + cor;
- cor = (sn - res) + cor;
- retval = (res == res + 1.096 * cor) ? res : slow1 (x);
+ /* Max ULP is 0.548. */
+ retval = do_sin (x, 0);
} /* else if (k < 0x3feb6000) */
/*----------------------- 0.855469 <|x|<2.426265 ----------------------*/
else if (k < 0x400368fd)
{
-
- y = (m > 0) ? hp0 - x : hp0 + x;
- if (y >= 0)
- {
- u.x = big + y;
- y = (y - (u.x - big)) + hp1;
- }
- else
- {
- u.x = big - y;
- y = (-hp1) - (y + (u.x - big));
- }
- res = do_cos (u, y, &cor);
- retval = (res == res + 1.020 * cor) ? ((m > 0) ? res : -res) : slow2 (x);
+ t = hp0 - fabs (x);
+ /* Max ULP is 0.51. */
+ retval = __copysign (do_cos (t, hp1), x);
} /* else if (k < 0x400368fd) */
-#ifndef IN_SINCOS
/*-------------------------- 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, 0);
+ n = reduce_sincos (x, &a, &da);
+ retval = do_sincos (a, da, n);
} /* else if (k < 0x419921FB ) */
-/*---------------------105414350 <|x|< 281474976710656 --------------------*/
- else if (k < 0x42F00000)
- {
- double a, da;
-
- int4 n = reduce_sincos_2 (x, &a, &da);
- retval = do_sincos_2 (a, da, x, n, 0);
- } /* else if (k < 0x42F00000 ) */
-
-/* -----------------281474976710656 <|x| <2^1024----------------------------*/
+/* --------------------105414350 <|x| <2^1024------------------------------*/
else if (k < 0x7ff00000)
- retval = reduce_and_compute (x, 0);
-
+ {
+ n = __branred (x, &a, &da);
+ retval = do_sincos (a, da, n);
+ }
/*--------------------- |x| > 2^1024 ----------------------------------*/
else
{
@@ -560,7 +253,6 @@ __sin (double x)
__set_errno (EDOM);
retval = x / x;
}
-#endif
return retval;
}
@@ -571,23 +263,17 @@ __sin (double x)
/* it computes the correctly rounded (to nearest) value of cos(x) */
/*******************************************************************/
-#ifdef IN_SINCOS
-static double
-#else
double
SECTION
-#endif
__cos (double x)
{
- double y, xx, res, cor, a, da;
+ double y, a, da;
mynumber u;
- int4 k, m;
+ int4 k, m, n;
double retval = 0;
-#ifndef IN_SINCOS
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
-#endif
u.x = x;
m = u.i[HIGH_HALF];
@@ -599,11 +285,8 @@ __cos (double x)
else if (k < 0x3feb6000)
{ /* 2^-27 < |x| < 0.855469 */
- y = fabs (x);
- u.x = big + y;
- y = y - (u.x - big);
- res = do_cos (u, y, &cor);
- retval = (res == res + 1.020 * cor) ? res : cslow2 (x);
+ /* Max ULP is 0.51. */
+ retval = do_cos (x, 0);
} /* else if (k < 0x3feb6000) */
else if (k < 0x400368fd)
@@ -611,55 +294,23 @@ __cos (double x)
y = hp0 - fabs (x);
a = y + hp1;
da = (y - a) + hp1;
- xx = a * a;
- if (xx < 0.01588)
- {
- res = TAYLOR_SIN (xx, a, da, cor);
- cor = (cor > 0) ? 1.02 * cor + 1.0e-31 : 1.02 * cor - 1.0e-31;
- retval = (res == res + cor) ? res : sloww (a, da, x, 1);
- }
- else
- {
- if (a > 0)
- {
- m = 1;
- }
- else
- {
- m = 0;
- a = -a;
- da = -da;
- }
- u.x = big + a;
- y = a - (u.x - big);
- res = do_sin (u, y, da, &cor);
- cor = (cor > 0) ? 1.035 * cor + 1.0e-31 : 1.035 * cor - 1.0e-31;
- retval = ((res == res + cor) ? ((m) ? res : -res)
- : sloww1 (a, da, x, m, 1));
- }
-
+ /* Max ULP is 0.501 if xx < 0.01588 or 0.518 otherwise.
