| 1 | /* |
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| 2 | * ==================================================== |
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| 3 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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| 4 | * |
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| 5 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
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| 6 | * Permission to use, copy, modify, and distribute this |
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| 7 | * software is freely granted, provided that this notice |
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| 8 | * is preserved. |
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| 9 | * ==================================================== |
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| 10 | */ |
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| 11 | |
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| 12 | /* |
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| 13 | * Modified for ALMOS-MKH OS at UPMC, France, August 2018. (Alain Greiner) |
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| 14 | */ |
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| 15 | |
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| 16 | /* |
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| 17 | * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) |
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| 18 | * double x[],y[]; int e0,nx,prec; int ipio2[]; |
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| 19 | * |
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| 20 | * __kernel_rem_pio2 return the last three digits of N with |
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| 21 | * y = x - N*pi/2 |
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| 22 | * so that |y| < pi/2. |
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| 23 | * |
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| 24 | * The method is to compute the integer (mod 8) and fraction parts of |
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| 25 | * (2/pi)*x without doing the full multiplication. In general we |
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| 26 | * skip the part of the product that are known to be a huge integer ( |
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| 27 | * more accurately, = 0 mod 8 ). Thus the number of operations are |
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| 28 | * independent of the exponent of the input. |
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| 29 | * |
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| 30 | * (2/pi) is represented by an array of 24-bit integers in ipio2[]. |
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| 31 | * |
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| 32 | * Input parameters: |
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| 33 | * x[] The input value (must be positive) is broken into nx |
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| 34 | * pieces of 24-bit integers in double precision format. |
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| 35 | * x[i] will be the i-th 24 bit of x. The scaled exponent |
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| 36 | * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 |
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| 37 | * match x's up to 24 bits. |
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| 38 | * |
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| 39 | * Example of breaking a double positive z into x[0]+x[1]+x[2]: |
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| 40 | * e0 = ilogb(z)-23 |
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| 41 | * z = scalbn(z,-e0) |
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| 42 | * for i = 0,1,2 |
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| 43 | * x[i] = floor(z) |
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| 44 | * z = (z-x[i])*2**24 |
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| 45 | * |
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| 46 | * |
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| 47 | * y[] ouput result in an array of double precision numbers. |
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| 48 | * The dimension of y[] is: |
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| 49 | * 24-bit precision 1 |
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| 50 | * 53-bit precision 2 |
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| 51 | * 64-bit precision 2 |
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| 52 | * 113-bit precision 3 |
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| 53 | * The actual value is the sum of them. Thus for 113-bit |
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| 54 | * precison, one may have to do something like: |
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| 55 | * |
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| 56 | * long double t,w,r_head, r_tail; |
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| 57 | * t = (long double)y[2] + (long double)y[1]; |
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| 58 | * w = (long double)y[0]; |
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| 59 | * r_head = t+w; |
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| 60 | * r_tail = w - (r_head - t); |
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| 61 | * |
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| 62 | * e0 The exponent of x[0] |
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| 63 | * |
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| 64 | * nx dimension of x[] |
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| 65 | * |
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| 66 | * prec an integer indicating the precision: |
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| 67 | * 0 24 bits (single) |
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| 68 | * 1 53 bits (double) |
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| 69 | * 2 64 bits (extended) |
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| 70 | * 3 113 bits (quad) |
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| 71 | * |
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| 72 | * ipio2[] |
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| 73 | * integer array, contains the (24*i)-th to (24*i+23)-th |
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| 74 | * bit of 2/pi after binary point. The corresponding |
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| 75 | * floating value is |
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| 76 | * |
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| 77 | * ipio2[i] * 2^(-24(i+1)). |
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| 78 | * |
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| 79 | * External function: |
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| 80 | * double scalbn(), floor(); |
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| 81 | * |
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| 82 | * |
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| 83 | * Here is the description of some local variables: |
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| 84 | * |
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| 85 | * jk jk+1 is the initial number of terms of ipio2[] needed |
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| 86 | * in the computation. The recommended value is 2,3,4, |
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| 87 | * 6 for single, double, extended,and quad. |
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| 88 | * |
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| 89 | * jz local integer variable indicating the number of |
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| 90 | * terms of ipio2[] used. |
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| 91 | * |
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| 92 | * jx nx - 1 |
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| 93 | * |
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| 94 | * jv index for pointing to the suitable ipio2[] for the |
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| 95 | * computation. In general, we want |
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| 96 | * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 |
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| 97 | * is an integer. Thus |
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| 98 | * e0-3-24*jv >= 0 or (e0-3)/24 >= jv |
