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 | 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 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 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 74209x 5x 5x 74204x 74204x 74204x 74204x 74204x 74204x 74209x 24722x 160x 160x 24722x 116x 116x 24722x 107x 107x 24722x 500x 500x 24722x 500x 500x 24722x 4117x 4117x 24722x 4098x 4098x 24722x 4111x 4111x 4111x 24722x 414x 414x 24722x 60081x 60081x 60081x 60081x 60081x 60081x 74209x 11141x 117x 117x 11141x 121x 121x 10903x 11141x 49x 11141x 49x 49x 11141x 303x 103x 103x 103x 303x 100x 100x 100x 100x 11141x 59491x 74209x 365x 74209x 100x 100x 100x 59391x 59391x 59391x 59391x 59391x 59391x 59391x 59391x 59391x 59391x 59391x 74209x 367x 74209x 59024x 59024x 59391x 59391x 59391x 74209x 1169x 1169x 4x 4x 1165x 1165x 1169x 331x 331x 167x 167x 167x 164x 164x 164x 1169x 334x 334x 171x 171x 171x 163x 163x 163x 500x 500x 500x 58222x 58222x 58222x 58222x 58722x 58722x 58722x 58722x 58722x 58722x 58722x 58722x 58722x 58722x 58722x 58722x 74209x 666x 666x 663x 663x 663x 3x 3x 3x 3x 666x 58056x 58056x 14131x 14131x 14129x 14129x 14129x 2x 2x 2x 2x 14131x 43925x 43925x 43925x 43925x 74209x 1x 1x 1x 1x 1x | /**
* @license Apache-2.0
*
* Copyright (c) 2018 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/s_pow.c}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright (C) 2004 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.
* ```
*/
'use strict';
// MODULES //
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var isOdd = require( '@stdlib/math/base/assert/is-odd' );
var isInfinite = require( '@stdlib/math/base/assert/is-infinite' );
var isInteger = require( '@stdlib/math/base/assert/is-integer' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var abs = require( '@stdlib/math/base/special/abs' );
var toWords = require( '@stdlib/number/float64/base/to-words' );
var setLowWord = require( '@stdlib/number/float64/base/set-low-word' );
var uint32ToInt32 = require( '@stdlib/number/uint32/base/to-int32' );
var NINF = require( '@stdlib/constants/float64/ninf' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var ABS_MASK = require( '@stdlib/constants/float64/high-word-abs-mask' );
var xIsZero = require( './x_is_zero.js' );
var yIsHuge = require( './y_is_huge.js' );
var yIsInfinite = require( './y_is_infinite.js' );
var log2ax = require( './log2ax.js' );
var logx = require( './logx.js' );
var pow2 = require( './pow2.js' );
// VARIABLES //
// 0x3fefffff = 1072693247 => 0 01111111110 11111111111111111111 => biased exponent: 1022 = -1+1023 => 2^-1
var HIGH_MAX_NEAR_UNITY = 0x3fefffff|0; // asm type annotation
// 0x41e00000 = 1105199104 => 0 10000011110 00000000000000000000 => biased exponent: 1054 = 31+1023 => 2^31
var HIGH_BIASED_EXP_31 = 0x41e00000|0; // asm type annotation
// 0x43f00000 = 1139802112 => 0 10000111111 00000000000000000000 => biased exponent: 1087 = 64+1023 => 2^64
var HIGH_BIASED_EXP_64 = 0x43f00000|0; // asm type annotation
// 0x40900000 = 1083179008 => 0 10000001001 00000000000000000000 => biased exponent: 1033 = 10+1023 => 2^10 = 1024
var HIGH_BIASED_EXP_10 = 0x40900000|0; // asm type annotation
// 0x3ff00000 = 1072693248 => 0 01111111111 00000000000000000000 => biased exponent: 1023 = 0+1023 => 2^0 = 1
var HIGH_BIASED_EXP_0 = 0x3ff00000|0; // asm type annotation
// 0x4090cc00 = 1083231232 => 0 10000001001 00001100110000000000
var HIGH_1075 = 0x4090cc00|0; // asm type annotation
// 0xc090cc00 = 3230714880 => 1 10000001001 00001100110000000000
var HIGH_NEG_1075 = 0xc090cc00>>>0; // asm type annotation
var HIGH_NUM_NONSIGN_BITS = 31|0; // asm type annotation
var HUGE = 1.0e300;
var TINY = 1.0e-300;
// -(1024-log2(ovfl+.5ulp))
var OVT = 8.0085662595372944372e-17;
// High/low words workspace:
var WORDS = [ 0|0, 0|0 ];
// Log workspace:
var LOG_WORKSPACE = [ 0.0, 0.0 ];
// MAIN //
/**
* Evaluates the exponential function.
