diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/README.md b/lib/node_modules/@stdlib/math/base/special/sincos/README.md new file mode 100644 index 000000000000..9099e029d250 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/README.md @@ -0,0 +1,75 @@ +# sincos + +> Simultaneously compute the [sine][@stdlib/math/base/special/sin] and [cosine][@stdlib/math/base/special/cos] of a number. + + +
+ +## Usage + +``` javascript +var sincos = require( '@stdlib/math/base/special/sincos' ); +``` + +#### sincos( \[out,\] x ) + +Simultaneously computes the [sine][@stdlib/math/base/special/sin] and [cosine][@stdlib/math/base/special/cos] of a `number` (in radians). + +``` javascript +var v = sincos( 0.0 ); +// returns [ ~0.0, ~1.0 ] + +v = sincos( Math.PI/2.0 ); +// returns [ ~0.0, ~1.0 ] + +v = sincos( -Math.PI/6.0 ); +// returns [ ~-0.5, ~0.866 ] +``` + +By default, the function returns the sine and cosine as a two-element `array`. To avoid extra memory allocation, the function supports providing an output (destination) object. + +``` javascript +var out = new Float64Array( 2 ); + +var v = sincos( out, 0.0 ); +// return [ ~0.0, ~1.0 ] + +var bool = ( v === out ); +// returns true +``` + + +
+ + + + +
+ +## Examples + +``` javascript +var linspace = require( '@stdlib/math/utils/linspace' ); +var sincos = require( '@stdlib/math/base/special/sincos' ); + +var x = linspace( 0.0, 2.0*Math.PI, 100 ); +var i; + +for ( i = 0; i < x.length; i++ ) { + console.log( sincos( x[ i ] ) ); +} +``` + +
+ + + + + + + diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/benchmark.js b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/benchmark.js new file mode 100644 index 000000000000..a166024c87fe --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/benchmark.js @@ -0,0 +1,146 @@ +'use strict'; + +// MODULES // + +var bench = require( '@stdlib/bench' ); +var randu = require( '@stdlib/math/base/random/randu' ); +var isnan = require( '@stdlib/math/base/assert/is-nan' ); +var sin = require( '@stdlib/math/base/special/sin' ); +var cos = require( '@stdlib/math/base/special/cos' ); +var pkg = require( './../package.json' ).name; +var sincos = require( './../lib' ); + + +// MAIN // + +bench( pkg, function benchmark( b ) { + var x; + var y; + var i; + + b.tic(); + for ( i = 0; i < b.iterations; i++ ) { + x = ( randu()*20.0 ) - 10.0; + y = sincos( x ); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + } + b.toc(); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + b.pass( 'benchmark finished' ); + b.end(); +}); + +bench( pkg+'::separate-evaluation', function benchmark( b ) { + var x; + var y; + var i; + + y = [ 0.0, 0.0 ]; + b.tic(); + for ( i = 0; i < b.iterations; i++ ) { + x = ( randu()*20.0 ) - 10.0; + y = [ sin( x ), cos( x ) ]; + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + } + b.toc(); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + b.pass( 'benchmark finished' ); + b.end(); +}); + +bench( pkg+'::in-place', function benchmark( b ) { + var x; + var y; + var i; + + y = [ 0.0, 0.0 ]; + b.tic(); + for ( i = 0; i < b.iterations; i++ ) { + x = ( randu()*20.0 ) - 10.0; + sincos( y, x ); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + } + b.toc(); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + b.pass( 'benchmark finished' ); + b.end(); +}); + +bench( pkg+'::separate-evaluation,in-place', function benchmark( b ) { + var x; + var y; + var i; + + y = [ 0.0, 0.0 ]; + b.tic(); + for ( i = 0; i < b.iterations; i++ ) { + x = ( randu()*20.0 ) - 10.0; + y[0] = sin( x ); + y[1] = cos( x ); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + } + b.toc(); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + b.pass( 'benchmark finished' ); + b.end(); +}); + +bench( pkg+'::built-in', function benchmark( b ) { + var x; + var y; + var i; + + b.tic(); + for ( i = 0; i < b.iterations; i++ ) { + x = ( randu()*20.0 ) - 10.0; + y = [ Math.sin( x ), Math.cos( x ) ]; + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + } + b.toc(); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + b.pass( 'benchmark finished' ); + b.end(); +}); + +bench( pkg+'::built-in,in-place', function benchmark( b ) { + var x; + var y; + var i; + + y = [ 0.0, 0.0 ]; + b.tic(); + for ( i = 0; i < b.iterations; i++ ) { + x = ( randu()*20.0 ) - 10.0; + y[0] = Math.sin( x ); + y[1] = Math.cos( x ); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + } + b.toc(); + if ( isnan( y[0] ) || isnan( y[1] ) ) { + b.fail( 'should not return NaN' ); + } + b.pass( 'benchmark finished' ); + b.end(); +}); diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/c/Makefile b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/c/Makefile new file mode 100644 index 000000000000..c93f7f1f19ad --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/c/Makefile @@ -0,0 +1,89 @@ + +# VARIABLES # + +ifndef VERBOSE + QUIET := @ +endif + +# Determine the OS: +# +# [1]: https://en.wikipedia.org/wiki/Uname#Examples +# [2]: http://stackoverflow.com/a/27776822/2225624 +OS ?= $(shell uname) +ifneq (, $(findstring MINGW,$(OS))) + OS := WINNT +else +ifneq (, $(findstring MSYS,$(OS))) + OS := WINNT +else +ifneq (, $(findstring CYGWIN,$(OS))) + OS := WINNT +endif +endif +endif + +# Define the program used for compiling C source files: +ifdef C_COMPILER + CC := $(C_COMPILER) +else + CC := gcc +endif + +# Define the command-line options when compiling C files: +CFLAGS ?= \ + -std=c99 \ + -O3 \ + -Wall \ + -pedantic + +# Determine whether to generate [position independent code][1]: +# +# [1]: https://gcc.gnu.org/onlinedocs/gcc/Code-Gen-Options.html#Code-Gen-Options +# [2]: http://stackoverflow.com/questions/5311515/gcc-fpic-option +ifeq ($(OS), WINNT) + fPIC ?= +else + fPIC ?= -fPIC +endif + +# List of C targets: +c_targets := benchmark.out + + +# TARGETS # + +# Default target. +# +# This target is the default target. + +all: $(c_targets) + +.PHONY: all + + +# Compile C source. +# +# This target compiles C source files. + +$(c_targets): %.out: %.c + $(QUIET) $(CC) $(CFLAGS) $(fPIC) -o $@ $< -lm + + +# Run a benchmark. +# +# This target runs a benchmark. + +run: $(c_targets) + $(QUIET) ./$< + +.PHONY: run + + +# Perform clean-up. +# +# This target removes generated files. + +clean: + $(QUIET) -rm -f *.o *.out + +.PHONY: clean diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/c/benchmark.c b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/c/benchmark.c new file mode 100644 index 000000000000..7f90824d4da7 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/c/benchmark.c @@ -0,0 +1,118 @@ +/** +* Benchmark `sincos`. +*/ +#include +#include +#include +#include + +#define NAME "sincos" +#define ITERATIONS 1000000 +#define REPEATS 3 + +/** +* Prints the TAP version. +*/ +void print_version() { + printf( "TAP version 13\n" ); +} + +/** +* Prints the TAP summary. +* +* @param total total number of tests +* @param passing total number of passing tests +*/ +void print_summary( int total, int passing ) { + printf( "#\n" ); + printf( "1..%d\n", total ); // TAP plan + printf( "# total %d\n", total ); + printf( "# pass %d\n", passing ); + printf( "#\n" ); + printf( "# ok\n" ); +} + +/** +* Prints benchmarks results. +* +* @param elapsed elapsed time in seconds +*/ +void print_results( double elapsed ) { + double rate = (double)ITERATIONS / elapsed; + printf( " ---\n" ); + printf( " iterations: %d\n", ITERATIONS ); + printf( " elapsed: %0.9f\n", elapsed ); + printf( " rate: %0.9f\n", rate ); + printf( " ...