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+/***************************************************************************
+ * Copyright (c) Johan Mabille, Sylvain Corlay, Wolf Vollprecht and *
+ * Martin Renou *
+ * Copyright (c) QuantStack *
+ * Copyright (c) Serge Guelton *
+ * *
+ * Distributed under the terms of the BSD 3-Clause License. *
+ * *
+ * The full license is in the file LICENSE, distributed with this software. *
+ ****************************************************************************/
+
+#include <cmath>
+#include <cstdint>
+#include <cstring>
+
+namespace xsimd
+{
+ namespace detail
+ {
+
+ /* origin: boost/simd/arch/common/scalar/function/rem_pio2.hpp */
+ /*
+ * ====================================================
+ * copyright 2016 NumScale SAS
+ *
+ * Distributed under the Boost Software License, Version 1.0.
+ * (See copy at http://boost.org/LICENSE_1_0.txt)
+ * ====================================================
+ */
+#if defined(_MSC_VER)
+#define ONCE0 \
+ __pragma(warning(push)) \
+ __pragma(warning(disable : 4127)) while (0) \
+ __pragma(warning(pop)) /**/
+#else
+#define ONCE0 while (0)
+#endif
+
+ /*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(__GNUC__) && defined(__BYTE_ORDER__)
+#if __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__
+#define XSIMD_LITTLE_ENDIAN
+#endif
+#elif defined(_WIN32)
+ // We can safely assume that Windows is always little endian
+#define XSIMD_LITTLE_ENDIAN
+#elif defined(i386) || defined(i486) || defined(intel) || defined(x86) || defined(i86pc) || defined(__alpha) || defined(__osf__)
+#define XSIMD_LITTLE_ENDIAN
+#endif
+
+#ifdef XSIMD_LITTLE_ENDIAN
+#define LOW_WORD_IDX 0
+#define HIGH_WORD_IDX sizeof(std::uint32_t)
+#else
+#define LOW_WORD_IDX sizeof(std::uint32_t)
+#define HIGH_WORD_IDX 0
+#endif
+
+#define GET_HIGH_WORD(i, d) \
+ do \
+ { \
+ double f = (d); \
+ std::memcpy(&(i), reinterpret_cast<char*>(&f) + HIGH_WORD_IDX, \
+ sizeof(std::uint32_t)); \
+ } \
+ ONCE0 \
+ /**/
+
+#define GET_LOW_WORD(i, d) \
+ do \
+ { \
+ double f = (d); \
+ std::memcpy(&(i), reinterpret_cast<char*>(&f) + LOW_WORD_IDX, \
+ sizeof(std::uint32_t)); \
+ } \
+ ONCE0 \
+ /**/
+
+#define SET_HIGH_WORD(d, v) \
+ do \
+ { \
+ double f = (d); \
+ std::uint32_t value = (v); \
+ std::memcpy(reinterpret_cast<char*>(&f) + HIGH_WORD_IDX, \
+ &value, sizeof(std::uint32_t)); \
+ (d) = f; \
+ } \
+ ONCE0 \
+ /**/
+
+#define SET_LOW_WORD(d, v) \
+ do \
+ { \
+ double f = (d); \
+ std::uint32_t value = (v); \
+ std::memcpy(reinterpret_cast<char*>(&f) + LOW_WORD_IDX, \
+ &value, sizeof(std::uint32_t)); \
+ (d) = f; \
+ } \
+ ONCE0 \
+ /**/
+
+ /*
+ * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
+ * 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
+ * so that |y| < pi/2.
+ *
+ * The method is to compute the integer (mod 8) and fraction parts of
+ * (2/pi)*x without doing the full multiplication. In general we
+ * skip the part of the product that are known to be a huge integer (
+ * more accurately, = 0 mod 8 ). Thus the number of operations are
+ * independent of the exponent of the input.
+ *
+ * (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.
+ *
+ * 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
+ * precison, 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);
+ *
+ * e0 The exponent of x[0]
+ *
+ * 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)
+ *
+ * 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)).
+ *
+ * External function:
+ * 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.
+ *
+ * jz local integer variable indicating the number of
+ * terms of ipio2[] used.
+ *
+ * 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).
+ *
+ * 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.
+ *
+ * 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.
+ *
+ * f[] ipio2[] in floating point
+ *
+ * iq[] integer array by breaking up q[] in 24-bits chunk.
