Press n or j to go to the next uncovered block, b, p or k for the previous block.
| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 | 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 63736x 63736x 63736x 63736x 63736x 20934x 20934x 20934x 20934x 20934x 20934x 20934x 1440x 1440x 1440x 1440x 1440x 1440x 19494x 240x 240x 19254x 9544x 9544x 20934x 20934x 10372x 10372x 10372x 10372x 10372x 31566x 31566x 10372x 10372x 10372x 10372x 31566x 21194x 21194x 31566x 10372x 708x 708x 708x 244x 244x 708x 464x 464x 708x 708x 10372x 9544x 9544x 9544x 9544x 9544x 10372x 20934x 20934x 1868x 1868x 934x 934x 1868x 934x 934x 934x 934x 934x 934x 934x 934x 934x 934x 934x 934x 934x 934x 934x 934x 934x 1868x 20934x 19066x 19066x 19066x 19066x 19066x 19066x 19066x 20000x 20000x 20934x 80000x 80000x 80000x 20000x 20934x 80000x 80000x 200000x 200000x 80000x 80000x 20000x 20000x 20934x 80000x 80000x 20934x 10078x 20934x 9922x 9922x 20000x 20934x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 20000x 80000x 80000x 80000x 80000x 80000x 80000x 20000x 20000x 80000x 80000x 80000x 80000x 80000x 80000x 20000x 20000x 20000x 1x 1x 1x 1x 1x | /**
* @license Apache-2.0
*
* Copyright (c) 2025 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
*
* ## Notice
*
* The following copyright and license were part of the original implementation available as part of [FreeBSD]{@link https://svnweb.freebsd.org/base/release/9.3.0/lib/msun/src/k_rem_pio2.c}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* 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.
* ```
*/
/* eslint-disable array-element-newline */
'use strict';
// MODULES //
var floor = require( '@stdlib/math/base/special/floor' );
var ldexp = require( '@stdlib/math/base/special/ldexp' );
var zeros = require( '@stdlib/array/base/zeros' );
// VARIABLES //
/*
* Table of constants for `2/π` (`396` hex digits, `476` decimal).
*
* Integer array which contains the (`24*i`)-th to (`24*i+23`)-th bit of `2/π` after binary point. The corresponding floating value is
*
* ```tex
* \operatorname{ipio2}[i] \cdot 2^{-24(i+1)}
* ```
*
* This table must have at least `(e0-3)/24 + jk` terms. For single precision (`e0 <= 127`, `jk = 3`), this is `9`.
*/
var IPIO2 = [
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
0x95993C, 0x439041, 0xFE5163
];
// Double precision array, obtained by cutting `π/2` into `24` bits chunks...
var 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
];
var TWO24 = 1.67772160000000000000e+07; // 0x41700000, 0x00000000
var TWON24 = 5.96046447753906250000e-08; // 0x3E700000, 0x00000000
// Arrays for storing temporary values (note that, in C, this is not thread safe):
var F = zeros( 20 );
var Q = zeros( 20 );
var FQ = zeros( 20 );
var IQ = zeros( 20 );
// FUNCTIONS //
/**
* Performs the computation for `kernelRempio2f()`.
*
* @private
* @param {PositiveNumber} x - input value
* @param {(Array|TypedArray|Object)} y - output object for storing double precision numbers
* @param {integer} jz - number of terms of `ipio2[]` used
* @param {Array<integer>} q - array with integral values, representing the 24-bits chunk of the product of `x` and `2/π`
* @param {integer} q0 - the corresponding exponent of `q[0]` (the exponent for `q[i]` would be `q0-24*i`)
* @param {integer} jk - `jk+1` is the initial number of terms of `IPIO2[]` needed in the computation
* @param {integer} jv - index for pointing to the suitable `ipio2[]` for the computation
* @param {integer} jx - `nx - 1`
* @param {Array<number>} f - `IPIO2[]` in floating point
* @returns {number} last three binary digits of `N`
*/
function compute( x, y, jz, q, q0, jk, jv, jx, f ) {
var carry;
var fw;
var ih;
var jp;
var i;
var k;
var n;
var j;
var z;
// `jp+1` is the number of terms in `PIO2[]` needed:
jp = jk;
// Distill `q[]` into `IQ[]` in reverse order...
z = q[ jz ];
j = jz;
for ( i = 0; j > 0; i++ ) {
fw = ( TWON24 * z )|0;
IQ[ i ] = ( z - (TWO24*fw) )|0;
z = q[ j-1 ] + fw;
j -= 1;
}
// Compute `n`...
z = ldexp( z, q0 );
z -= 8.0 * floor( z*0.125 ); // Trim off integer >= 8
n = z|0;
z -= 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;
}
// Case: q > 0.5
if ( ih > 0 ) {
n += 1;
carry = 0;
// Compute `1-q`:
for ( i = 0; i < jz; i++ ) {
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 ) { // eslint-disable-line default-case
case 1:
IQ[ jz-1 ] &= 0x7fffff;
break;
case 2:
IQ[ jz-1 ] &= 0x3fffff;
break;
}
}
if ( ih === 2 ) {
z = 1.0 - z;
if ( carry !== 0 ) {
z -= ldexp( 1.0, q0 );
}
}
}
// Check if re-computation is needed...
if ( z === 0.0 ) {
j = 0;
for ( i = jz-1; i >= jk; i-- ) {
j |= IQ[ i ];
}
if ( j === 0 ) {
// Need re-computation...
for ( k = 1; IQ[ jk-k ] === 0; k++ ) {
// `k` is the number of terms needed...