\n" ); +} + +/** +* Returns a clock time. +* +* @return clock time +*/ +double tic() { + struct timeval now; + gettimeofday( &now, NULL ); + return (double)now.tv_sec + (double)now.tv_usec/1.0e6; +} + +/** +* Generates a random double on the interval [0,1]. +* +* @return random double +*/ +double rand_double() { + int r = rand(); + return (double)r / ( (double)RAND_MAX + 1.0 ); +} + +/** +* Runs a benchmark. +* +* @return elapsed time in seconds +*/ +double benchmark() { + double elapsed; + double x; + double y; + double z; + double t; + int i; + + t = tic(); + for ( i = 0; i < ITERATIONS; i++ ) { + x = ( 20.0 * rand_double() ) - 10.0; + y = sin( x ); + z = cos( x ); + if ( y != y || z != z ) { + printf( "should not return NaN\n" ); + break; + } + } + elapsed = tic() - t; + if ( y != y || z != z ) { + printf( "should not return NaN\n" ); + } + return elapsed; +} + +/** +* Main execution sequence. +*/ +int main( void ) { + double elapsed; + int i; + + // Use the current time to seed the random number generator: + srand( time( NULL ) ); + + print_version(); + for ( i = 0; i < REPEATS; i++ ) { + printf( "# c::%s\n", NAME ); + elapsed = benchmark(); + print_results( elapsed ); + printf( "ok %d benchmark finished\n", i+1 ); + } + print_summary( REPEATS, REPEATS ); +} diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/julia/REQUIRE b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/julia/REQUIRE new file mode 100644 index 000000000000..11b2b096cae4 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/julia/REQUIRE @@ -0,0 +1,2 @@ +julia 0.5 +BenchmarkTools 0.0.8 diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/julia/benchmark.jl b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/julia/benchmark.jl new file mode 100755 index 000000000000..eb6f0a4c0e72 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/julia/benchmark.jl @@ -0,0 +1,146 @@ +#!/usr/bin/env julia + +import BenchmarkTools + +# Benchmark variables: +name = "sincos"; +repeats = 3; + +""" + print_version() + +Prints the TAP version. + +# Examples + +``` julia +julia> print_version() +``` +""" +function print_version() + @printf( "TAP version 13\n" ); +end + +""" + print_summary( total, passing ) + +Print the benchmark summary. + +# Arguments + +* `total`: total number of tests +* `passing`: number of passing tests + +# Examples + +``` julia +julia> print_summary( 3, 3 ) +``` +""" +function print_summary( total, passing ) + @printf( "#\n" ); + @printf( "1..%d\n", total ); # TAP plan + @printf( "# total %d\n", total ); + @printf( "# pass %d\n", passing ); + @printf( "#\n" ); + @printf( "# ok\n" ); +end + +""" + print_results( iterations, elapsed ) + +Print benchmark results. + +# Arguments + +* `iterations`: number of iterations +* `elapsed`: elapsed time (in seconds) + +# Examples + +``` julia +julia> print_results( 1000000, 0.131009101868 ) +``` +""" +function print_results( iterations, elapsed ) + rate = iterations / elapsed + + @printf( " ---\n" ); + @printf( " iterations: %d\n", iterations ); + @printf( " elapsed: %0.9f\n", elapsed ); + @printf( " rate: %0.9f\n", rate ); + @printf( " ...\n" ); +end + +""" + sincos( x ) + +Compute the sine and cosine of a number (since sincos is not directly exposed in julia). + +# Arguments + +* `x`: argument, in radians + +# Examples + +``` julia +julia> sincos( 0.0 ) +``` +""" +function sincos( x ) + [ sin( x ), cos( x ) ]; +end + +""" + benchmark() + +Run a benchmark. + +# Notes + +* Benchmark results are returned as a two-element array: [ iterations, elapsed ]. +* The number of iterations is not the true number of iterations. Instead, an 'iteration' is defined as a 'sample', which is a computed estimate for a single evaluation. +* The elapsed time is in seconds. + +# Examples + +``` julia +julia> benchmark(); +``` +""" +function benchmark() + t = BenchmarkTools.