+ *
+ * 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.
+ *
+ */
+
+ inline int32_t __kernel_rem_pio2(double* x, double* y, int32_t e0, int32_t nx, int32_t prec, const int32_t* ipio2) noexcept
+ {
+ static const int32_t 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 */
+ };
+
+ static const double
+ zero
+ = 0.0,
+ one = 1.0,
+ two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
+ twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
+
+ int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
+ double z, fw, f[20], fq[20], q[20];
+
+ /* initialize jk*/
+ jk = init_jk[prec];
+ jp = jk;
+
+ /* determine jx,jv,q0, note that 3>q0 */
+ jx = nx - 1;
+ jv = (e0 - 3) / 24;
+ if (jv < 0)
+ jv = 0;
+ q0 = e0 - 24 * (jv + 1);
+
+ /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
+ j = jv - jx;
+ m = jx + jk;
+ for (i = 0; i <= m; i++, j++)
+ f[i] = (j < 0) ? zero : (double)ipio2[j];
+
+ /* compute q[0],q[1],...q[jk] */
+ for (i = 0; i <= jk; i++)
+ {
+ for (j = 0, fw = 0.0; j <= jx; j++)
+ fw += x[j] * f[jx + i - j];
+ q[i] = fw;
+ }
+
+ jz = jk;
+
+ recompute:
+ /* distill q[] into iq[] reversingly */
+ for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
+ {
+ fw = (double)((int32_t)(twon24 * z));
+ iq[i] = (int)(z - two24 * fw);
+ z = q[j - 1] + fw;
+ }
+
+ /* compute n */
+ z = std::scalbn(z, q0); /* actual value of z */
+ z -= 8.0 * std::floor(z * 0.125); /* trim off integer >= 8 */
+ n = (int32_t)z;
+ z -= (double)n;
+ ih = 0;
+ if (q0 > 0)
+ { /* need iq[jz-1] to determine n */
+ i = (iq[jz - 1] >> (24 - q0));
+ n += i;
+ iq[jz - 1] -= i << (24 - q0);
+ ih = iq[jz - 1] >> (23 - q0);
+ }
+ else if (q0 == 0)
+ ih = iq[jz - 1] >> 23;
+ else if (z >= 0.5)
+ ih = 2;
+
+ if (ih > 0)
+ { /* 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;
+ }
+ 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;
+ }
+ }
+ if (ih == 2)
+ {
+ z = one - z;
+ if (carry != 0)
+ z -= std::scalbn(one, q0);
+ }
+ }
+
+ /* check if recomputation is needed */
+ if (z == zero)
+ {
+ j = 0;
+ for (i = jz - 1; i >= jk; 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;
+ }
+ }
+
+ /* chop off zero terms */
+ if (z == 0.0)
+ {
+ jz -= 1;
+ q0 -= 24;
+ while (iq[jz] == 0)
+ {
+ jz--;
+ q0 -= 24;
+ }
+ }
+ else
+ { /* break z into 24-bit if necessary */
+ z = std::scalbn(z, -q0);
+ if (z >= two24)
+ {
+ fw = (double)((int32_t)(twon24 * z));
+ iq[jz] = (int32_t)(z - two24 * fw);
+ jz += 1;
+ q0 += 24;
+ iq[jz] = (int32_t)fw;
+ }
+ else
+ iq[jz] = (int32_t)z;
+ }
+
+ /* convert integer "bit" chunk to floating-point value */
+ fw = scalbn(one, q0);
+ for (i = jz; i >= 0; i--)
+ {
+ q[i] = fw * (double)iq[i];
+ fw *= twon24;
+ }
+
+ /* compute PIo2[0,...,jp]*q[jz,...,0] */
+ for (i = jz; i >= 0; i--)
+ {
+ for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
+ fw += PIo2[k] * q[i + k];
+ fq[jz - i] = fw;
+ }
+
