/* Quad-precision floating point argument reduction.
- Copyright (C) 1999, 2007 Free Software Foundation, Inc.
+ Copyright (C) 1999, 2007, 2009-2013 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by Jakub Jelinek <jj@ultra.linux.cz>
#include <config.h>
/* Specification. */
-#include <math.h>
+#include "trigl.h"
#include <float.h>
+#include <math.h>
+
+/* Code based on glibc/sysdeps/ieee754/ldbl-128/e_rem_pio2l.c
+ and glibc/sysdeps/ieee754/dbl-64/k_rem_pio2.c. */
/* Table of constants for 2/pi, 5628 hexadecimal digits of 2/pi */
static const int two_over_pi[] = {
static const long double c[] = {
/* 93 bits of pi/2 */
#define PI_2_1 c[0]
- 1.57079632679489661923132169155131424e+00L, /* 3fff921fb54442d18469898cc5100000 */
+ 1.57079632679489661923132169155131424e+00L, /* 3fff921fb54442d18469898cc5100000 */
/* pi/2 - PI_2_1 */
#define PI_2_1t c[1]
- 8.84372056613570112025531863263659260e-29L, /* 3fa1c06e0e68948127044533e63a0106 */
+ 8.84372056613570112025531863263659260e-29L, /* 3fa1c06e0e68948127044533e63a0106 */
};
static int kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec,
- const int *ipio2);
+ const int *ipio2);
int
ieee754_rem_pio2l (long double x, long double *y)
int exp, n;
if (x >= -0.78539816339744830961566084581987572104929234984377
- && x < 0.78539816339744830961566084581987572104929234984377)
+ && x <= 0.78539816339744830961566084581987572104929234984377)
/* x in <-pi/4, pi/4> */
{
y[0] = x;
return 0;
}
- if (x >= 2.35619449019234492884698253745962716314787704953131
- && x < 2.35619449019234492884698253745962716314787704953131)
- if (x > 0)
+ if (x > 0 && x < 2.35619449019234492884698253745962716314787704953131)
{
- /* 113 + 93 bit PI is ok */
- z = x - PI_2_1;
- y[0] = z - PI_2_1t;
- y[1] = (z - y[0]) - PI_2_1t;
- return 1;
+ /* 113 + 93 bit PI is ok */
+ z = x - PI_2_1;
+ y[0] = z - PI_2_1t;
+ y[1] = (z - y[0]) - PI_2_1t;
+ return 1;
}
- else
+
+ if (x < 0 && x > -2.35619449019234492884698253745962716314787704953131)
{
- /* 113 + 93 bit PI is ok */
- z = x + PI_2_1;
- y[0] = z + PI_2_1t;
- y[1] = (z - y[0]) + PI_2_1t;
- return -1;
+ /* 113 + 93 bit PI is ok */
+ z = x + PI_2_1;
+ y[0] = z + PI_2_1t;
+ y[1] = (z - y[0]) + PI_2_1t;
+ return -1;
}
- if (x + x == x) /* x is ±oo */
+ if (x + x == x) /* x is ±oo */
{
y[0] = x - x;
y[1] = y[0];
* double x[],y[]; int e0,nx,prec; int ipio2[];
*
* kernel_rem_pio2 return the last three digits of N with
- * y = x - N*pi/2
+ * y = x - N*pi/2
* so that |y| < pi/2.
*
* The method is to compute the integer (mod 8) and fraction parts of
* (2/pi) is represented by an array of 24-bit integers in ipio2[].
*
* Input parameters:
- * x[] The input value (must be positive) is broken into nx
- * pieces of 24-bit integers in double precision format.
- * x[i] will be the i-th 24 bit of x. The scaled exponent
- * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
- * match x's up to 24 bits.
+ * x[] The input value (must be positive) is broken into nx
+ * pieces of 24-bit integers in double precision format.
+ * x[i] will be the i-th 24 bit of x. The scaled exponent
+ * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
+ * match x's up to 24 bits.
