/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
- * Copyright (C) 2001-2012 Free Software Foundation
+ * 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
/* bsloww1 */
/* bsloww2 */
/* cslow2 */
-/* csloww */
-/* csloww1 */
-/* csloww2 */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
/* branred.c sincos32.c dosincos.c mpa.c */
/* sincos.tbl */
#include <errno.h>
+#include <float.h>
#include "endian.h"
#include "mydefs.h"
#include "usncs.h"
#include "MathLib.h"
+#include <math.h>
#include <math_private.h>
+#include <libm-alias-double.h>
#include <fenv.h>
+/* Helper macros to compute sin of the input values. */
+#define POLYNOMIAL2(xx) ((((s5 * (xx) + s4) * (xx) + s3) * (xx) + s2) * (xx))
+
+#define POLYNOMIAL(xx) (POLYNOMIAL2 (xx) + s1)
+
+/* The computed polynomial is a variation of the Taylor series expansion for
+ sin(a):
+
+ 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) \
+({ \
+ 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; \
+})
+
+#define SINCOS_TABLE_LOOKUP(u, sn, ssn, cs, ccs) \
+({ \
+ int4 k = u.i[LOW_HALF] << 2; \
+ sn = __sincostab.x[k]; \
+ ssn = __sincostab.x[k + 1]; \
+ cs = __sincostab.x[k + 2]; \
+ ccs = __sincostab.x[k + 3]; \
+})
+
#ifndef SECTION
# define SECTION
#endif
} __sincostab attribute_hidden;
static const double
- sn3 = -1.66666666666664880952546298448555E-01,
- sn5 = 8.33333214285722277379541354343671E-03,
- cs2 = 4.99999999999999999999950396842453E-01,
- cs4 = -4.16666666666664434524222570944589E-02,
- cs6 = 1.38888874007937613028114285595617E-03;
-
-void __dubsin(double x, double dx, double w[]);
-void __docos(double x, double dx, double w[]);
-double __mpsin(double x, double dx);
-double __mpcos(double x, double dx);
-double __mpsin1(double x);
-double __mpcos1(double x);
-static double slow(double x);
-static double slow1(double x);
-static double slow2(double x);
-static double sloww(double x, double dx, double orig);
-static double sloww1(double x, double dx, double orig);
-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);
-static double csloww(double x, double dx, double orig);
-static double csloww1(double x, double dx, double orig);
-static double csloww2(double x, double dx, double orig, int n);
+ sn3 = -1.66666666666664880952546298448555E-01,
+ sn5 = 8.33333214285722277379541354343671E-03,
+ cs2 = 4.99999999999999999999950396842453E-01,
+ 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, bool shift_quadrant);
+static double sloww1 (double x, double dx, double orig, bool shift_quadrant);
+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 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 in RES and a correction value in COR. */
+static inline double
+__always_inline
+do_cos (double x, double dx, double *corp)
+{
+ 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, res, 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;
+}
+
+/* A more precise variant of DO_COS. EPS is the adjustment to the correction
+ COR. */
+static inline double
+__always_inline
+do_cos_slow (double x, double dx, double eps, double *corp)
+{
+ mynumber u;
+
+ if (x <= 0)
+ dx = -dx;
+
+ u.x = big + fabs (x);
+ x = fabs (x) - (u.x - big);
+
+ 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;
+ cor = 1.0005 * cor + __copysign (eps, cor);
+ *corp = cor;
+ return res;
+}
+
+/* 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 in RES and a correction value in COR. */
+static inline double
+__always_inline
+do_sin (double x, double dx, double *corp)
+{
+ 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, res;
+ 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 DO_SIN. EPS is the adjustment to the correction
+ COR. */
+static inline double
+__always_inline
+do_sin_slow (double x, double dx, double eps, double *corp)
+{
+ mynumber u;
+
+ if (x <= 0)
+ dx = -dx;
+ u.x = big + fabs (x);
+ x = fabs (x) - (u.x - big);
+
+ 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;
+ cor = 1.0005 * cor + __copysign (eps, cor);
+ *corp = cor;
+ return res;
+}
+
+/* Reduce range of X and compute sin of a + da. When SHIFT_QUADRANT is true,
+ the routine returns the cosine of a + da by rotating the quadrant once and
+ computing the sine of the result. */
+static inline double
+__always_inline
+reduce_and_compute (double x, bool shift_quadrant)
+{
+ double retval = 0, a, da;
+ unsigned int n = __branred (x, &a, &da);
+ int4 k = (n + shift_quadrant) % 4;
+ switch (k)
+ {
+ case 2:
+ a = -a;
+ da = -da;
+ /* Fall through. */
+ case 0:
+ 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;
+}
+
+static inline int4
+__always_inline
+reduce_sincos_1 (double x, double *a, double *da)
+{
+ mynumber v;
+
+ double t = (x * hpinv + toint);
+ double xn = t - toint;
