@@ -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. */
/* */
@@ -74,29 +63,6 @@
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; \
@@ -123,23 +89,7 @@ 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, 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
@@ -166,40 +116,6 @@ do_cos (double x, double dx)
return cs + cor;
}
-/* 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
@@ -224,70 +140,6 @@ do_sin (double x, double dx)
return sn + cor;
}
-/* 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;
-}
-
/* 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
@@ -508,299 +360,6 @@ __cos (double x)
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 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 */
-/* and if result still doesn't accurate enough by mpsin or dubsin */
-/*******************************************************************************/
-
-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 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);
-}
-
-/* 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;
- 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);
-}
-
-/* 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 */
-/* and quarter(n= 1or 3)of x for computing error of result.And if result not*/
-/* accurate enough routine calls mpsin1 or dubsin */
-/***************************************************************************/
-
-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) */
-/* 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 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);
-}
-
-/***************************************************************************/
-/* 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 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);
-}
-
-/***************************************************************************/
-/* 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 inline double
-__always_inline
-bsloww2 (double x, double dx, double orig, int n)
-{
- double w[2], cor, res;
-
- res = do_cos_slow (x, dx, 1.1e-24, &cor);
- if (res == res + cor)
- return (n & 2) ? -res : res;
-
- dx = (x > 0) ? dx : -dx;
- __docos (fabs (x), dx, w);
-
- cor = 1.000000005 * w[1] + __copysign (1.1e-24, 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 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 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;
-
- __docos (fabs (x), 0, w);
- if (w[0] == w[0] + 1.000000005 * w[1])
- return w[0];
-
- return __mpcos (x, 0, false);
-}
-
#ifndef __cos
libm_alias_double (__cos, cos)
#endif