@benchmark $sincos( (20.0*rand()) - 10.0 ) samples=1e6 + + # Compute the total "elapsed" time and convert from nanoseconds to seconds: + s = sum( t.times ) / 1.0e9; + + # Determine the number of "iterations": + iter = length( t.times ); + + # Return the results: + [ iter, s ]; +end + +""" + main() + +Run benchmarks. + +# Examples + +``` julia +julia> main(); +``` +""" +function main() + print_version(); + for i in 1:3 + @printf( "# julia::%s\n", name ); + results = benchmark(); + print_results( results[ 1 ], results[ 2 ] ); + @printf( "ok %d benchmark finished\n", i ); + end + print_summary( repeats, repeats ); +end + +main(); diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/python/benchmark.py b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/python/benchmark.py new file mode 100644 index 000000000000..809b6b7cf016 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/benchmark/python/benchmark.py @@ -0,0 +1,80 @@ +#!/usr/bin/env python +"""Benchmark sincos.""" + +from __future__ import print_function +import timeit + +NAME = "sincos" +REPEATS = 3 +ITERATIONS = 1000000 + + +def print_version(): + """Print the TAP version.""" + print("TAP version 13") + + +def print_summary(total, passing): + """Print the benchmark summary. + + # Arguments + + * `total`: total number of tests + * `passing`: number of passing tests + + """ + print("#") + print("1.." + str(total)) # TAP plan + print("# total " + str(total)) + print("# pass " + str(passing)) + print("#") + print("# ok") + + +def print_results(elapsed): + """Print benchmark results. + + # Arguments + + * `elapsed`: elapsed time (in seconds) + + # Examples + + ``` python + python> print_results(0.131009101868) + ``` + """ + rate = ITERATIONS / elapsed + + print(" ---") + print(" iterations: " + str(ITERATIONS)) + print(" elapsed: " + str(elapsed)) + print(" rate: " + str(rate)) + print(" ...") + + +def benchmark(): + """Run the benchmark and print benchmark results.""" + setup = "from math import sin, cos; from random import random;" + stmt = "x = 20.0*random() - 10.0; y = [ sin( x ), cos( x ) ]" + + t = timeit.Timer(stmt, setup=setup) + + print_version() + + for i in xrange(REPEATS): + print("# python::" + NAME) + elapsed = t.timeit(number=ITERATIONS) + print_results(elapsed) + print("ok " + str(i+1) + " benchmark finished") + + print_summary(REPEATS, REPEATS) + + +def main(): + """Run the benchmark.""" + benchmark() + + +if __name__ == "__main__": + main() diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/docs/repl.txt b/lib/node_modules/@stdlib/math/base/special/sincos/docs/repl.txt new file mode 100644 index 000000000000..4e402cff2e9a --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/docs/repl.txt @@ -0,0 +1,37 @@ + +{{alias}}( [out,] x ) + Simultaneously computes the sine and cosine of a number. + + Parameters + ---------- + out: Array|TypedArray|Object (optional) + Destination array. + + x: number + Input value. + + Returns + ------- + y: Array|TypedArray|Object + Sine and cosine (in radians). + + Examples + -------- + > var y = {{alias}}( 0.0 ) + [ ~0.0, ~1.0 ] + > y = {{alias}}( {{alias:@stdlib/math/constants/float64-pi}}/2.0 ) + [ ~1.0, ~0.0 ] + > y = {{alias}}( -{{alias:@stdlib/math/constants/float64-pi}}/6.0 ) + [ ~-0.5, ~0.866 ] + > y = {{alias}}( NaN ) + [ NaN, NaN ] + + > var out = new Float64Array( 2 ); + > var v = {{alias}}( out, 0.0 ) + [ ~0.0, ~1.0 ] + > var bool = ( v === out ) + true + + See Also + -------- + diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/examples/index.js b/lib/node_modules/@stdlib/math/base/special/sincos/examples/index.js new file mode 100644 index 000000000000..02dd8fc976eb --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/examples/index.js @@ -0,0 +1,14 @@ +'use strict'; + +var linspace = require( '@stdlib/math/utils/linspace' ); +var PI = require( '@stdlib/math/constants/float64-pi' ); +var sincos = require( './../lib' ); + +var x = linspace( 0.0, 2.0*PI, 100 ); +var i; +var y; + +for ( i = 0; i < x.length; i++ ) { + y = sincos( x[ i ] ); + console.log( 'sincos(%d) = [ %d, %d ]', x[ i ], y[ 0 ], y[ 1 ] ); +} diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/lib/index.js b/lib/node_modules/@stdlib/math/base/special/sincos/lib/index.js new file mode 100644 index 000000000000..4ad3af8b44a4 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/lib/index.js @@ -0,0 +1,42 @@ +'use