+ 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 passing SHIFT_QUADRANT as
+ true, which results in shifting the quadrant N clockwise. */
+static double
+__always_inline
+do_sincos_1 (double a, double da, double x, int4 n, bool shift_quadrant)
+{
+ double xx, retval, res, cor;
+ double eps = fabs (x) * 1.2e-30;
+
+ 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 (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);
+ }
+ 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;
+ }
+
+ return retval;
+}
+
+static inline int4
+__always_inline
+reduce_sincos_2 (double x, double *a, double *da)
+{
+ mynumber v;
+
+ 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;
+
+ *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. */
+static double
+__always_inline
+do_sincos_2 (double a, double da, double x, int4 n, bool shift_quadrant)
+{
+ double res, retval, cor, xx;
+
+ double eps = 1.0e-24;
+
+ int4 k = (n + shift_quadrant) & 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 = 1.02 * cor + __copysign (eps, cor);
+ retval = (res == res + cor) ? res : bsloww (a, da, x, n);
+ }
+ else
+ {
+ res = do_sin (a, da, &cor);
+ cor = 1.035 * cor + __copysign (eps, cor);
+ retval = ((res == res + cor) ? __copysign (res, a)
+ : bsloww1 (a, da, x, n));
+ }
+ 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)
+ : bsloww2 (a, da, x, n));
+ break;
+ }
+
+ return 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
double
SECTION
-__sin(double x){
- double xx,res,t,cor,y,s,c,sn,ssn,cs,ccs,xn,a,da,db,eps,xn1,xn2;
-#if 0
- double w[2];
#endif
- mynumber u,v;
- int4 k,m,n;
-#if 0
- int4 nn;
-#endif
- double retval = 0;
+__sin (double x)
+{
+ double xx, res, t, cor;
+ mynumber u;
+ int4 k, m;
+ double retval = 0;
- SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
+#ifndef IN_SINCOS
+ SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
+#endif
- u.x = x;
- m = u.i[HIGH_HALF];
- k = 0x7fffffff&m; /* no sign */
- if (k < 0x3e500000) /* if x->0 =>sin(x)=x */
- { retval = x; goto ret; }
+ u.x = x;
+ m = u.i[HIGH_HALF];
+ k = 0x7fffffff & m; /* no sign */
+ if (k < 0x3e500000) /* if x->0 =>sin(x)=x */
+ {
+ math_check_force_underflow (x);
+ retval = x;
+ }
/*---------------------------- 2^-26 < |x|< 0.25 ----------------------*/
- else if (k < 0x3fd00000){
- xx = x*x;
- /*Taylor series */
- t = ((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*(xx*x);
- res = x+t;
- cor = (x-res)+t;
- retval = (res == res + 1.07*cor)? res : slow(x);
- goto ret;
- } /* else if (k < 0x3fd00000) */
+ 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---------------------- */
- else if (k < 0x3feb6000) {
- u.x=(m>0)?big.x+x:big.x-x;
- y=(m>0)?x-(u.x-big.x):x+(u.x-big.x);
- xx=y*y;
- s = y + y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=(m>0)?__sincostab.x[k]:-__sincostab.x[k];
- ssn=(m>0)?__sincostab.x[k+1]:-__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- cor=(ssn+s*ccs-sn*c)+cs*s;
- res=sn+cor;
- cor=(sn-res)+cor;
- retval = (res==res+1.096*cor)? res : slow1(x);
- goto ret;
- } /* else if (k < 0x3feb6000) */
+ else if (k < 0x3feb6000)
+ {
+ res = do_sin (x, 0, &cor);
+ retval = (res == res + 1.096 * cor) ? res : slow1 (x);
+ retval = __copysign (retval, x);
+ } /* else if (k < 0x3feb6000) */
/*----------------------- 0.855469 <|x|<2.426265 ----------------------*/
- else if (k < 0x400368fd ) {
-
- y = (m>0)? hp0.x-x:hp0.x+x;
- if (y>=0) {
- u.x = big.x+y;
- y = (y-(u.x-big.x))+hp1.x;
- }
- else {
- u.x = big.x-y;
- y = (-hp1.x) - (y+(u.x-big.x));
- }
- xx=y*y;
- s = y + y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- cor=(ccs-s*ssn-cs*c)-sn*s;
- res=cs+cor;
- cor=(cs-res)+cor;
- retval = (res==res+1.020*cor)? ((m>0)?res:-res) : slow2(x);
- goto ret;
- } /* else if (k < 0x400368fd) */
+ 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);
+ } /* else if (k < 0x400368fd) */
+#ifndef IN_SINCOS
/*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
- else if (k < 0x419921FB ) {
- t = (x*hpinv.x + toint.x);
- xn = t - toint.x;
- v.x = t;
- y = (x - xn*mp1.x) - xn*mp2.x;
- n =v.i[LOW_HALF]&3;
- da = xn*mp3.x;
- a=y-da;
- da = (y-a)-da;
- eps = ABS(x)*1.2e-30;
-
- switch (n) { /* quarter of unit circle */
- case 0:
- case 2:
- xx = a*a;
- if (n) {a=-a;da=-da;}
- if (xx < 0.01588) {
- /*Taylor series */
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
- res = a+t;
- cor = (a-res)+t;
- cor = (cor>0)? 1.02*cor+eps : 1.02*cor -eps;
- retval = (res == res + cor)? res : sloww(a,da,x);
- goto ret;
- }
- else {
- if (a>0)
- {m=1;t=a;db=da;}
- else
- {m=0;t=-a;db=-da;}
- u.x=big.x+t;
- y=t-(u.x-big.x);
- xx=y*y;
- s = y + (db+y*xx*(sn3 +xx*sn5));
- c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- cor=(ssn+s*ccs-sn*c)+cs*s;