+ Range reduction uses 106 bits here which is sufficient. */
+ retval = do_sin (a, da);
} /* else if (k < 0x400368fd) */
-
-#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, 1);
+ n = reduce_sincos (x, &a, &da);
+ retval = do_sincos (a, da, n + 1);
} /* else if (k < 0x419921FB ) */
- else if (k < 0x42F00000)
- {
- double a, da;
-
- int4 n = reduce_sincos_2 (x, &a, &da);
- retval = do_sincos_2 (a, da, x, n, 1);
- } /* else if (k < 0x42F00000 ) */
-
- /* 281474976710656 <|x| <2^1024 */
+ /* 105414350 <|x| <2^1024 */
else if (k < 0x7ff00000)
- retval = reduce_and_compute (x, 1);
+ {
+ n = __branred (x, &a, &da);
+ retval = do_sincos (a, da, n + 1);
+ }
else
{
@@ -667,366 +318,15 @@ __cos (double x)
__set_errno (EDOM);
retval = x / x; /* |x| > 2^1024 */
}
-#endif
return retval;
}
-/************************************************************************/
-/* Routine compute sin(x) for 2^-26 < |x|< 0.25 by Taylor with more */
-/* precision and if still doesn't accurate enough by mpsin or dubsin */
-/************************************************************************/
-
-static double
-SECTION
-slow (double x)
-{
- double res, cor, w[2];
- res = TAYLOR_SLOW (x, 0, cor);
- if (res == res + 1.0007 * cor)
- return res;
-
- __dubsin (fabs (x), 0, w);
- if (w[0] == w[0] + 1.000000001 * w[1])
- return (x > 0) ? w[0] : -w[0];
-
- return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
-}
-
-/*******************************************************************************/
-/* Routine compute sin(x) for 0.25<|x|< 0.855469 by __sincostab.tbl and Taylor */
-/* and if result still doesn't accurate enough by mpsin or dubsin */
-/*******************************************************************************/
-
-static double
-SECTION
-slow1 (double x)
-{
- mynumber u;
- double w[2], y, cor, res;
- y = fabs (x);
- u.x = big + y;
- y = y - (u.x - big);
- res = do_sin_slow (u, y, 0, 0, &cor);
- if (res == res + cor)
- return (x > 0) ? res : -res;
-
- __dubsin (fabs (x), 0, w);
- if (w[0] == w[0] + 1.000000005 * w[1])
- return (x > 0) ? w[0] : -w[0];
-
- return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
-}
-
-/**************************************************************************/
-/* Routine compute sin(x) for 0.855469 <|x|<2.426265 by __sincostab.tbl */
-/* and if result still doesn't accurate enough by mpsin or dubsin */
-/**************************************************************************/
-static double
-SECTION
-slow2 (double x)
-{
- mynumber u;
- double w[2], y, y1, y2, cor, res, del;
-
- y = fabs (x);
- y = hp0 - y;
- if (y >= 0)
- {
- u.x = big + y;
- y = y - (u.x - big);
- del = hp1;
- }
- else
- {
- u.x = big - y;
- y = -(y + (u.x - big));
- del = -hp1;
- }
- res = do_cos_slow (u, y, del, 0, &cor);
- if (res == res + cor)
- return (x > 0) ? res : -res;
-
- y = fabs (x) - hp0;
- y1 = y - hp1;
- y2 = (y - y1) - hp1;
- __docos (y1, y2, w);
- if (w[0] == w[0] + 1.000000005 * w[1])
- return (x > 0) ? w[0] : -w[0];
-
- return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
-}
-
-/***************************************************************************/
-/* Routine compute sin(x+dx) (Double-Length number) where x is small enough*/
-/* to use Taylor series around zero and (x+dx) */
-/* in first or third quarter of unit circle.Routine receive also */
-/* (right argument) the original value of x for computing error of */
-/* result.And if result not accurate enough routine calls mpsin1 or dubsin */
-/***************************************************************************/
-
-static double
-SECTION
-sloww (double x, double dx, double orig, int k)