strict'; + +/** +* Simultaneously compute the sine and cosine of a number. +* +* @module @stdlib/math/base/special/sincos +* +* @example +* var sincos = require( '@stdlib/math/base/special/sincos' ); +* +* var v = sincos( 0.0 ); +* // returns [ ~0.0, ~1.0 ] +* +* v = sincos( Math.PI/2.0 ); +* // returns [ ~1.0, ~0.0 ] +* +* v = sincos( -Math.PI/6.0 ); +* // returns [ ~-0.5, ~0.866 ] +* +* v = sincos( NaN ); +* // returns [ NaN, NaN ] +* +* @example +* var sincos = require( '@stdlib/math/base/special/sincos' ); +* +* var out = new Float64Array( 2 ); +* +* var v = sincos( out, 0.0 ); +* // return [ ~0.0, ~1.0 ] +* +* var bool = ( v === out ); +* // returns true +*/ + +// MODULES // + +var sincos = require( './main.js' ); + + +// EXPORTS // + +module.exports = sincos; diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/lib/kernel_rem_pio2.js b/lib/node_modules/@stdlib/math/base/special/sincos/lib/kernel_rem_pio2.js new file mode 100644 index 000000000000..866ab319d629 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/lib/kernel_rem_pio2.js @@ -0,0 +1,378 @@ +'use strict'; + +/* +* The following copyright, license, and long comment 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?view=co}. +* +* The implementation follows the original, but has been modified for JavaScript. +*/ + +/* +* ==================================================== +* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. +* +* Developed at SunSoft, a Sun Microsystems, Inc. business. +* Permission to use, copy, modify, and distribute this +* software is freely granted, provided that this notice +* is preserved. +* ==================================================== +*/ + +// MODULES // + +var floor = require( '@stdlib/math/base/special/floor' ); +var ldexp = require( '@stdlib/math/base/special/ldexp' ); + + +// VARIABLES // + +// Initial value for `jk`: +var INIT_JK = [ 3, 4, 4, 6 ]; + +/* +* Table of constants for `2/pi` (`396` hex digits, `476` decimal). +* +* Integer array which contains the (24*i)-th to (24*i+23)-th bit of `2/pi` after binary point. The corresponding floating value is +* +* ``` text +* ipio2[i] * 2^(-24(i+1)) +* ``` +* +* This table must have at least `(e0-3)/24 + jk` terms. For quad precision (e0 <= 16360, jk = 6), this is `686`. +*/ +var IPIO2 = [ + 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 +]; + +// Double precision array, obtained by cutting `pi/2` into `24` bit 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 = zero( new Array( 20 ) ); +var Q = zero( new Array( 20 ) ); +var FQ = zero( new Array( 20 ) ); +var IQ = zero( new Array( 20 ) ); + + +// FUNCTIONS // + +/** +* Zeros an array. +* +* @private +* @param {Array} arr - array to zero +* @returns {Array} input array +*/ +function zero( arr ) { + var len = arr.length; + var i; + for ( i = 0; i < len; i++ ) { + arr[ i ] = 0.0; + } + return arr; +} // end FUNCTION zero() + +/** +* Performs the computation for `kernelRemPio2()`. +* +* @private +* @param {PositiveNumber} x - input value +* @param {Array} y - output result in an array of double precision numbers +* @param {integer} jz - number of terms of `ipio2[]` used +* @param {Array} q - array with integral values, representing the 24-bits chunk of the product of `x` and `2/pi` +* @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} f - `IPIO2[]` in floating point +* @param {PositiveInteger} prec - precision in bits (can be 24 (single), 53 (double), 64 (extended), 113 (quad)) +* @returns {number} last three digits of `N` +*/ +function compute( x, y, jz, q, q0, jk, jv, jx, f, prec ) { + 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++, j-- ) { // eslint-disable-line no-plusplus + fw = ( TWON24 * z )|0; + IQ[ i ] = ( z - (TWO24*fw) )|0; + z = q[ j-1 ] + fw; + } + // 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 ) { + case 1: + IQ[ jz-1 ] &= 0x7fffff; + break; + case 2: + IQ[ jz-1 ] &= 0x3fffff; + break; + default: + 