*
- * Example of breaking a double positive z into x[0]+x[1]+x[2]:
- * e0 = ilogb(z)-23
- * z = scalbn(z,-e0)
- * for i = 0,1,2
- * x[i] = floor(z)
- * z = (z-x[i])*2**24
+ * Example of breaking a double positive z into x[0]+x[1]+x[2]:
+ * e0 = ilogb(z)-23
+ * z = scalbn(z,-e0)
+ * for i = 0,1,2
+ * x[i] = floor(z)
+ * z = (z-x[i])*2**24
*
*
- * y[] ouput result in an array of double precision numbers.
- * The dimension of y[] is:
- * 24-bit precision 1
- * 53-bit precision 2
- * 64-bit precision 2
- * 113-bit precision 3
- * The actual value is the sum of them. Thus for 113-bit
- * precision, one may have to do something like:
+ * y[] ouput result in an array of double precision numbers.
+ * The dimension of y[] is:
+ * 24-bit precision 1
+ * 53-bit precision 2
+ * 64-bit precision 2
+ * 113-bit precision 3
+ * The actual value is the sum of them. Thus for 113-bit
+ * precision, one may have to do something like:
*
- * long double t,w,r_head, r_tail;
- * t = (long double)y[2] + (long double)y[1];
- * w = (long double)y[0];
- * r_head = t+w;
- * r_tail = w - (r_head - t);
+ * long double t,w,r_head, r_tail;
+ * t = (long double)y[2] + (long double)y[1];
+ * w = (long double)y[0];
+ * r_head = t+w;
+ * r_tail = w - (r_head - t);
*
- * e0 The exponent of x[0]
+ * e0 The exponent of x[0]
*
- * nx dimension of x[]
+ * nx dimension of x[]
*
- * prec an integer indicating the precision:
- * 0 24 bits (single)
- * 1 53 bits (double)
- * 2 64 bits (extended)
- * 3 113 bits (quad)
+ * prec an integer indicating the precision:
+ * 0 24 bits (single)
+ * 1 53 bits (double)
+ * 2 64 bits (extended)
+ * 3 113 bits (quad)
*
- * ipio2[]
- * integer array, contains the (24*i)-th to (24*i+23)-th
- * bit of 2/pi after binary point. The corresponding
- * floating value is
+ * ipio2[]
+ * integer array, contains the (24*i)-th to (24*i+23)-th
+ * bit of 2/pi after binary point. The corresponding
+ * floating value is
*
- * ipio2[i] * 2^(-24(i+1)).
+ * ipio2[i] * 2^(-24(i+1)).
*
* External function:
- * double scalbn(), floor();
+ * double scalbn(), floor();
*
*
* Here is the description of some local variables:
*
- * jk jk+1 is the initial number of terms of ipio2[] needed
- * in the computation. The recommended value is 2,3,4,
- * 6 for single, double, extended,and quad.
+ * jk jk+1 is the initial number of terms of ipio2[] needed
+ * in the computation. The recommended value is 2,3,4,
+ * 6 for single, double, extended,and quad.
*
- * jz local integer variable indicating the number of
- * terms of ipio2[] used.
+ * jz local integer variable indicating the number of
+ * terms of ipio2[] used.
*
- * jx nx - 1
+ * jx nx - 1
*
- * jv index for pointing to the suitable ipio2[] for the
- * computation. In general, we want
- * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
- * is an integer. Thus
- * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
- * Hence jv = max(0,(e0-3)/24).
+ * jv index for pointing to the suitable ipio2[] for the
+ * computation. In general, we want
+ * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
+ * is an integer. Thus
+ * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
+ * Hence jv = max(0,(e0-3)/24).
*
- * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
+ * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
*
- * q[] double array with integral value, representing the
- * 24-bits chunk of the product of x and 2/pi.
+ * q[] double array with integral value, representing the
+ * 24-bits chunk of the product of x and 2/pi.
*
- * q0 the corresponding exponent of q[0]. Note that the
- * exponent for q[i] would be q0-24*i.