*
* ## Method
*
* 1. Let \\(x = 2^n (1+f)\\).
*
* 2. Compute \\(\operatorname{log2}(x)\\) as
*
* ```tex
* \operatorname{log2}(x) = w_1 + w_2
* ```
*
* where \\(w_1\\) has \\(53 - 24 = 29\\) bit trailing zeros.
*
* 3. Compute
*
* ```tex
* y \cdot \operatorname{log2}(x) = n + y^\prime
* ```
*
* by simulating multi-precision arithmetic, where \\(|y^\prime| \leq 0.5\\).
*
* 4. Return
*
* ```tex
* x^y = 2^n e^{y^\prime \cdot \mathrm{log2}}
* ```
*
* ## Special Cases
*
* ```tex
* \begin{align*}
* x^{\mathrm{NaN}} &= \mathrm{NaN} & \\
* (\mathrm{NaN})^y &= \mathrm{NaN} & \\
* 1^y &= 1 & \\
* x^0 &= 1 & \\
* x^1 &= x & \\
* (\pm 0)^\infty &= +0 & \\
* (\pm 0)^{-\infty} &= +\infty & \\
* (+0)^y &= +0 & \mathrm{if}\ y > 0 \\
* (+0)^y &= +\infty & \mathrm{if}\ y < 0 \\
* (-0)^y &= -\infty & \mathrm{if}\ y\ \mathrm{is\ an\ odd\ integer\ and}\ y < 0 \\
* (-0)^y &= +\infty & \mathrm{if}\ y\ \mathrm{is\ not\ an\ odd\ integer\ and}\ y < 0 \\
* (-0)^y &= -0 & \mathrm{if}\ y\ \mathrm{is\ an\ odd\ integer\ and}\ y > 0 \\
* (-0)^y &= +0 & \mathrm{if}\ y\ \mathrm{is\ not\ an\ odd\ integer\ and}\ y > 0 \\
* (-1)^{\pm\infty} &= \mathrm{NaN} & \\
* x^{\infty} &= +\infty & |x| > 1 \\
* x^{\infty} &= +0 & |x| < 1 \\
* x^{-\infty} &= +0 & |x| > 1 \\
* x^{-\infty} &= +\infty & |x| < 1 \\
* (-\infty)^y &= (-0)^y & \\
* \infty^y &= +0 & y < 0 \\
* \infty^y &= +\infty & y > 0 \\
* x^y &= \mathrm{NaN} & \mathrm{if}\ y\ \mathrm{is\ not\ a\ finite\ integer\ and}\ x < 0
* \end{align*}
* ```
*
* ## Notes
*
* - \\(\operatorname{pow}(x,y)\\) returns \\(x^y\\) nearly rounded. In particular, \\(\operatorname{pow}(<\mathrm{integer}>,<\mathrm{integer}>)\\) **always** returns the correct integer, provided the value is representable.
* - The hexadecimal values shown in the source code are the intended values for used constants. Decimal values may be used, provided the compiler will accurately convert decimal to binary in order to produce the hexadecimal values.
*
* @param {number} x - base
* @param {number} y - exponent
* @returns {number} function value
*
* @example
* var v = pow( 2.0, 3.0 );
* // returns 8.0
*
* @example
* var v = pow( 4.0, 0.5 );
* // returns 2.0
*
* @example
* var v = pow( 100.0, 0.0 );
* // returns 1.0
*
* @example
* var v = pow( 3.141592653589793, 5.0 );
* // returns ~306.0197
*
* @example
* var v = pow( 3.141592653589793, -0.2 );
* // returns ~0.7954
*
* @example
* var v = pow( NaN, 3.0 );
* // returns NaN
*
* @example
* var v = pow( 5.0, NaN );
* // returns NaN
*
* @example
* var v = pow( NaN, NaN );
* // returns NaN
*/
function pow( x, y ) {
var ahx; // absolute value high word `x`
var ahy; // absolute value high word `y`
var ax; // absolute value `x`
var hx; // high word `x`
var lx; // low word `x`
var hy; // high word `y`
var ly; // low word `y`
var sx; // sign `x`
var sy; // sign `y`
var y1;
var hp;
var lp;
var t;
var z; // y prime
var j;
var i;
if ( isnan( x ) || isnan( y ) ) {
return NaN;
}
// Split `y` into high and low words:
toWords.assign( y, WORDS, 1, 0 );
hy = WORDS[ 0 ];
ly = WORDS[ 1 ];
// Special cases `y`...