+ /* compress fq[] into y[] */
+ switch (prec)
+ {
+ case 0:
+ fw = 0.0;
+ for (i = jz; i >= 0; 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];
+ y[0] = (ih == 0) ? fw : -fw;
+ fw = fq[0] - fw;
+ for (i = 1; i <= jz; i++)
+ fw += fq[i];
+ y[1] = (ih == 0) ? fw : -fw;
+ break;
+ 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;
+ }
+ for (i = jz; i > 1; i--)
+ {
+ 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];
+ if (ih == 0)
+ {
+ y[0] = fq[0];
+ y[1] = fq[1];
+ y[2] = fw;
+ }
+ else
+ {
+ y[0] = -fq[0];
+ y[1] = -fq[1];
+ y[2] = -fw;
+ }
+ }
+ return n & 7;
+ }
+
+ inline std::int32_t __ieee754_rem_pio2(double x, double* y) noexcept
+ {
+ static const std::int32_t two_over_pi[] = {
+ 0xA2F983,
+ 0x6E4E44,
+ 0x1529FC,
+ 0x2757D1,
+ 0xF534DD,
+ 0xC0DB62,
+ 0x95993C,
+ 0x439041,
+ 0xFE5163,
+ 0xABDEBB,
+ 0xC561B7,
+ 0x246E3A,
+ 0x424DD2,
+ 0xE00649,
+ 0x2EEA09,
+ 0xD1921C,
+ 0xFE1DEB,
+ 0x1CB129,
+ 0xA73EE8,
+ 0x8235F5,
+ 0x2EBB44,
+ 0x84E99C,
+ 0x7026B4,
+ 0x5F7E41,
+ 0x3991D6,
+ 0x398353,
+ 0x39F49C,
+ 0x845F8B,
+ 0xBDF928,
+ 0x3B1FF8,
+ 0x97FFDE,
+ 0x05980F,
+ 0xEF2F11,
+ 0x8B5A0A,
+ 0x6D1F6D,
+ 0x367ECF,
+ 0x27CB09,
+ 0xB74F46,
+ 0x3F669E,
+ 0x5FEA2D,
+ 0x7527BA,
+ 0xC7EBE5,
+ 0xF17B3D,
+ 0x0739F7,
+ 0x8A5292,
+ 0xEA6BFB,
+ 0x5FB11F,
+ 0x8D5D08,
+ 0x560330,
+ 0x46FC7B,
+ 0x6BABF0,
+ 0xCFBC20,
+ 0x9AF436,
+ 0x1DA9E3,
+ 0x91615E,
+ 0xE61B08,
+ 0x659985,
+ 0x5F14A0,
+ 0x68408D,
+ 0xFFD880,
+ 0x4D7327,
+ 0x310606,
+ 0x1556CA,
+ 0x73A8C9,
+ 0x60E27B,
+ 0xC08C6B,
+ };
+
+ static const std::int32_t npio2_hw[] = {
+ 0x3FF921FB,
+ 0x400921FB,
+ 0x4012D97C,
+ 0x401921FB,
+ 0x401F6A7A,
+ 0x4022D97C,
+ 0x4025FDBB,
+ 0x402921FB,
+ 0x402C463A,
+ 0x402F6A7A,
+ 0x4031475C,
+ 0x4032D97C,
+ 0x40346B9C,
+ 0x4035FDBB,
+ 0x40378FDB,
+ 0x403921FB,
+ 0x403AB41B,
+ 0x403C463A,
+ 0x403DD85A,
+ 0x403F6A7A,
+ 0x40407E4C,
+ 0x4041475C,
+ 0x4042106C,
+ 0x4042D97C,
+ 0x4043A28C,
+ 0x40446B9C,
+ 0x404534AC,
+ 0x4045FDBB,
+ 0x4046C6CB,
+ 0x40478FDB,
+ 0x404858EB,
+ 0x404921FB,
+ };
+
+ /*
+ * invpio2: 53 bits of 2/pi
+ * pio2_1: first 33 bit of pi/2
+ * pio2_1t: pi/2 - pio2_1
+ * pio2_2: second 33 bit of pi/2
+ * pio2_2t: pi/2 - (pio2_1+pio2_2)
+ * pio2_3: third 33 bit of pi/2
+ * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
+ */
+
+ static const double
+ zero
+ = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
+ half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
+ two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
+ invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
+ pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
+ pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
+ pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
+ pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
+ pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
+ pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
+