}
for ( i = jz+1; i <= jz+k; i++ ) {
// Add `q[jz+1]` to `q[jz+k]`...
f[ jx+i ] = IPIO2[ jv+i ];
fw = 0.0;
for ( j = 0; j <= jx; j++ ) {
fw += x[ j ] * f[ jx + (i-j) ];
}
q[ i ] = fw;
}
jz += k;
return compute( x, y, jz, q, q0, jk, jv, jx, f );
}
// Chop off zero terms...
jz -= 1;
q0 -= 24;
while ( IQ[ jz ] === 0 ) {
jz -= 1;
q0 -= 24;
}
} else {
// Break `z` into 24-bit if necessary...
z = ldexp( z, -q0 );
if ( z >= TWO24 ) {
fw = (TWON24*z)|0;
IQ[ jz ] = ( z - (TWO24*fw) )|0;
jz += 1;
q0 += 24;
IQ[ jz ] = fw;
} else {
IQ[ jz ] = z|0;
}
}
// Convert integer "bit" chunk to floating-point value...
fw = ldexp( 1.0, q0 );
for ( i = jz; i >= 0; i-- ) {
q[ i ] = fw * IQ[i];
fw *= TWON24;
}
// Compute `PIO2[0,...,jp]*q[jz,...,0]`...
for ( i = jz; i >= 0; i-- ) {
fw = 0.0;
for ( k = 0; k <= jp && k <= jz-i; k++ ) {
fw += PIO2[ k ] * q[ i+k ];
}
FQ[ jz-i ] = fw;
}
// Compress `FQ[]` into `y[]`...
fw = 0.0;
for ( i = jz; i >= 0; i-- ) {
fw += FQ[ i ];
}
if ( ih === 0 ) {
y[ 0 ] = fw;
} else {
y[ 0 ] = -fw;
}
return ( n & 7 );
}
// MAIN //
/**
* Returns the last three binary digits of `N` with `y = x - Nπ/2` so that `|y| < π/2` (single precision).
*
* ## Method
*
* - The method is to compute the integer (mod 8) and fraction parts of (2/π) * 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/π) 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]`:
*
* ```tex
* e0 = \mathrm{ilogb}(z) - 23
* z = \mathrm{scalbn}(z, -e0)
* ```
*
* for `i = 0,1,2`
*
* ```tex
* x[i] = \lfloor z \rfloor
* z = (z - x[i]) \times 2^{24}
* ```
*
* - `y[]` output 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:
*
* ```tex
* \mathrm{long\ double} \: t, w, r_{\text{head}}, r_{\text{tail}}; \\
* t &= (\mathrm{long\ double}) y[2] + (\mathrm{long\ double}) y[1]; \\
* w &= (\mathrm{long\ double}) y[0]; \\
* r_{\text{head}} &= t + w; \\
* r_{\text{tail}} &= w - (r_{\text{head}} - t);
* ```
*
* - `e0` The exponent of `x[0]`. Must be <= 16360 or you need to expand the `ipio2` table.
*
* - `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)
*
* - 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 minimum and recommended value for `jk` is 3,4,4,6 for single, double, extended, and quad. `jk+1` must be 2 larger than you might expect so that our recomputation test works. (Up to 24 bits in the integer part (the 24 bits of it that we compute) and 23 bits in the fraction part may be lost to cancellation before we recompute.)
*
* - `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
*
* ```tex
* \frac{{2^{e0} \cdot x[0] \cdot \mathrm{ipio2}[jv-1] \cdot 2^{-24jv}}}{{8}}
* ```
*
* to be an integer. Thus
*
* ```tex
* e0 - 3 - 24 \cdot jv \geq 0 \quad \text{or} \quad \frac{{e0 - 3}}{{24}} \geq jv
* ```
*
* Hence
*
* ```tex
* jv = \max(0, \frac{{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/π`.
*
* - `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 `π/2` into 24 bits chunks.
*
* - `f[]` `ipso2[]` in floating point
*
* - `iq[]` integer array by breaking up `q[]` in 24-bits chunk.
*
* - `fq[]` final product of `x*(2/π)` in `fq[0],..,fq[jk]`
*
* - `ih` integer. If >0 it indicates `q[]` is >= 0.5, hence it also indicates the _sign_ of the result.
*
* - Constants:
*
* - The hexadecimal values are the intended ones for the following constants. The decimal values may be used, provided that the compiler will convert from decimal to binary accurately enough to produce the hexadecimal values shown.
*
* @private
* @param {PositiveNumber} x - input value
* @param {(Array|TypedArray|Object)} y - remainder element
* @param {PositiveInteger} e0 - the exponent of `x[0]` (must be <= 127)
* @param {PositiveInteger} nx - dimension of `x[]`
* @returns {number} last three binary digits of `N`
*/
function kernelRempio2f( x, y, e0, nx ) {
var fw;
var jk;
var jv;
var jx;
var jz;
var q0;
var i;
var j;
var m;
// Initialize `jk` for single-precision floating-point numbers:
jk = 3;
// Determine `jx`, `jv`, `q0` (note that `q0 < 3`):
jx = nx - 1;
jv = ( (e0 - 3) / 24 )|0;
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++ ) {
if ( j < 0 ) {
F[ i ] = 0.0;
} else {
F[ i ] = IPIO2[ j ];
}
j += 1;
}
// Compute `Q[0],Q[1],...,Q[jk]`:
for ( i = 0; i <= jk; i++ ) {
fw = 0.0;
for ( j = 0; j <= jx; j++ ) {
fw += x[ j ] * F[ jx + (i-j) ];
}
Q[ i ] = fw;
}
jz = jk;
return compute( x, y, jz, Q, q0, jk, jv, jx, F );
}
// EXPORTS //
module.exports = kernelRempio2f;
|