+ * q0 the corresponding exponent of q[0]. Note that the
+ * exponent for q[i] would be q0-24*i.
*
- * PIo2[] double precision array, obtained by cutting pi/2
- * into 24 bits chunks.
+ * PIo2[] double precision array, obtained by cutting pi/2
+ * into 24 bits chunks.
*
- * f[] ipio2[] in floating point
+ * f[] ipio2[] in floating point
*
- * iq[] integer array by breaking up q[] in 24-bits chunk.
+ * iq[] integer array by breaking up q[] in 24-bits chunk.
*
- * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
+ * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
*
- * ih integer. If >0 it indicates q[] is >= 0.5, hence
- * it also indicates the *sign* of the result.
+ * ih integer. If >0 it indicates q[] is >= 0.5, hence
+ * it also indicates the *sign* of the result.
*
*/
* to produce the hexadecimal values shown.
*/
-static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
+static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
static const double PIo2[] = {
- 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
- 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
- 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
- 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
- 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
- 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
- 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
- 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
+ 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
+ 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
+ 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
+ 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
+ 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
+ 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
+ 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
+ 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
};
-static const double zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
- twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
+static const double zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
+ twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
static int
kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec,
- const int *ipio2)
+ const int *ipio2)
{
int jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
double z, fw, f[20], fq[20], q[20];
for (i = 0; i <= jk; i++)
{
for (j = 0, fw = 0.0; j <= jx; j++)
- fw += x[j] * f[jx + i - j];
+ fw += x[j] * f[jx + i - j];
q[i] = fw;
}
jz = jk;
recompute:
- /* distill q[] into iq[] reversingly */
+ /* distill q[] into iq[] in reverse order */
for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
{
fw = (double) ((int) (twon24 * z));
}
/* compute n */
- z = ldexp (z, q0); /* actual value of z */
- z -= 8.0 * floor (z * 0.125); /* trim off integer >= 8 */
+ z = ldexp (z, q0); /* actual value of z */
+ z -= 8.0 * floor (z * 0.125); /* trim off integer >= 8 */
n = (int) z;
z -= (double) n;
ih = 0;
if (q0 > 0)
- { /* need iq[jz-1] to determine n */
+ { /* need iq[jz-1] to determine n */
i = (iq[jz - 1] >> (24 - q0));
n += i;
iq[jz - 1] -= i << (24 - q0);
ih = 2;
if (ih > 0)
- { /* q > 0.5 */
+ { /* q > 0.5 */
n += 1;
carry = 0;
for (i = 0; i < jz; i++)
- { /* compute 1-q */
- j = iq[i];
- if (carry == 0)
- {
- if (j != 0)
- {
- carry = 1;