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 = 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, prec ); + } + } + // Chop off zero terms... + if ( z === 0.0 ) { + 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[]`... + switch ( prec ) { + case 0: + fw = 0.0; + for ( i = jz; i >= 0; i-- ) { + fw += FQ[ i ]; + } + if ( ih === 0 ) { + y[ 0 ] = fw; + } else { + y[ 0 ] = -fw; + } + break; + case 1: + case 2: + fw = 0.0; + for ( i = jz; i >= 0; i-- ) { + fw += FQ[ i ]; + } + if ( ih === 0 ) { + y[ 0 ] = fw; + } else { + y[ 0 ] = -fw; + } + fw = FQ[ 0 ] - fw; + for ( i = 1; i <= jz; i++ ) { + fw += FQ[i]; + } + if ( ih === 0 ) { + y[ 1 ] = fw; + } else { + y[ 1 ] = -fw; + } + break; + case 3: + 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; + } + fw = 0.0; + for ( 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; + } + break; + default: + break; + } + return ( n & 7 ); +} // end FUNCTION compute() + + +// MAIN // + +/** +* Returns the last three digits of `N` with `y = x - N*pi/2` so that `|y| < pi/2`. +* +* ## Method +* +* 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 is known to be a huge integer (more accurately, equals 0 mod 8 ). Thus, the number of operations are independent of the exponent of the input. +* +* @private +* @param {PositiveNumber} x - input value +* @param {Array} y - output result in an array of double precision numbers +* @param {PositiveInteger} e0 - the exponent of `x[0]` (must be <= 16360) +* @param {PositiveInteger} nx - dimension of `x[]` +* @param {PositiveInteger} prec - precision in bits (can be 24 (single), 53 (double), 64 (extended), 113 (quad)) +* @returns {number} last three digits of `N` +*/ +function kernelRemPio2( x, y, e0, nx, prec ) { + var fw; + var jk; + var jv; + var jx; + var jz; + var q0; + var i; + var j; + var m; + + // Initialize `jk`: + jk = INIT_JK[ prec ]; + + // 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++, j++ ) { // eslint-disable-line no-plusplus + if ( j < 0 ) { + F[ i ] = 0.0; + } else { + F[ i ] = IPIO2[ j ]; + } + } + // 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, prec ); +} // end FUNCTION kernelRemPio2() + + +// EXPORTS // + +module.exports = kernelRemPio2; diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/lib/kernel_sincos.js b/lib/node_modules/@stdlib/math/base/special/sincos/lib/kernel_sincos.js new file mode 100644 index 000000000000..b3d0f7165785 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/lib/kernel_sincos.js @@ -0,0 +1,77 @@ +'use strict'; + +/* +* The following copyright and license were part of the original implementation available as part of FreeBSD implementations of [k_sin.c]{@link https://svnweb.freebsd.org/base/release/9.3.0/lib/msun/src/k_sin.c?view=co} and [k_cos.c]{@link https://svnweb.freebsd.org/base/release/9.3.0/lib/msun/src/k_cos.c?view=co}.. +* +* The implementation follows the original sine and cosine kernels, but has been modified for JavaScript and combined into a single function. +*/ + +/* +* ==================================================== +* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. +* +* Developed at SunSoft, a Sun Microsystems, Inc. business. +* Permission to use, copy, modify, and distribute this +* software is freely granted, provided that this notice +* is preserved. +* ==================================================== +*/ + +// VARIABLES // + +var S1 = -1.66666666666666324348e-01; // 0xBFC55555, 0x55555549 +var S2 = 8.33333333332248946124e-03; // 0x3F811111, 0x1110F8A6 +var S3 = -1.98412698298579493134e-04; // 0xBF2A01A0, 0x19C161D5 +var S4 = 2.75573137070700676789e-06; // 0x3EC71DE3, 0x57B1FE7D +var S5 = -2.50507602534068634195e-08; // 0xBE5AE5E6, 0x8A2B9CEB +var S6 = 1.58969099521155010221e-10; // 0x3DE5D93A, 0x5ACFD57C + +var C1 = 4.16666666666666019037e-02; // 0x3FA55555, 0x5555554C +var C2 = -1.38888888888741095749e-03; // 0xBF56C16C, 0x16C15177 +var C3 = 2.48015872894767294178e-05; // 