-{
- double y, t, res, cor, w[2], a, da, xn;
- mynumber v;
- int4 n;
- res = TAYLOR_SLOW (x, dx, cor);
-
- if (cor > 0)
- cor = 1.0005 * cor + fabs (orig) * 3.1e-30;
- else
- cor = 1.0005 * cor - fabs (orig) * 3.1e-30;
-
- if (res == res + cor)
- return res;
-
- (x > 0) ? __dubsin (x, dx, w) : __dubsin (-x, -dx, w);
- if (w[1] > 0)
- cor = 1.000000001 * w[1] + fabs (orig) * 1.1e-30;
- else
- cor = 1.000000001 * w[1] - fabs (orig) * 1.1e-30;
-
- if (w[0] == w[0] + cor)
- return (x > 0) ? w[0] : -w[0];
-
- t = (orig * hpinv + toint);
- xn = t - toint;
- v.x = t;
- y = (orig - xn * mp1) - xn * mp2;
- n = (v.i[LOW_HALF] + k) & 3;
- da = xn * pp3;
- t = y - da;
- da = (y - t) - da;
- y = xn * pp4;
- a = t - y;
- da = ((t - a) - y) + da;
-
- if (n == 2 || n == 1)
- {
- a = -a;
- da = -da;
- }
- (a > 0) ? __dubsin (a, da, w) : __dubsin (-a, -da, w);
- if (w[1] > 0)
- cor = 1.000000001 * w[1] + fabs (orig) * 1.1e-40;
- else
- cor = 1.000000001 * w[1] - fabs (orig) * 1.1e-40;
-
- if (w[0] == w[0] + cor)
- return (a > 0) ? w[0] : -w[0];
-
- return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
-}
-
-/***************************************************************************/
-/* Routine compute sin(x+dx) (Double-Length number) where x in first or */
-/* third quarter of unit circle.Routine receive also (right argument) the */
-/* original value of x for computing error of result.And if result not */
-/* accurate enough routine calls mpsin1 or dubsin */
-/***************************************************************************/
-
-static double
-SECTION
-sloww1 (double x, double dx, double orig, int m, int k)
-{
- mynumber u;
- double w[2], y, cor, res;
-
- u.x = big + x;
- y = x - (u.x - big);
- res = do_sin_slow (u, y, dx, 3.1e-30 * fabs (orig), &cor);
-
- if (res == res + cor)
- return (m > 0) ? res : -res;
-
- __dubsin (x, dx, w);
-
- if (w[1] > 0)
- cor = 1.000000005 * w[1] + 1.1e-30 * fabs (orig);
- else
- cor = 1.000000005 * w[1] - 1.1e-30 * fabs (orig);
-
- if (w[0] == w[0] + cor)
- return (m > 0) ? w[0] : -w[0];
-
- return (k == 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
-}
-
-/***************************************************************************/
-/* Routine compute sin(x+dx) (Double-Length number) where x in second or */
-/* fourth quarter of unit circle.Routine receive also the original value */
-/* and quarter(n= 1or 3)of x for computing error of result.And if result not*/
-/* accurate enough routine calls mpsin1 or dubsin */
-/***************************************************************************/
-
-static double
-SECTION
-sloww2 (double x, double dx, double orig, int n)
-{
- mynumber u;
- double w[2], y, cor, res;
-
- u.x = big + x;
- y = x - (u.x - big);
- res = do_cos_slow (u, y, dx, 3.1e-30 * fabs (orig), &cor);
-
- if (res == res + cor)
- return (n & 2) ? -res : res;
-
- __docos (x, dx, w);
-
- if (w[1] > 0)
- cor = 1.000000005 * w[1] + 1.1e-30 * fabs (orig);
- else
- cor = 1.000000005 * w[1] - 1.1e-30 * fabs (orig);
-
- if (w[0] == w[0] + cor)
- return (n & 2) ? -w[0] : w[0];
-
- return (n & 1) ? __mpsin (orig, 0, true) : __mpcos (orig, 0, true);
-}
-
-/***************************************************************************/
-/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
-/* is small enough to use Taylor series around zero and (x+dx) */
-/* in first or third quarter of unit circle.Routine receive also */
-/* (right argument) the original value of x for computing error of */
-/* result.And if result not accurate enough routine calls other routines */
-/***************************************************************************/