+ double z = 0., w, t, r, fn;
+ double tx[3];
+ std::int32_t e0, i, j, nx, n, ix, hx;
+ std::uint32_t low;
+
+ GET_HIGH_WORD(hx, x); /* high word of x */
+ ix = hx & 0x7fffffff;
+ if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
+ {
+ y[0] = x;
+ y[1] = 0;
+ return 0;
+ }
+ if (ix < 0x4002d97c)
+ { /* |x| < 3pi/4, special case with n=+-1 */
+ if (hx > 0)
+ {
+ z = x - pio2_1;
+ if (ix != 0x3ff921fb)
+ { /* 33+53 bit pi is good enough */
+ y[0] = z - pio2_1t;
+ y[1] = (z - y[0]) - pio2_1t;
+ }
+ else
+ { /* near pi/2, use 33+33+53 bit pi */
+ z -= pio2_2;
+ y[0] = z - pio2_2t;
+ y[1] = (z - y[0]) - pio2_2t;
+ }
+ return 1;
+ }
+ else
+ { /* negative x */
+ z = x + pio2_1;
+ if (ix != 0x3ff921fb)
+ { /* 33+53 bit pi is good enough */
+ y[0] = z + pio2_1t;
+ y[1] = (z - y[0]) + pio2_1t;
+ }
+ else
+ { /* near pi/2, use 33+33+53 bit pi */
+ z += pio2_2;
+ y[0] = z + pio2_2t;
+ y[1] = (z - y[0]) + pio2_2t;
+ }
+
+ return -1;
+ }
+ }
+ if (ix <= 0x413921fb)
+ { /* |x| ~<= 2^19*(pi/2), medium_ size */
+ t = std::fabs(x);
+ n = (std::int32_t)(t * invpio2 + half);
+ fn = (double)n;
+ r = t - fn * pio2_1;
+ w = fn * pio2_1t; /* 1st round good to 85 bit */
+ if ((n < 32) && (n > 0) && (ix != npio2_hw[n - 1]))
+ {
+ y[0] = r - w; /* quick check no cancellation */
+ }
+ else
+ {
+ std::uint32_t high;
+ j = ix >> 20;
+ y[0] = r - w;
+ GET_HIGH_WORD(high, y[0]);
+ i = j - static_cast<int32_t>((high >> 20) & 0x7ff);
+ if (i > 16)
+ { /* 2nd iteration needed, good to 118 */
+ t = r;
+ w = fn * pio2_2;
+ r = t - w;
+ w = fn * pio2_2t - ((t - r) - w);
+ y[0] = r - w;
+ GET_HIGH_WORD(high, y[0]);
+ i = j - static_cast<int32_t>((high >> 20) & 0x7ff);
+ if (i > 49)
+ { /* 3rd iteration need, 151 bits acc */
+ t = r; /* will cover all possible cases */
+ w = fn * pio2_3;
+ r = t - w;
+ w = fn * pio2_3t - ((t - r) - w);
+ y[0] = r - w;
+ }
+ }
+ }
+ y[1] = (r - y[0]) - w;
+ if (hx < 0)
+ {
+ y[0] = -y[0];
+ y[1] = -y[1];
+ return -n;
+ }
+ else
+ return n;
+ }
+ /*
+ * all other (large) arguments
+ */
+ if (ix >= 0x7ff00000)
+ { /* x is inf or NaN */
+ y[0] = y[1] = x - x;
+ return 0;
+ }
+ /* set z = scalbn(|x|,ilogb(x)-23) */
+ GET_LOW_WORD(low, x);
+ SET_LOW_WORD(z, low);
+ e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
+ SET_HIGH_WORD(z, static_cast<uint32_t>(ix - (e0 << 20)));
+ for (i = 0; i < 2; i++)
+ {
+ tx[i] = (double)((std::int32_t)(z));
+ z = (z - tx[i]) * two24;
+ }
+ tx[2] = z;
+ nx = 3;
+ while (tx[nx - 1] == zero)
+ nx--; /* skip zero term */
+ n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
+ if (hx < 0)
+ {
+ y[0] = -y[0];
+ y[1] = -y[1];
+ return -n;
+ }
+ return n;
+ }
+ }
+
+#undef XSIMD_LITTLE_ENDIAN
+#undef SET_LOW_WORD
+#undef SET_HIGH_WORD
+#undef GET_LOW_WORD
+#undef GET_HIGH_WORD
+#undef HIGH_WORD_IDX
+#undef LOW_WORD_IDX
+#undef ONCE0
+}
generated by cgit v1.2.3 (git 2.39.1) at 2024-06-09 14:24:51 +0000