if ( ly === 0 ) {
if ( y === 0.0 ) {
return 1.0;
}
if ( y === 1.0 ) {
return x;
}
if ( y === -1.0 ) {
return 1.0 / x;
}
if ( y === 0.5 ) {
return sqrt( x );
}
if ( y === -0.5 ) {
return 1.0 / sqrt( x );
}
if ( y === 2.0 ) {
return x * x;
}
if ( y === 3.0 ) {
return x * x * x;
}
if ( y === 4.0 ) {
x *= x;
return x * x;
}
if ( isInfinite( y ) ) {
return yIsInfinite( x, y );
}
}
// Split `x` into high and low words:
toWords.assign( x, WORDS, 1, 0 );
hx = WORDS[ 0 ];
lx = WORDS[ 1 ];
// Special cases `x`...
if ( lx === 0 ) {
if ( hx === 0 ) {
return xIsZero( x, y );
}
if ( x === 1.0 ) {
return 1.0;
}
if (
x === -1.0 &&
isOdd( y )
) {
return -1.0;
}
if ( isInfinite( x ) ) {
if ( x === NINF ) {
// `pow( 1/x, -y )`
return pow( -0.0, -y );
}
if ( y < 0.0 ) {
return 0.0;
}
return PINF;
}
}
if (
x < 0.0 &&
isInteger( y ) === false
) {
// Signal NaN...
return (x-x)/(x-x);
}
ax = abs( x );
// Remove the sign bits (i.e., get absolute values):
ahx = (hx & ABS_MASK)|0; // asm type annotation
ahy = (hy & ABS_MASK)|0; // asm type annotation
// Extract the sign bits:
sx = (hx >>> HIGH_NUM_NONSIGN_BITS)|0; // asm type annotation
sy = (hy >>> HIGH_NUM_NONSIGN_BITS)|0; // asm type annotation
// Determine the sign of the result...
if ( sx && isOdd( y ) ) {
sx = -1.0;
} else {
sx = 1.0;
}
// Case 1: `|y|` is huge...
// |y| > 2^31
if ( ahy > HIGH_BIASED_EXP_31 ) {
// `|y| > 2^64`, then must over- or underflow...
if ( ahy > HIGH_BIASED_EXP_64 ) {
return yIsHuge( x, y );
}
// Over- or underflow if `x` is not close to unity...
if ( ahx < HIGH_MAX_NEAR_UNITY ) {
// y < 0
if ( sy === 1 ) {
// Signal overflow...
return sx * HUGE * HUGE;
}
// Signal underflow...
return sx * TINY * TINY;
}
if ( ahx > HIGH_BIASED_EXP_0 ) {
// y > 0
if ( sy === 0 ) {
// Signal overflow...
return sx * HUGE * HUGE;
}
// Signal underflow...
return sx * TINY * TINY;
}
// At this point, `|1-x|` is tiny (`<= 2^-20`). Suffice to compute `log(x)` by `x - x^2/2 + x^3/3 - x^4/4`.
t = logx( LOG_WORKSPACE, ax );
}
// Case 2: `|y|` is not huge...
else {
t = log2ax( LOG_WORKSPACE, ax, ahx );
}
// Split `y` into `y1 + y2` and compute `(y1+y2) * (t1+t2)`...
y1 = setLowWord( y, 0 );
lp = ( (y-y1)*t[0] ) + ( y*t[1] );
hp = y1 * t[0];
z = lp + hp;
// Note: *can* be more performant to use `getHighWord` and `getLowWord` directly, but using `toWords` looks cleaner.
toWords.assign( z, WORDS, 1, 0 );
j = uint32ToInt32( WORDS[0] );
i = uint32ToInt32( WORDS[1] );
// z >= 1024
if ( j >= HIGH_BIASED_EXP_10 ) {
// z > 1024
if ( ((j-HIGH_BIASED_EXP_10)|i) !== 0 ) {
// Signal overflow...
return sx * HUGE * HUGE;
}
if ( (lp+OVT) > (z-hp) ) {
// Signal overflow...
return sx * HUGE * HUGE;
}
}
// z <= -1075
else if ( (j&ABS_MASK) >= HIGH_1075 ) {
// z < -1075
if ( ((j-HIGH_NEG_1075)|i) !== 0 ) {
// Signal underflow...
return sx * TINY * TINY;
}
if ( lp <= (z-hp) ) {
// Signal underflow...
return sx * TINY * TINY;
}
}
// Compute `2^(hp+lp)`...
z = pow2( j, hp, lp );
return sx * z;
}
// EXPORTS //
module.exports = pow;
|