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| 99 | * Hence jv = max(0,(e0-3)/24). |
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| 100 | * |
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| 101 | * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. |
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| 102 | * |
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| 103 | * q[] double array with integral value, representing the |
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| 104 | * 24-bits chunk of the product of x and 2/pi. |
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| 105 | * |
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| 106 | * q0 the corresponding exponent of q[0]. Note that the |
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| 107 | * exponent for q[i] would be q0-24*i. |
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| 108 | * |
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| 109 | * PIo2[] double precision array, obtained by cutting pi/2 |
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| 110 | * into 24 bits chunks. |
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| 111 | * |
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| 112 | * f[] ipio2[] in floating point |
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| 113 | * |
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| 114 | * iq[] integer array by breaking up q[] in 24-bits chunk. |
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| 115 | * |
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| 116 | * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] |
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| 117 | * |
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| 118 | * ih integer. If >0 it indicates q[] is >= 0.5, hence |
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| 119 | * it also indicates the *sign* of the result. |
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| 120 | * |
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| 121 | */ |
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| 122 | |
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| 123 | |
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| 124 | /* |
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| 125 | * Constants: |
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| 126 | * The hexadecimal values are the intended ones for the following |
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| 127 | * constants. The decimal values may be used, provided that the |
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| 128 | * compiler will convert from decimal to binary accurately enough |
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| 129 | * to produce the hexadecimal values shown. |
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| 130 | */ |
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| 131 | |
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| 132 | #include "math.h" |
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| 133 | #include "math_private.h" |
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| 134 | |
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| 135 | static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ |
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| 136 | |
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| 137 | static const double PIo2[] = { |
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| 138 | 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ |
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| 139 | 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ |
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| 140 | 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ |
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| 141 | 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ |
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| 142 | 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ |
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| 143 | 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ |
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| 144 | 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ |
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| 145 | 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ |
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| 146 | }; |
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| 147 | |
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| 148 | static const double |
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| 149 | zero = 0.0, |
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| 150 | one = 1.0, |
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| 151 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
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| 152 | twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ |
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| 153 | |
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| 154 | int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) |
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| 155 | { |
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| 156 | int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; |
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| 157 | double z,fw,f[20],fq[20],q[20]; |
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| 158 | |
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| 159 | /* initialize jk*/ |
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| 160 | jk = init_jk[prec]; |
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| 161 | jp = jk; |
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| 162 | |
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| 163 | /* determine jx,jv,q0, note that 3>q0 */ |
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| 164 | jx = nx-1; |
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| 165 | jv = (e0-3)/24; if(jv<0) jv=0; |
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| 166 | q0 = e0-24*(jv+1); |
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| 167 | |
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| 168 | /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ |
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| 169 | j = jv-jx; m = jx+jk; |
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| 170 | for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; |
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| 171 | |
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| 172 | /* compute q[0],q[1],...q[jk] */ |
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| 173 | for (i=0;i<=jk;i++) { |
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| 174 | for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; |
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| 175 | } |
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| 176 | |
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| 177 | jz = jk; |
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| 178 | recompute: |
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| 179 | /* distill q[] into iq[] reversingly */ |
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| 180 | for(i=0,j=jz,z=q[jz];j>0;i++,j--) { |
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| 181 | fw = (double)((int32_t)(twon24* z)); |
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| 182 | iq[i] = (int32_t)(z-two24*fw); |
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| 183 | z = q[j-1]+fw; |
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| 184 | } |
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| 185 | |
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| 186 | /* compute n */ |
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| 187 | z = scalbn(z,q0); /* actual value of z */ |
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| 188 | z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ |
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| 189 | n = (int32_t) z; |
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| 190 | z -= (double)n; |
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| 191 | ih = 0; |
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| 192 | if(q0>0) { /* need iq[jz-1] to determine n */ |
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| 193 | i = (iq[jz-1]>>(24-q0)); n += i; |
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| 194 | iq[jz-1] -= i<<(24-q0); |
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| 195 | ih = iq[jz-1]>>(23-q0); |
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| 196 | } |