- res=sn+cor;
- cor=(sn-res)+cor;
- cor = (cor>0)? 1.035*cor+eps : 1.035*cor-eps;
- retval = (res==res+cor)? ((m)?res:-res) : sloww1(a,da,x);
- goto ret;
- }
- break;
-
- case 1:
- case 3:
- if (a<0)
- {a=-a;da=-da;}
- u.x=big.x+a;
- y=a-(u.x-big.x)+da;
- xx=y*y;
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- s = y + y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- cor=(ccs-s*ssn-cs*c)-sn*s;
- res=cs+cor;
- cor=(cs-res)+cor;
- cor = (cor>0)? 1.025*cor+eps : 1.025*cor-eps;
- retval = (res==res+cor)? ((n&2)?-res:res) : sloww2(a,da,x,n);
- goto ret;
-
- break;
-
- }
-
- } /* else if (k < 0x419921FB ) */
+ else if (k < 0x419921FB)
+ {
+ double a, da;
+ int4 n = reduce_sincos_1 (x, &a, &da);
+ retval = do_sincos_1 (a, da, x, n, false);
+ } /* else if (k < 0x419921FB ) */
/*---------------------105414350 <|x|< 281474976710656 --------------------*/
- else if (k < 0x42F00000 ) {
- t = (x*hpinv.x + toint.x);
- xn = t - toint.x;
- v.x = t;
- xn1 = (xn+8.0e22)-8.0e22;
- xn2 = xn - xn1;
- y = ((((x - xn1*mp1.x) - xn1*mp2.x)-xn2*mp1.x)-xn2*mp2.x);
- n =v.i[LOW_HALF]&3;
- da = xn1*pp3.x;
- t=y-da;
- da = (y-t)-da;
- da = (da - xn2*pp3.x) -xn*pp4.x;
- a = t+da;
- da = (t-a)+da;
- eps = 1.0e-24;
-
- switch (n) {
- case 0:
- case 2:
- xx = a*a;
- if (n) {a=-a;da=-da;}
- if (xx < 0.01588) {
- /* Taylor series */
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
- res = a+t;
- cor = (a-res)+t;
- cor = (cor>0)? 1.02*cor+eps : 1.02*cor -eps;
- retval = (res == res + cor)? res : bsloww(a,da,x,n);
- goto ret;
- }
- else {
- if (a>0) {m=1;t=a;db=da;}
- else {m=0;t=-a;db=-da;}
- u.x=big.x+t;
- y=t-(u.x-big.x);
- xx=y*y;
- s = y + (db+y*xx*(sn3 +xx*sn5));
- c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- cor=(ssn+s*ccs-sn*c)+cs*s;
- res=sn+cor;
- cor=(sn-res)+cor;
- cor = (cor>0)? 1.035*cor+eps : 1.035*cor-eps;
- retval = (res==res+cor)? ((m)?res:-res) : bsloww1(a,da,x,n);
- goto ret;
- }
- break;
-
- case 1:
- case 3:
- if (a<0)
- {a=-a;da=-da;}
- u.x=big.x+a;
- y=a-(u.x-big.x)+da;
- xx=y*y;
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- s = y + y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- cor=(ccs-s*ssn-cs*c)-sn*s;
- res=cs+cor;
- cor=(cs-res)+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);
- goto ret;
-
- break;
-
- }
-
- } /* else if (k < 0x42F00000 ) */
+ else if (k < 0x42F00000)
+ {
+ double a, da;
+
+ int4 n = reduce_sincos_2 (x, &a, &da);
+ retval = do_sincos_2 (a, da, x, n, false);
+ } /* else if (k < 0x42F00000 ) */
/* -----------------281474976710656 <|x| <2^1024----------------------------*/
- else if (k < 0x7ff00000) {
-
- n = __branred(x,&a,&da);
- switch (n) {
- case 0:
- if (a*a < 0.01588) retval = bsloww(a,da,x,n);
- else retval = bsloww1(a,da,x,n);
- goto ret;
- break;
- case 2:
- if (a*a < 0.01588) retval = bsloww(-a,-da,x,n);
- else retval = bsloww1(-a,-da,x,n);
- goto ret;
- break;
-
- case 1:
- case 3:
- retval = bsloww2(a,da,x,n);
- goto ret;
- break;
- }
-
- } /* else if (k < 0x7ff00000 ) */
+ else if (k < 0x7ff00000)
+ retval = reduce_and_compute (x, false);
/*--------------------- |x| > 2^1024 ----------------------------------*/
- else {
- if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
- __set_errno (EDOM);
- retval = x / x;
- goto ret;
- }
+ else
+ {
+ if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
+ __set_errno (EDOM);
+ retval = x / x;
+ }
+#endif
- ret:
- return retval;
+ return retval;
}
/* it computes the correctly rounded (to nearest) value of cos(x) */
/*******************************************************************/
+#ifdef IN_SINCOS
+static double
+#else
double
SECTION
-__cos(double x)
+#endif
+__cos (double x)
{
- double y,xx,res,t,cor,s,c,sn,ssn,cs,ccs,xn,a,da,db,eps,xn1,xn2;
- mynumber u,v;
- int4 k,m,n;
+ double y, xx, res, cor, a, da;
+ mynumber u;
+ int4 k, m;
double retval = 0;
+#ifndef IN_SINCOS
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
+#endif
u.x = x;
m = u.i[HIGH_HALF];
- k = 0x7fffffff&m;
-
- if (k < 0x3e400000 ) { retval = 1.0; goto ret; } /* |x|<2^-27 => cos(x)=1 */
-
- else if (k < 0x3feb6000 ) {/* 2^-27 < |x| < 0.855469 */
- y=ABS(x);
- u.x = big.x+y;
- y = y-(u.x-big.x);
- xx=y*y;
- s = y + y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- cor=(ccs-s*ssn-cs*c)-sn*s;
- res=cs+cor;
- cor=(cs-res)+cor;
- retval = (res==res+1.020*cor)? res : cslow2(x);
- goto ret;
-
-} /* else if (k < 0x3feb6000) */
-
- else if (k < 0x400368fd ) {/* 0.855469 <|x|<2.426265 */;
- y=hp0.x-ABS(x);
- a=y+hp1.x;
- da=(y-a)+hp1.x;
- xx=a*a;
- if (xx < 0.01588) {
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
- res = a+t;
- cor = (a-res)+t;
- cor = (cor>0)? 1.02*cor+1.0e-31 : 1.02*cor -1.0e-31;