-
-static double
-SECTION
-bsloww (double x, double dx, double orig, int n)
-{
- double res, cor, w[2];
-
- res = TAYLOR_SLOW (x, dx, cor);
- cor = (cor > 0) ? 1.0005 * cor + 1.1e-24 : 1.0005 * cor - 1.1e-24;
- if (res == res + cor)
- return res;
-
- (x > 0) ? __dubsin (x, dx, w) : __dubsin (-x, -dx, w);
- if (w[1] > 0)
- cor = 1.000000001 * w[1] + 1.1e-24;
- else
- cor = 1.000000001 * w[1] - 1.1e-24;
-
- if (w[0] == w[0] + cor)
- return (x > 0) ? w[0] : -w[0];
-
- return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
-}
-
-/***************************************************************************/
-/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
-/* in first or third quarter of unit circle.Routine receive also */
-/* (right argument) the original value of x for computing error of result.*/
-/* And if result not accurate enough routine calls other routines */
-/***************************************************************************/
-
-static double
-SECTION
-bsloww1 (double x, double dx, double orig, int n)
-{
- mynumber u;
- double w[2], y, cor, res;
-
- y = fabs (x);
- u.x = big + y;
- y = y - (u.x - big);
- dx = (x > 0) ? dx : -dx;
- res = do_sin_slow (u, y, dx, 1.1e-24, &cor);
- if (res == res + cor)
- return (x > 0) ? res : -res;
-
- __dubsin (fabs (x), dx, w);
-
- if (w[1] > 0)
- cor = 1.000000005 * w[1] + 1.1e-24;
- else
- cor = 1.000000005 * w[1] - 1.1e-24;
-
- if (w[0] == w[0] + cor)
- return (x > 0) ? w[0] : -w[0];
-
- return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
-}
-
-/***************************************************************************/
-/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
-/* in second or fourth quarter of unit circle.Routine receive also the */
-/* original value and quarter(n= 1or 3)of x for computing error of result. */
-/* And if result not accurate enough routine calls other routines */
-/***************************************************************************/
-
-static double
-SECTION
-bsloww2 (double x, double dx, double orig, int n)
-{
- mynumber u;
- double w[2], y, cor, res;
-
- y = fabs (x);
- u.x = big + y;
- y = y - (u.x - big);
- dx = (x > 0) ? dx : -dx;
- res = do_cos_slow (u, y, dx, 1.1e-24, &cor);
- if (res == res + cor)
- return (n & 2) ? -res : res;
-
- __docos (fabs (x), dx, w);
-
- if (w[1] > 0)
- cor = 1.000000005 * w[1] + 1.1e-24;
- else
- cor = 1.000000005 * w[1] - 1.1e-24;
-
- if (w[0] == w[0] + cor)
- return (n & 2) ? -w[0] : w[0];
-
- return (n & 1) ? __mpsin (orig, 0, true) : __mpcos (orig, 0, true);
-}
-
-/************************************************************************/
-/* Routine compute cos(x) for 2^-27 < |x|< 0.25 by Taylor with more */
-/* precision and if still doesn't accurate enough by mpcos or docos */
-/************************************************************************/
-
-static double
-SECTION
-cslow2 (double x)
-{
- mynumber u;
- double w[2], y, cor, res;
-
- y = fabs (x);
- u.x = big + y;
- y = y - (u.x - big);
- res = do_cos_slow (u, y, 0, 0, &cor);
- if (res == res + cor)
- return res;
-
- y = fabs (x);
- __docos (y, 0, w);
- if (w[0] == w[0] + 1.000000005 * w[1])
- return w[0];
-
- return __mpcos (x, 0, false);
-}
-
#ifndef __cos
-weak_alias (__cos, cos)
-# ifdef NO_LONG_DOUBLE
-strong_alias (__cos, __cosl)
-weak_alias (__cos, cosl)
-# endif
+libm_alias_double (__cos, cos)
#endif
#ifndef __sin
-weak_alias (__sin, sin)
-# ifdef NO_LONG_DOUBLE
-strong_alias (__sin, __sinl)
-weak_alias (__sin, sinl)
-# endif
+libm_alias_double (__sin, sin)
+#endif
+
#endif
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