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| 197 | else if(q0==0) ih = iq[jz-1]>>23; |
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| 198 | else if(z>=0.5) ih=2; |
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| 199 | |
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| 200 | if(ih>0) { /* q > 0.5 */ |
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| 201 | n += 1; carry = 0; |
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| 202 | for(i=0;i<jz ;i++) { /* compute 1-q */ |
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| 203 | j = iq[i]; |
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| 204 | if(carry==0) { |
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| 205 | if(j!=0) { |
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| 206 | carry = 1; iq[i] = 0x1000000- j; |
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| 207 | } |
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| 208 | } else iq[i] = 0xffffff - j; |
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| 209 | } |
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| 210 | if(q0>0) { /* rare case: chance is 1 in 12 */ |
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| 211 | switch(q0) { |
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| 212 | case 1: |
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| 213 | iq[jz-1] &= 0x7fffff; break; |
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| 214 | case 2: |
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| 215 | iq[jz-1] &= 0x3fffff; break; |
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| 216 | } |
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| 217 | } |
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| 218 | if(ih==2) { |
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| 219 | z = one - z; |
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| 220 | if(carry!=0) z -= scalbn(one,q0); |
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| 221 | } |
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| 222 | } |
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| 223 | |
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| 224 | /* check if recomputation is needed */ |
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| 225 | if(z==zero) { |
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| 226 | j = 0; |
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| 227 | for (i=jz-1;i>=jk;i--) j |= iq[i]; |
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| 228 | if(j==0) { /* need recomputation */ |
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| 229 | for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ |
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| 230 | |
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| 231 | for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ |
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| 232 | f[jx+i] = (double) ipio2[jv+i]; |
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| 233 | for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; |
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| 234 | q[i] = fw; |
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| 235 | } |
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| 236 | jz += k; |
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| 237 | goto recompute; |
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| 238 | } |
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| 239 | } |
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| 240 | |
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| 241 | /* chop off zero terms */ |
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| 242 | if(z==0.0) { |
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| 243 | jz -= 1; q0 -= 24; |
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| 244 | while(iq[jz]==0) { jz--; q0-=24;} |
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| 245 | } else { /* break z into 24-bit if necessary */ |
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| 246 | z = scalbn(z,-q0); |
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| 247 | if(z>=two24) { |
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| 248 | fw = (double)((int32_t)(twon24*z)); |
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| 249 | iq[jz] = (int32_t)(z-two24*fw); |
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| 250 | jz += 1; q0 += 24; |
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| 251 | iq[jz] = (int32_t) fw; |
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| 252 | } else iq[jz] = (int32_t) z ; |
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| 253 | } |
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| 254 | |
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| 255 | /* convert integer "bit" chunk to floating-point value */ |
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| 256 | fw = scalbn(one,q0); |
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| 257 | for(i=jz;i>=0;i--) { |
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| 258 | q[i] = fw*(double)iq[i]; fw*=twon24; |
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| 259 | } |
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| 260 | |
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| 261 | /* compute PIo2[0,...,jp]*q[jz,...,0] */ |
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| 262 | for(i=jz;i>=0;i--) { |
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| 263 | for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; |
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| 264 | fq[jz-i] = fw; |
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| 265 | } |
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| 266 | |
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| 267 | /* compress fq[] into y[] */ |
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| 268 | switch(prec) { |
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| 269 | case 0: |
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| 270 | fw = 0.0; |
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| 271 | for (i=jz;i>=0;i--) fw += fq[i]; |
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| 272 | y[0] = (ih==0)? fw: -fw; |
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| 273 | break; |
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| 274 | case 1: |
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| 275 | case 2: |
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| 276 | fw = 0.0; |
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| 277 | for (i=jz;i>=0;i--) fw += fq[i]; |
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| 278 | y[0] = (ih==0)? fw: -fw; |
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| 279 | fw = fq[0]-fw; |
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| 280 | for (i=1;i<=jz;i++) fw += fq[i]; |
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| 281 | y[1] = (ih==0)? fw: -fw; |
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| 282 | break; |
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| 283 | case 3: /* painful */ |
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| 284 | for (i=jz;i>0;i--) { |
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| 285 | fw = fq[i-1]+fq[i]; |
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| 286 | fq[i] += fq[i-1]-fw; |
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| 287 | fq[i-1] = fw; |
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| 288 | } |
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| 289 | for (i=jz;i>1;i--) { |
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| 290 | fw = fq[i-1]+fq[i]; |
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| 291 | fq[i] += fq[i-1]-fw; |
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| 292 | fq[i-1] = fw; |
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| 293 | } |
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| 294 | for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; |
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| 295 | if(ih==0) { |
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| 296 | y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; |
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| 297 | } else { |
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| 298 | y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; |
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| 299 | } |
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| 300 | } |
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| 301 | return n&7; |
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| 302 | } |
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