- retval = (res == res + cor)? res : csloww(a,da,x);
- goto ret;
- }
- else {
- if (a>0) {m=1;t=a;db=da;}
- else {m=0;t=-a;db=-da;}
- u.x=big.x+t;
- y=t-(u.x-big.x);
- xx=y*y;
- s = y + (db+y*xx*(sn3 +xx*sn5));
- c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- cor=(ssn+s*ccs-sn*c)+cs*s;
- res=sn+cor;
- cor=(sn-res)+cor;
- cor = (cor>0)? 1.035*cor+1.0e-31 : 1.035*cor-1.0e-31;
- retval = (res==res+cor)? ((m)?res:-res) : csloww1(a,da,x);
- goto ret;
-}
-
-} /* else if (k < 0x400368fd) */
-
-
- else if (k < 0x419921FB ) {/* 2.426265<|x|< 105414350 */
- t = (x*hpinv.x + toint.x);
- xn = t - toint.x;
- v.x = t;
- y = (x - xn*mp1.x) - xn*mp2.x;
- n =v.i[LOW_HALF]&3;
- da = xn*mp3.x;
- a=y-da;
- da = (y-a)-da;
- eps = ABS(x)*1.2e-30;
-
- switch (n) {
- case 1:
- case 3:
- xx = a*a;
- if (n == 1) {a=-a;da=-da;}
- if (xx < 0.01588) {
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
- res = a+t;
- cor = (a-res)+t;
- cor = (cor>0)? 1.02*cor+eps : 1.02*cor -eps;
- retval = (res == res + cor)? res : csloww(a,da,x);
- goto ret;
- }
- else {
- if (a>0) {m=1;t=a;db=da;}
- else {m=0;t=-a;db=-da;}
- u.x=big.x+t;
- y=t-(u.x-big.x);
- xx=y*y;
- s = y + (db+y*xx*(sn3 +xx*sn5));
- c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- cor=(ssn+s*ccs-sn*c)+cs*s;
- res=sn+cor;
- cor=(sn-res)+cor;
- cor = (cor>0)? 1.035*cor+eps : 1.035*cor-eps;
- retval = (res==res+cor)? ((m)?res:-res) : csloww1(a,da,x);
- goto ret;
- }
- break;
-
- case 0:
- case 2:
- if (a<0) {a=-a;da=-da;}
- u.x=big.x+a;
- y=a-(u.x-big.x)+da;
- xx=y*y;
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- s = y + y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- cor=(ccs-s*ssn-cs*c)-sn*s;
- res=cs+cor;
- cor=(cs-res)+cor;
- cor = (cor>0)? 1.025*cor+eps : 1.025*cor-eps;
- retval = (res==res+cor)? ((n)?-res:res) : csloww2(a,da,x,n);
- goto ret;
-
- break;
-
- }
+ k = 0x7fffffff & m;
+
+ /* |x|<2^-27 => cos(x)=1 */
+ if (k < 0x3e400000)
+ retval = 1.0;
+
+ else if (k < 0x3feb6000)
+ { /* 2^-27 < |x| < 0.855469 */
+ res = do_cos (x, 0, &cor);
+ retval = (res == res + 1.020 * cor) ? res : cslow2 (x);
+ } /* else if (k < 0x3feb6000) */
+
+ else if (k < 0x400368fd)
+ { /* 0.855469 <|x|<2.426265 */ ;
+ 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 = 1.02 * cor + __copysign (1.0e-31, cor);
+ retval = (res == res + cor) ? res : sloww (a, da, x, true);
+ }
+ 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));
+ }
- } /* else if (k < 0x419921FB ) */
-
-
- else if (k < 0x42F00000 ) {
- t = (x*hpinv.x + toint.x);
- xn = t - toint.x;
- v.x = t;
- xn1 = (xn+8.0e22)-8.0e22;
- xn2 = xn - xn1;
- y = ((((x - xn1*mp1.x) - xn1*mp2.x)-xn2*mp1.x)-xn2*mp2.x);
- n =v.i[LOW_HALF]&3;
- da = xn1*pp3.x;
- t=y-da;
- da = (y-t)-da;
- da = (da - xn2*pp3.x) -xn*pp4.x;
- a = t+da;
- da = (t-a)+da;
- eps = 1.0e-24;
-
- switch (n) {
- case 1:
- case 3:
- xx = a*a;
- if (n==1) {a=-a;da=-da;}
- if (xx < 0.01588) {
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
- res = a+t;
- cor = (a-res)+t;
- cor = (cor>0)? 1.02*cor+eps : 1.02*cor -eps;
- retval = (res == res + cor)? res : bsloww(a,da,x,n);
- goto ret;
- }
- else {
- if (a>0) {m=1;t=a;db=da;}
- else {m=0;t=-a;db=-da;}
- u.x=big.x+t;
- y=t-(u.x-big.x);
- xx=y*y;
- s = y + (db+y*xx*(sn3 +xx*sn5));
- c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- cor=(ssn+s*ccs-sn*c)+cs*s;
- res=sn+cor;
- cor=(sn-res)+cor;
- cor = (cor>0)? 1.035*cor+eps : 1.035*cor-eps;
- retval = (res==res+cor)? ((m)?res:-res) : bsloww1(a,da,x,n);
- goto ret;
- }
- break;
+ } /* else if (k < 0x400368fd) */
- case 0:
- case 2:
- if (a<0) {a=-a;da=-da;}
- u.x=big.x+a;
- y=a-(u.x-big.x)+da;
- xx=y*y;
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- s = y + y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- cor=(ccs-s*ssn-cs*c)-sn*s;
- res=cs+cor;
- cor=(cs-res)+cor;
- cor = (cor>0)? 1.025*cor+eps : 1.025*cor-eps;
- retval = (res==res+cor)? ((n)?-res:res) : bsloww2(a,da,x,n);
- goto ret;
- break;
- }
+#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);
+ } /* else if (k < 0x419921FB ) */
- } /* else if (k < 0x42F00000 ) */
+ else if (k < 0x42F00000)
+ {
+ double a, da;
- else if (k < 0x7ff00000) {/* 281474976710656 <|x| <2^1024 */
+ int4 n = reduce_sincos_2 (x, &a, &da);
+ retval = do_sincos_2 (a, da, x, n, true);
+ } /* else if (k < 0x42F00000 ) */
- n = __branred(x,&a,&da);
- switch (n) {
- case 1:
- if (a*a < 0.01588) retval = bsloww(-a,-da,x,n);
- else retval = bsloww1(-a,-da,x,n);
- goto ret;
- break;
- case 3:
- if (a*a < 0.01588) retval = bsloww(a,da,x,n);
- else retval = bsloww1(a,da,x,n);
- goto ret;
- break;
+ /* 281474976710656 <|x| <2^1024 */
+ else if (k < 0x7ff00000)
+ retval = reduce_and_compute (x, true);
- case 0:
- case 2:
- retval = bsloww2(a,da,x,n);
- goto ret;
- break;
+ else
+ {
+ if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
+ __set_errno (EDOM);
+ retval = x / x; /* |x| > 2^1024 */
}
+#endif
- } /* else if (k < 0x7ff00000 ) */
-
-
-
-
- else {
- if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
- __set_errno (EDOM);
- retval = x / x; /* |x| > 2^1024 */
- goto ret;
- }
-
- ret:
return retval;
}
/* precision and if still doesn't accurate enough by mpsin or dubsin */
/************************************************************************/
-static double
-SECTION
-slow(double x) {
-static const double th2_36 = 206158430208.0; /* 1.5*2**37 */
- double y,x1,x2,xx,r,t,res,cor,w[2];
- x1=(x+th2_36)-th2_36;
- y = aa.x*x1*x1*x1;
- r=x+y;
- x2=x-x1;
- xx=x*x;
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + bb.x)*xx + 3.0*aa.x*x1*x2)*x +aa.x*x2*x2*x2;
- t=((x-r)+y)+t;
- res=r+t;
- cor = (r-res)+t;
- if (res == res + 1.0007*cor) return res;
- else {
- __dubsin(ABS(x),0,w);
- if (w[0] == w[0]+1.000000001*w[1]) return (x>0)?w[0]:-w[0];
- else return (x>0)?__mpsin(x,0):-__mpsin(-x,0);
- }
+static inline double
+__always_inline
+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 __copysign (w[0], x);
+
+ return __copysign (__mpsin (fabs (x), 0, false), x);
}
+
/*******************************************************************************/
-/* Routine compute sin(x) for 0.25<|x|< 0.855469 by __sincostab.tbl and Taylor */
+/* 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 sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,c1,c2,xx,cor,res;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- u.x=big.x+y;
- y=y-(u.x-big.x);
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k]; /* Data */
- ssn=__sincostab.x[k+1]; /* from */
- cs=__sincostab.x[k+2]; /* tables */
- ccs=__sincostab.x[k+3]; /* __sincostab.tbl */
- y1 = (y+t22)-t22;
- y2 = y - y1;
- c1 = (cs+t22)-t22;
- c2=(cs-c1)+ccs;
- cor=(ssn+s*ccs+cs*s+c2*y+c1*y2)-sn*c;
- y=sn+c1*y1;
- cor = cor+((sn-y)+c1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- if (res == res+1.0005*cor) return (x>0)?res:-res;
- else {
- __dubsin(ABS(x),0,w);
- if (w[0] == w[0]+1.000000005*w[1]) return (x>0)?w[0]:-w[0];
- else return (x>0)?__mpsin(x,0):-__mpsin(-x,0);
- }
+static inline double
+__always_inline
+slow1 (double x)
+{
+ double w[2], cor, res;
+
+ res = do_sin_slow (x, 0, 0, &cor);
+ if (res == res + cor)
+ return res;
+
+ __dubsin (fabs (x), 0, w);
+ if (w[0] == w[0] + 1.000000005 * w[1])
+ return w[0];
+
+ return __mpsin (fabs (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 sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res,del;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- y = hp0.x-y;
- if (y>=0) {
- u.x = big.x+y;
- y = y-(u.x-big.x);
- del = hp1.x;
- }
- else {
- u.x = big.x-y;
- y = -(y+(u.x-big.x));
- del = -hp1.x;
- }
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = y*del+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- y1 = (y+t22)-t22;
- y2 = (y - y1)+del;
- e1 = (sn+t22)-t22;
- e2=(sn-e1)+ssn;
- cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
- y=cs-e1*y1;
- cor = cor+((cs-y)-e1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- if (res == res+1.0005*cor) return (x>0)?res:-res;
- else {
- y=ABS(x)-hp0.x;
- y1=y-hp1.x;
- y2=(y-y1)-hp1.x;
- __docos(y1,y2,w);
- if (w[0] == w[0]+1.000000005*w[1]) return (x>0)?w[0]:-w[0];
- else return (x>0)?__mpsin(x,0):-__mpsin(-x,0);
- }
+static inline double
+__always_inline
+slow2 (double x)
+{
+ double w[2], y, y1, y2, cor, res;
+
+ double t = hp0 - fabs (x);
+ res = do_cos_slow (t, hp1, 0, &cor);
+ if (res == res + cor)
+ return 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 w[0];
+
+ return __mpsin (fabs (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) {
- static const double th2_36 = 206158430208.0; /* 1.5*2**37 */
- double y,x1,x2,xx,r,t,res,cor,w[2],a,da,xn;
- union {int4 i[2]; double x;} v;
+/* Compute sin(x + dx) where X is small enough to use Taylor series around zero
+ and (x + dx) in the first or third quarter of the unit circle. ORIG is the
+ original value of X for computing error of the result. If the result is not
+ accurate enough, the routine calls mpsin or dubsin. SHIFT_QUADRANT rotates
+ the unit circle by 1 to compute the cosine instead of sine. */
+static inline double
+__always_inline
+sloww (double x, double dx, double orig, bool shift_quadrant)
+{
+ double y, t, res, cor, w[2], a, da, xn;
+ mynumber v;
int4 n;
- x1=(x+th2_36)-th2_36;
- y = aa.x*x1*x1*x1;
- r=x+y;
- x2=(x-x1)+dx;
- xx=x*x;
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + bb.x)*xx + 3.0*aa.x*x1*x2)*x +aa.x*x2*x2*x2+dx;
- t=((x-r)+y)+t;
- res=r+t;
- cor = (r-res)+t;