- iq[i] = 0x1000000 - j;
- }
- }
- else
- iq[i] = 0xffffff - j;
- }
+ { /* compute 1-q */
+ j = iq[i];
+ if (carry == 0)
+ {
+ if (j != 0)
+ {
+ carry = 1;
+ iq[i] = 0x1000000 - j;
+ }
+ }
+ else
+ iq[i] = 0xffffff - j;
+ }
if (q0 > 0)
- { /* rare case: chance is 1 in 12 */
- switch (q0)
- {
- case 1:
- iq[jz - 1] &= 0x7fffff;
- break;
- case 2:
- iq[jz - 1] &= 0x3fffff;
- break;
- }
- }
+ { /* rare case: chance is 1 in 12 */
+ switch (q0)
+ {
+ case 1:
+ iq[jz - 1] &= 0x7fffff;
+ break;
+ case 2:
+ iq[jz - 1] &= 0x3fffff;
+ break;
+ }
+ }
if (ih == 2)
- {
- z = one - z;
- if (carry != 0)
- z -= ldexp (one, q0);
- }
+ {
+ z = one - z;
+ if (carry != 0)
+ z -= ldexp (one, q0);
+ }
}
/* check if recomputation is needed */
{
j = 0;
for (i = jz - 1; i >= jk; i--)
- j |= iq[i];
+ j |= iq[i];
if (j == 0)
- { /* need recomputation */
- for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */
-
- for (i = jz + 1; i <= jz + k; i++)
- { /* add q[jz+1] to q[jz+k] */
- f[jx + i] = (double) ipio2[jv + i];
- for (j = 0, fw = 0.0; j <= jx; j++)
- fw += x[j] * f[jx + i - j];
- q[i] = fw;
- }
- jz += k;
- goto recompute;
- }
+ { /* need recomputation */
+ for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */
+
+ for (i = jz + 1; i <= jz + k; i++)
+ { /* add q[jz+1] to q[jz+k] */
+ f[jx + i] = (double) ipio2[jv + i];
+ for (j = 0, fw = 0.0; j <= jx; j++)
+ fw += x[j] * f[jx + i - j];
+ q[i] = fw;
+ }
+ jz += k;
+ goto recompute;
+ }
}
/* chop off zero terms */
jz -= 1;
q0 -= 24;
while (iq[jz] == 0)
- {
- jz--;
- q0 -= 24;
- }
+ {
+ jz--;
+ q0 -= 24;
+ }
}
else
- { /* break z into 24-bit if necessary */
+ { /* break z into 24-bit if necessary */
z = ldexp (z, -q0);
if (z >= two24)
- {
- fw = (double) ((int) (twon24 * z));
- iq[jz] = (int) (z - two24 * fw);
- jz += 1;
- q0 += 24;
- iq[jz] = (int) fw;
- }
+ {
+ fw = (double) ((int) (twon24 * z));
+ iq[jz] = (int) (z - two24 * fw);
+ jz += 1;
+ q0 += 24;
+ iq[jz] = (int) fw;
+ }
else
- iq[jz] = (int) z;
+ iq[jz] = (int) z;
}
/* convert integer "bit" chunk to floating-point value */
for (i = jz; i >= 0; i--)
{
for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
- fw += PIo2[k] * q[i + k];
+ fw += PIo2[k] * q[i + k];
fq[jz - i] = fw;
}
case 0:
fw = 0.0;
for (i = jz; i >= 0; i--)
- fw += fq[i];
+ fw += fq[i];
y[0] = (ih == 0) ? fw : -fw;
break;
case 1:
case 2:
fw = 0.0;
for (i = jz; i >= 0; i--)
- fw += fq[i];
+ fw += fq[i];
y[0] = (ih == 0) ? fw : -fw;
fw = fq[0] - fw;
for (i = 1; i <= jz; i++)
- fw += fq[i];
+ fw += fq[i];
y[1] = (ih == 0) ? fw : -fw;
break;
- case 3: /* painful */
+ case 3: /* painful */
for (i = jz; i > 0; i--)
- {
- fw = fq[i - 1] + fq[i];
- fq[i] += fq[i - 1] - fw;
- fq[i - 1] = fw;
- }
+ {
+ fw = fq[i - 1] + fq[i];
+ fq[i] += fq[i - 1] - fw;
+ fq[i - 1] = fw;
+ }
for (i = jz; i > 1; i--)
- {
- fw = fq[i - 1] + fq[i];
- fq[i] += fq[i - 1] - fw;
- fq[i - 1] = fw;
- }
+ {
+ fw = fq[i - 1] + fq[i];
+ fq[i] += fq[i - 1] - fw;
+ fq[i - 1] = fw;
+ }
for (fw = 0.0, i = jz; i >= 2; i--)
- fw += fq[i];
+ fw += fq[i];
if (ih == 0)
- {
- y[0] = fq[0];
- y[1] = fq[1];
- y[2] = fw;
- }
+ {
+ y[0] = fq[0];
+ y[1] = fq[1];
+ y[2] = fw;
+ }
else
- {
- y[0] = -fq[0];
- y[1] = -fq[1];
- y[2] = -fw;
- }
+ {
+ y[0] = -fq[0];
+ y[1] = -fq[1];
+ y[2] = -fw;
+ }
}
return n & 7;
}
projects / gnulib.git / blobdiff