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* @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 getHighWord = require( '@stdlib/number/float64/base/get-high-word' );
var setLowWord = require( '@stdlib/number/float64/base/set-low-word' );
var setHighWord = require( '@stdlib/number/float64/base/set-high-word' );
var BIAS = require( '@stdlib/constants/float64/exponent-bias' );
var HIGH_NUM_SIGNIFICAND_BITS = require( '@stdlib/constants/float64/num-high-word-significand-bits' );
var polyvalL = require( './polyval_l.js' );
// VARIABLES //
// 0x000fffff = 1048575 => 0 00000000000 11111111111111111111
var HIGH_SIGNIFICAND_MASK = 0x000fffff|0; // asm type annotation
// 0x00100000 = 1048576 => 0 00000000001 00000000000000000000 => biased exponent: 1 = -1022+1023 => 2^-1022
var HIGH_MIN_NORMAL_EXP = 0x00100000|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
// 0x20000000 = 536870912 => 0 01000000000 00000000000000000000 => biased exponent: 512 = -511+1023
var HIGH_BIASED_EXP_NEG_512 = 0x20000000|0; // asm type annotation
// 0x00080000 = 524288 => 0 00000000000 10000000000000000000
var HIGH_SIGNIFICAND_HALF = 0x00080000|0; // asm type annotation
var TWO53 = 9007199254740992.0; // 0x43400000, 0x00000000
// 2/(3*LN2)
var CP = 9.61796693925975554329e-01; // 0x3FEEC709, 0xDC3A03FD
// (float)CP
var CP_HI = 9.61796700954437255859e-01; // 0x3FEEC709, 0xE0000000
// Low: CP_HI
var CP_LO = -7.02846165095275826516e-09; // 0xBE3E2FE0, 0x145B01F5
var BP = [
1.0,
1.5
];
var DP_HI = [
0.0,
5.84962487220764160156e-01 // 0x3FE2B803, 0x40000000
];
var DP_LO = [
0.0,
1.35003920212974897128e-08 // 0x3E4CFDEB, 0x43CFD006
];
// MAIN //
/**
* Computes \\(\operatorname{log2}(ax)\\).
*
* @private
* @param {Array} out - output array
* @param {number} ax - absolute value of `x`
* @param {number} ahx - high word of `ax`
* @returns {Array} output array containing a tuple comprised of high and low parts
*
* @example
* var t = log2ax( [ 0.0, 0.0 ], 9.0, 1075970048 ); // => [ t1, t2 ]
* // returns [ 3.169923782348633, 0.0000012190936795504075 ]
*/
function log2ax( out, ax, ahx ) {
var tmp;
var ss; // `hs + ls`
var s2; // `ss` squared
var hs;
var ls;
var ht;
var lt;
var bp; // `BP` constant
var dp; // `DP` constant
var hp;
var lp;
var hz;
var lz;
var t1;
var t2;
var t;
var r;
var u;
var v;
var n;
var j;
var k;
n = 0|0; // asm type annotation
// Check if `x` is subnormal...
if ( ahx < HIGH_MIN_NORMAL_EXP ) {
ax *= TWO53;
n -= 53|0; // asm type annotation
ahx = getHighWord( ax );
}
// Extract the unbiased exponent of `x`:
n += ((ahx >> HIGH_NUM_SIGNIFICAND_BITS) - BIAS)|0; // asm type annotation
// Isolate the significand bits of `x`:
j = (ahx & HIGH_SIGNIFICAND_MASK)|0; // asm type annotation
// Normalize `ahx` by setting the (biased) exponent to `1023`:
ahx = (j | HIGH_BIASED_EXP_0)|0; // asm type annotation
// Determine the interval of `|x|` by comparing significand bits...
// |x| < sqrt(3/2)
if ( j <= 0x3988E ) { // 0 00000000000 00111001100010001110
k = 0;
}
// |x| < sqrt(3)
else if ( j < 0xBB67A ) { // 0 00000000000 10111011011001111010
k = 1;
}
// |x| >= sqrt(3)
else {
k = 0;
n += 1|0; // asm type annotation
ahx -= HIGH_MIN_NORMAL_EXP;
}
// Load the normalized high word into `|x|`:
ax = setHighWord( ax, ahx );
// Compute `ss = hs + ls = (x-1)/(x+1)` or `(x-1.5)/(x+1.5)`:
bp = BP[ k ]; // BP[0] = 1.0, BP[1] = 1.5
u = ax - bp; // (x-1) || (x-1.5)
v = 1.0 / (ax + bp); // 1/(x+1) || 1/(x+1.5)
ss = u * v;
hs = setLowWord( ss, 0 ); // set all low word (less significant significand) bits to 0s
// Compute `ht = ax + bp` (via manipulation, i.e., bit flipping, of the high word):
tmp = ((ahx>>1) | HIGH_BIASED_EXP_NEG_512) + HIGH_SIGNIFICAND_HALF;
tmp += (k << 18); // `(k<<18)` can be considered the word equivalent of `1.0` or `1.5`
ht = setHighWord( 0.0, tmp );
lt = ax - (ht - bp);
ls = v * ( ( u - (hs*ht) ) - ( hs*lt ) );
// Compute `log(ax)`...
s2 = ss * ss;
r = s2 * s2 * polyvalL( s2 );
r += ls * (hs + ss);
s2 = hs * hs;
ht = 3.0 + s2 + r;
ht = setLowWord( ht, 0 );
lt = r - ((ht-3.0) - s2);
// u+v = ss*(1+...):
u = hs * ht;
v = ( ls*ht ) + ( lt*ss );
// 2/(3LN2) * (ss+...):
hp = u + v;
hp = setLowWord( hp, 0 );
lp = v - (hp - u);
hz = CP_HI * hp; // CP_HI+CP_LO = 2/(3*LN2)
lz = ( CP_LO*hp ) + ( lp*CP ) + DP_LO[ k ];
// log2(ax) = (ss+...)*2/(3*LN2) = n + dp + hz + lz
dp = DP_HI[ k ];
t = n;
t1 = ((hz+lz) + dp) + t; // log2(ax)
t1 = setLowWord( t1, 0 );
t2 = lz - (((t1-t) - dp) - hz);
out[ 0 ] = t1;
out[ 1 ] = t2;
return out;
}
// EXPORTS //
module.exports = log2ax;
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