- cor = (cor>0)? 1.0005*cor+ABS(orig)*3.1e-30 : 1.0005*cor-ABS(orig)*3.1e-30;
- if (res == res + cor) return res;
- else {
- (x>0)? __dubsin(x,dx,w) : __dubsin(-x,-dx,w);
- cor = (w[1]>0)? 1.000000001*w[1] + ABS(orig)*1.1e-30 : 1.000000001*w[1] - ABS(orig)*1.1e-30;
- if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
- else {
- t = (orig*hpinv.x + toint.x);
- xn = t - toint.x;
- v.x = t;
- y = (orig - xn*mp1.x) - xn*mp2.x;
- n =v.i[LOW_HALF]&3;
- da = xn*pp3.x;
- t=y-da;
- da = (y-t)-da;
- y = xn*pp4.x;
- a = t - y;
- da = ((t-a)-y)+da;
- if (n&2) {a=-a; da=-da;}
- (a>0)? __dubsin(a,da,w) : __dubsin(-a,-da,w);
- cor = (w[1]>0)? 1.000000001*w[1] + ABS(orig)*1.1e-40 : 1.000000001*w[1] - ABS(orig)*1.1e-40;
- if (w[0] == w[0]+cor) return (a>0)?w[0]:-w[0];
- else return __mpsin1(orig);
+ res = TAYLOR_SLOW (x, dx, cor);
+
+ double eps = fabs (orig) * 3.1e-30;
+
+ cor = 1.0005 * cor + __copysign (eps, cor);
+
+ if (res == res + cor)
+ return res;
+
+ a = fabs (x);
+ da = (x > 0) ? dx : -dx;
+ __dubsin (a, da, w);
+ eps = fabs (orig) * 1.1e-30;
+ cor = 1.000000001 * w[1] + __copysign (eps, w[1]);
+
+ if (w[0] == w[0] + cor)
+ return __copysign (w[0], x);
+
+ t = (orig * hpinv + toint);
+ xn = t - toint;
+ v.x = t;
+ y = (orig - xn * mp1) - xn * mp2;
+ n = (v.i[LOW_HALF] + shift_quadrant) & 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)
+ {
+ a = -a;
+ da = -da;
}
- }
+ x = fabs (a);
+ dx = (a > 0) ? da : -da;
+ __dubsin (x, dx, w);
+ eps = fabs (orig) * 1.1e-40;
+ cor = 1.000000001 * w[1] + __copysign (eps, w[1]);
+
+ if (w[0] == w[0] + cor)
+ return __copysign (w[0], a);
+
+ return shift_quadrant ? __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) {
- mynumber u;
- double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,c1,c2,xx,cor,res;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- u.x=big.x+y;
- y=y-(u.x-big.x);
- dx=(x>0)?dx:-dx;
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- y1 = (y+t22)-t22;
- y2 = (y - y1)+dx;
- c1 = (cs+t22)-t22;
- c2=(cs-c1)+ccs;
- cor=(ssn+s*ccs+cs*s+c2*y+c1*y2-sn*y*dx)-sn*c;
- y=sn+c1*y1;
- cor = cor+((sn-y)+c1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- cor = (cor>0)? 1.0005*cor+3.1e-30*ABS(orig) : 1.0005*cor-3.1e-30*ABS(orig);
- if (res == res + cor) return (x>0)?res:-res;
- else {
- __dubsin(ABS(x),dx,w);
- cor = (w[1]>0)? 1.000000005*w[1]+1.1e-30*ABS(orig) : 1.000000005*w[1]-1.1e-30*ABS(orig);
- if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
- else return __mpsin1(orig);
- }
+/* Compute sin(x + dx) where X is in the first or third quarter of the unit
+ circle. ORIG is the original value of X for computing error of the result.
+ If the result is not accurate enough, the routine calls mpsin or dubsin.
+ SHIFT_QUADRANT rotates the unit circle by 1 to compute the cosine instead of
+ sine. */
+static inline double
+__always_inline
+sloww1 (double x, double dx, double orig, bool shift_quadrant)
+{
+ double w[2], cor, res;
+
+ res = do_sin_slow (x, dx, 3.1e-30 * fabs (orig), &cor);
+
+ if (res == res + cor)
+ return __copysign (res, x);
+
+ dx = (x > 0 ? dx : -dx);
+ __dubsin (fabs (x), dx, w);
+
+ double eps = 1.1e-30 * fabs (orig);
+ cor = 1.000000005 * w[1] + __copysign (eps, w[1]);
+
+ if (w[0] == w[0] + cor)
+ return __copysign (w[0], x);
+
+ return shift_quadrant ? __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 */
/* accurate enough routine calls mpsin1 or dubsin */
/***************************************************************************/
-static double
-SECTION
-sloww2(double x, double dx, double orig, int n) {
- mynumber u;
- double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- u.x=big.x+y;
- y=y-(u.x-big.x);
- dx=(x>0)?dx:-dx;
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = y*dx+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
-
- y1 = (y+t22)-t22;
- y2 = (y - y1)+dx;
- e1 = (sn+t22)-t22;
- e2=(sn-e1)+ssn;
- cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
- y=cs-e1*y1;
- cor = cor+((cs-y)-e1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- cor = (cor>0)? 1.0005*cor+3.1e-30*ABS(orig) : 1.0005*cor-3.1e-30*ABS(orig);
- if (res == res + cor) return (n&2)?-res:res;
- else {
- __docos(ABS(x),dx,w);
- cor = (w[1]>0)? 1.000000005*w[1]+1.1e-30*ABS(orig) : 1.000000005*w[1]-1.1e-30*ABS(orig);
- if (w[0] == w[0]+cor) return (n&2)?-w[0]:w[0];
- else return __mpsin1(orig);
- }
+static inline double
+__always_inline
+sloww2 (double x, double dx, double orig, int n)
+{
+ double w[2], cor, res;
+
+ res = do_cos_slow (x, dx, 3.1e-30 * fabs (orig), &cor);
+
+ if (res == res + cor)
+ return (n & 2) ? -res : res;
+
+ dx = x > 0 ? dx : -dx;
+ __docos (fabs (x), dx, w);
+
+ double eps = 1.1e-30 * fabs (orig);
+ cor = 1.000000005 * w[1] + __copysign (eps, w[1]);
+
+ 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) */
/* result.And if result not accurate enough routine calls other routines */
/***************************************************************************/
-static double
-SECTION
-bsloww(double x,double dx, double orig,int n) {
- static const double th2_36 = 206158430208.0; /* 1.5*2**37 */
- double y,x1,x2,xx,r,t,res,cor,w[2];
-#if 0
- double a,da,xn;
- union {int4 i[2]; double x;} v;
-#endif
- x1=(x+th2_36)-th2_36;
- y = aa.x*x1*x1*x1;
- r=x+y;
- x2=(x-x1)+dx;
- xx=x*x;
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + bb.x)*xx + 3.0*aa.x*x1*x2)*x +aa.x*x2*x2*x2+dx;
- t=((x-r)+y)+t;
- res=r+t;
- cor = (r-res)+t;
- cor = (cor>0)? 1.0005*cor+1.1e-24 : 1.0005*cor-1.1e-24;
- if (res == res + cor) return res;
- else {
- (x>0)? __dubsin(x,dx,w) : __dubsin(-x,-dx,w);
- cor = (w[1]>0)? 1.000000001*w[1] + 1.1e-24 : 1.000000001*w[1] - 1.1e-24;
- if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
- else return (n&1)?__mpcos1(orig):__mpsin1(orig);
- }
+static inline double
+__always_inline
+bsloww (double x, double dx, double orig, int n)
+{
+ double res, cor, w[2], a, da;
+
+ res = TAYLOR_SLOW (x, dx, cor);
+ cor = 1.0005 * cor + __copysign (1.1e-24, cor);
+ if (res == res + cor)
+ return res;
+
+ a = fabs (x);
+ da = (x > 0) ? dx : -dx;
+ __dubsin (a, da, w);
+ cor = 1.000000001 * w[1] + __copysign (1.1e-24, w[1]);
+
+ if (w[0] == w[0] + cor)
+ return __copysign (w[0], x);
+
+ return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
}
/***************************************************************************/
/* 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 sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,c1,c2,xx,cor,res;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- u.x=big.x+y;
- y=y-(u.x-big.x);
- dx=(x>0)?dx:-dx;
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- y1 = (y+t22)-t22;
- y2 = (y - y1)+dx;
- c1 = (cs+t22)-t22;
- c2=(cs-c1)+ccs;
- cor=(ssn+s*ccs+cs*s+c2*y+c1*y2-sn*y*dx)-sn*c;
- y=sn+c1*y1;
- cor = cor+((sn-y)+c1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- cor = (cor>0)? 1.0005*cor+1.1e-24 : 1.0005*cor-1.1e-24;
- if (res == res + cor) return (x>0)?res:-res;
- else {
- __dubsin(ABS(x),dx,w);
- cor = (w[1]>0)? 1.000000005*w[1]+1.1e-24: 1.000000005*w[1]-1.1e-24;
- if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
- else return (n&1)?__mpcos1(orig):__mpsin1(orig);
- }
+static inline double
+__always_inline
+bsloww1 (double x, double dx, double orig, int n)
+{
+ double w[2], cor, res;
+
+ res = do_sin_slow (x, dx, 1.1e-24, &cor);
+ if (res == res + cor)
+ return (x > 0) ? res : -res;
+
+ dx = (x > 0) ? dx : -dx;
+ __dubsin (fabs (x), dx, w);
+
+ cor = 1.000000005 * w[1] + __copysign (1.1e-24, w[1]);
+
+ if (w[0] == w[0] + cor)
+ return __copysign (w[0], x);
+
+ return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
}
/***************************************************************************/
/* 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 sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- u.x=big.x+y;
- y=y-(u.x-big.x);
- dx=(x>0)?dx:-dx;
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = y*dx+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
-
- y1 = (y+t22)-t22;
- y2 = (y - y1)+dx;
- e1 = (sn+t22)-t22;
- e2=(sn-e1)+ssn;
- cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
- y=cs-e1*y1;
- cor = cor+((cs-y)-e1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- cor = (cor>0)? 1.0005*cor+1.1e-24 : 1.0005*cor-1.1e-24;
- if (res == res + cor) return (n&2)?-res:res;
- else {
- __docos(ABS(x),dx,w);
- cor = (w[1]>0)? 1.000000005*w[1]+1.1e-24 : 1.000000005*w[1]-1.1e-24;
- if (w[0] == w[0]+cor) return (n&2)?-w[0]:w[0];
- else return (n&1)?__mpsin1(orig):__mpcos1(orig);
- }
-}
+static inline double
+__always_inline
+bsloww2 (double x, double dx, double orig, int n)
+{
+ double w[2], cor, res;
-/************************************************************************/
-/* 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 */
-/************************************************************************/
+ res = do_cos_slow (x, dx, 1.1e-24, &cor);
+ if (res == res + cor)
+ return (n & 2) ? -res : res;
-static double
-SECTION
-cslow2(double x) {
- mynumber u;
- double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- u.x = big.x+y;
- y = y-(u.x-big.x);
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- y1 = (y+t22)-t22;
- y2 = y - y1;
- e1 = (sn+t22)-t22;
- e2=(sn-e1)+ssn;
- cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
- y=cs-e1*y1;
- cor = cor+((cs-y)-e1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- if (res == res+1.0005*cor)
- return res;
- else {
- y=ABS(x);
- __docos(y,0,w);
- if (w[0] == w[0]+1.000000005*w[1]) return w[0];
- else return __mpcos(x,0);
- }
-}
+ dx = (x > 0) ? dx : -dx;
+ __docos (fabs (x), dx, w);
-/***************************************************************************/
-/* Routine compute cos(x+dx) (Double-Length number) where x is small enough*/
-/* to use Taylor series around zero and (x+dx) .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 */
-/***************************************************************************/
+ cor = 1.000000005 * w[1] + __copysign (1.1e-24, w[1]);
+ if (w[0] == w[0] + cor)
+ return (n & 2) ? -w[0] : w[0];
-static double
-SECTION
-csloww(double x,double dx, double orig) {
- static const double th2_36 = 206158430208.0; /* 1.5*2**37 */
- double y,x1,x2,xx,r,t,res,cor,w[2],a,da,xn;
- union {int4 i[2]; double x;} v;
- int4 n;
- x1=(x+th2_36)-th2_36;
- y = aa.x*x1*x1*x1;
- r=x+y;
- x2=(x-x1)+dx;
- xx=x*x;
- /* Taylor series */
- t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + bb.x)*xx + 3.0*aa.x*x1*x2)*x +aa.x*x2*x2*x2+dx;
- t=((x-r)+y)+t;
- res=r+t;
- cor = (r-res)+t;
- cor = (cor>0)? 1.0005*cor+ABS(orig)*3.1e-30 : 1.0005*cor-ABS(orig)*3.1e-30;
- if (res == res + cor) return res;
- else {
- (x>0)? __dubsin(x,dx,w) : __dubsin(-x,-dx,w);
- cor = (w[1]>0)? 1.000000001*w[1] + ABS(orig)*1.1e-30 : 1.000000001*w[1] - ABS(orig)*1.1e-30;
- if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
- else {
- t = (orig*hpinv.x + toint.x);
- xn = t - toint.x;
- v.x = t;
- y = (orig - xn*mp1.x) - xn*mp2.x;
- n =v.i[LOW_HALF]&3;
- da = xn*pp3.x;
- t=y-da;
- da = (y-t)-da;
- y = xn*pp4.x;
- a = t - y;
- da = ((t-a)-y)+da;
- if (n==1) {a=-a; da=-da;}
- (a>0)? __dubsin(a,da,w) : __dubsin(-a,-da,w);
- cor = (w[1]>0)? 1.000000001*w[1] + ABS(orig)*1.1e-40 : 1.000000001*w[1] - ABS(orig)*1.1e-40;
- if (w[0] == w[0]+cor) return (a>0)?w[0]:-w[0];
- else return __mpcos1(orig);
- }
- }
+ return (n & 1) ? __mpsin (orig, 0, true) : __mpcos (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 other routines */
-/***************************************************************************/
+/************************************************************************/
+/* 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
-csloww1(double x, double dx, double orig) {
- mynumber u;
- double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,c1,c2,xx,cor,res;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- u.x=big.x+y;
- y=y-(u.x-big.x);
- dx=(x>0)?dx:-dx;
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
- y1 = (y+t22)-t22;
- y2 = (y - y1)+dx;
- c1 = (cs+t22)-t22;
- c2=(cs-c1)+ccs;
- cor=(ssn+s*ccs+cs*s+c2*y+c1*y2-sn*y*dx)-sn*c;
- y=sn+c1*y1;
- cor = cor+((sn-y)+c1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- cor = (cor>0)? 1.0005*cor+3.1e-30*ABS(orig) : 1.0005*cor-3.1e-30*ABS(orig);
- if (res == res + cor) return (x>0)?res:-res;
- else {
- __dubsin(ABS(x),dx,w);
- cor = (w[1]>0)? 1.000000005*w[1]+1.1e-30*ABS(orig) : 1.000000005*w[1]-1.1e-30*ABS(orig);
- if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
- else return __mpcos1(orig);
- }
-}
+static inline double
+__always_inline
+cslow2 (double x)
+{
+ double w[2], cor, res;
+ res = do_cos_slow (x, 0, 0, &cor);
+ if (res == res + cor)
+ return res;
-/***************************************************************************/
-/* 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 other routines */
-/***************************************************************************/
+ __docos (fabs (x), 0, w);
+ if (w[0] == w[0] + 1.000000005 * w[1])
+ return w[0];
-static double
-SECTION
-csloww2(double x, double dx, double orig, int n) {
- mynumber u;
- double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res;
- static const double t22 = 6291456.0;
- int4 k;
- y=ABS(x);
- u.x=big.x+y;
- y=y-(u.x-big.x);
- dx=(x>0)?dx:-dx;
- xx=y*y;
- s = y*xx*(sn3 +xx*sn5);
- c = y*dx+xx*(cs2 +xx*(cs4 + xx*cs6));
- k=u.i[LOW_HALF]<<2;
- sn=__sincostab.x[k];
- ssn=__sincostab.x[k+1];
- cs=__sincostab.x[k+2];
- ccs=__sincostab.x[k+3];
-
- y1 = (y+t22)-t22;
- y2 = (y - y1)+dx;
- e1 = (sn+t22)-t22;
- e2=(sn-e1)+ssn;
- cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
- y=cs-e1*y1;
- cor = cor+((cs-y)-e1*y1);
- res=y+cor;
- cor=(y-res)+cor;
- cor = (cor>0)? 1.0005*cor+3.1e-30*ABS(orig) : 1.0005*cor-3.1e-30*ABS(orig);
- if (res == res + cor) return (n)?-res:res;
- else {
- __docos(ABS(x),dx,w);
- cor = (w[1]>0)? 1.000000005*w[1]+1.1e-30*ABS(orig) : 1.000000005*w[1]-1.1e-30*ABS(orig);
- if (w[0] == w[0]+cor) return (n)?-w[0]:w[0];
- else return __mpcos1(orig);
- }
+ 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
projects / thirdparty / glibc.git / blobdiff