0x3EFA01A0, 0x19CB1590 +var C4 = -2.75573143513906633035e-07; // 0xBE927E4F, 0x809C52AD +var C5 = 2.08757232129817482790e-09; // 0x3E21EE9E, 0xBDB4B1C4 +var C6 = -1.13596475577881948265e-11; // 0xBDA8FAE9, 0xBE8838D4 + + +// MAIN // + +/** +* Computes the sine and cosine on \\( \approx [-\pi/4, \pi/4] \\) (except on \\(-0\\)), where \\( \pi/4 \approx 0.7854 \\). +* +* @private +* @param {(Array|TypedArray|Object)} out - destination array +* @param {number} x - input value (assumed to be bounded by `~pi/4` in magnitude) +* @param {number} y - tail of `x` +* @param {number} iy - indicates whether `y` is `0` (if `iy = 0`, `y` assumed to be `0`) +* @returns {(Array|TypedArray|Object)} sine and cosine (in radians) +*/ +function kernelSincos( out, x, y, iy ) { + var hz; + var r; + var v; + var w; + var z; + z = x * x; + w = z * z; + r = S2 + (z * (S3 + (z*S4))) + (z * w * (S5 + (z*S6))); + v = z * x; + if ( iy === 0 ) { + out[0] = x + (v * (S1 + (z*r))); + } else { + out[0] = x - (((z*((0.5*y) - (v*r))) - y) - (v*S1)); + } + + r = z * (C1 + (z * (C2 + (z*C3)))); + r += w * w * (C4 + (z * (C5 + (z*C6)))); + hz = 0.5 * z; + w = 1.0 - hz; + out[1] = w + ( ((1.0-w) - hz) + ((z*r) - (x*y)) ); + + return out; +} // end FUNCTION kernelSincos() + + +// EXPORTS // + +module.exports = kernelSincos; diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/lib/main.js b/lib/node_modules/@stdlib/math/base/special/sincos/lib/main.js new file mode 100644 index 000000000000..b0f223ffbe8e --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/lib/main.js @@ -0,0 +1,52 @@ +'use strict'; + +// MODULES // + +var computeSincos = require( './sincos.js' ); + + +// MAIN // + +/** +* Simultaneously computes the sine and cosine of a number. +* +* @param {(Array|TypedArray|Object)} [out] - destination array +* @param {number} x - input value +* @returns {(Array|TypedArray|Object)} sine and cosine (in radians) +* +* @example +* var v = sincos( 0.0 ); +* // returns ~0.0 +* +* @example +* var v = sincos( Math.PI/2.0 ); +* // returns ~1.0 +* +* @example +* var v = sincos( -Math.PI/6.0 ); +* // returns ~-0.5 +* +* @example +* var v = sincos( NaN ); +* // returns NaN +* +* @example +* var out = new Float64Array( 2 ); +* +* var v = sincos( out, 0.0 ); +* // return [ ~0.0, ~1.0 ] +* +* var bool = ( v === out ); +* // returns true +*/ +function sincos( out, x ) { + if ( arguments.length === 1 ) { + return computeSincos( new Array( 2 ), out ); + } + return computeSincos( out, x ); +} // end FUNCTION sincos() + + +// EXPORTS // + +module.exports = sincos; diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/lib/rem_pio2.js b/lib/node_modules/@stdlib/math/base/special/sincos/lib/rem_pio2.js new file mode 100644 index 000000000000..198dffd0616d --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/lib/rem_pio2.js @@ -0,0 +1,223 @@ +'use strict'; + +/* +* The following copyright, license, and long comment were part of the original implementation available as part of [FreeBSD]{@link https://svnweb.freebsd.org/base/release/9.3.0/lib/msun/src/e_rem_pio2.c?view=co}. +* +* The implementation follows the original, but has been modified for JavaScript. +*/ + +/* +* ==================================================== +* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. +* +* Developed at SunSoft, a Sun Microsystems, Inc. business. +* Permission to use, copy, modify, and distribute this +* software is freely granted, provided that this notice +* is preserved. +* ==================================================== +* +* Optimized by Bruce D. Evans. +*/ + +// MODULES // + +var getHighWord = require( '@stdlib/math/base/utils/float64-get-high-word' ); +var getLowWord = require( '@stdlib/math/base/utils/float64-get-low-word' ); +var fromWords = require( '@stdlib/math/base/utils/float64-from-words' ); +var remPio2Kernel = require( './kernel_rem_pio2.js' ); +var remPio2Medium = require( './rem_pio2_medium.js' ); + + +// VARIABLES // + +var ZERO = 0.00000000000000000000e+00; // 0x00000000, 0x00000000 +var TWO24 = 1.67772160000000000000e+07; // 0x41700000, 0x00000000 + +// 33 bits of PI/2: +var PIO2_1 = 1.57079632673412561417e+00; // 0x3FF921FB, 0x54400000 + +// PIO2_1T = PI/2 - PIO2_1: +var PIO2_1T = 6.07710050650619224932e-11; // 0x3DD0B461, 0x1A626331 +var TWO_PIO2_1T = 2.0 * PIO2_1T; +var THREE_PIO2_1T = 3.0 * PIO2_1T; +var FOUR_PIO2_1T = 4.0 * PIO2_1T; + +// Absolute value mask: 0x7fffffff = 2147483647 => 01111111111111111111111111111111 +var ABS_MASK = 0x7fffffff; + +// Exponent mask: 0x7ff00000 = 2146435072 => 01111111111100000000000000000000 +var EXPONENT_MASK = 0x7ff00000; + +// High word significand mask: 0xfffff = 1048575 => 00000000000011111111111111111111 +var SIGNIFICAND_MASK = 0xfffff; + +// High word significand for PI and PI/2: 0x921fb = 598523 => 00000000000010010010000111111011 +var PI_HIGH_WORD_SIGNIFICAND = 0x921fb; + +// High word for PI/4: 0x3fe921fb = 1072243195 => 00111111111010010010000111111011 +var PIO4_HIGH_WORD = 0x3fe921fb; + +// High word for 3*PI/4: 0x4002d97c = 1073928572 => 01000000000000101101100101111100 +var THREE_PIO4_HIGH_WORD = 0x4002d97c; + +// High word for 5*PI/4: 0x400f6a7a = 1074752122 => 01000000000011110110101001111010 +var FIVE_PIO4_HIGH_WORD = 0x400f6a7a; + +// High word for 6*PI/4: 0x4012d97c = 1074977148 => 01000000000100101101100101111100 +var THREE_PIO2_HIGH_WORD = 0x4012d97c; + +// High word for 7*PI/4: 0x4015fdbc = 1075183036 => 01000000000101011111110110111100 +var SEVEN_PIO4_HIGH_WORD = 0x4015fdbc; + +// High word for 8*PI/4: 0x401921fb = 1075388923 => 01000000000110010010000111111011 +var TWO_PI_HIGH_WORD = 0x401921fb; + +// High word for 9*PI/4: 0x401c463b = 1075594811 => 01000000000111000100011000111011 +var NINE_PIO4_HIGH_WORD = 0x401c463b; + +// 2^20*pi/2 = 1647099.3291652855 => 0100000100111001001000011111101101010100010001000010110100011000 => high word => 0x413921fb = 1094263291 => 01000001001110010010000111111011 +var MEDIUM = 0x413921fb; + +// Arrays for storing temporary values (note that, in C, this would not be thread-safe): +var TX = new Array( 3 ); +var TY = new Array( 2 ); + + +// MAIN // + +/** +* Computes `x - n*pi/2 = r`. +* +* ## Notes +* +* * Returns `n` and stores the remainder `r` as two numbers `y[0]` and `y[1]`, such that `y[0]+y[1] = r`. +* +* +* @private +* @param {number} x - input value +* @param {Array} y - remainder elements +* @returns {integer} factor of `pi/2` +*/ +function remPio2( x, y ) { + var low; + var e0; + var hx; + var ix; + var nx; + var i; + var n; + var z; + + hx = getHighWord( x ); + ix = hx & ABS_MASK; + + // Case: |x| ~<= pi/4 (no need for reduction) + if ( ix <= PIO4_HIGH_WORD ) { + y[ 0 ] = x; + y[ 1 ] = 0.0; + return 0; + } + // Case: |x| ~<= 5pi/4 + if ( ix <= FIVE_PIO4_HIGH_WORD ) { + // Case: |x| ~= pi/2 or pi + if ( (ix & SIGNIFICAND_MASK) === PI_HIGH_WORD_SIGNIFICAND ) { + // Cancellation => use medium case + return remPio2Medium( x, ix, y ); + } + // Case: |x| ~<= 3pi/4 + if ( ix <= THREE_PIO4_HIGH_WORD ) { + if ( x > 0.0 ) { + z = x - PIO2_1; + y[ 0 ] = z - PIO2_1T; + y[ 1 ] = (z - y[0]) - PIO2_1T; + return 1; + } + z = x + PIO2_1; + y[ 0 ] = z + PIO2_1T; + y[ 1 ] = (z - y[0]) + PIO2_1T; + return -1; + } + if ( x > 0.0 ) { + z = x - ( 2.0*PIO2_1 ); + y[ 0 ] = z - TWO_PIO2_1T; + y[ 1 ] = (z - y[0]) - TWO_PIO2_1T; + return 2; + } + z = x + ( 2.0*PIO2_1 ); + y[ 0 ] = z + TWO_PIO2_1T; + y[ 1 ] = (z - y[0]) + TWO_PIO2_1T; + return -2; + } + // Case: |x| ~<= 9pi/4 + if ( ix <= NINE_PIO4_HIGH_WORD ) { + // Case: |x| ~<= 7pi/4 + if ( ix <= SEVEN_PIO4_HIGH_WORD ) { + // Case: |x| ~= 3pi/2 + if ( ix === THREE_PIO2_HIGH_WORD ) { + return remPio2Medium( x, ix, y ); + } + if ( x > 0.0 ) { + z = x - ( 3.0*PIO2_1 ); + y[ 0 ] = z - THREE_PIO2_1T; + y[ 1 ] = (z - y[0]) - THREE_PIO2_1T; + return 3; + } + z = x + ( 3.0*PIO2_1 ); + y[ 0 ] = z + THREE_PIO2_1T; + y[ 1 ] = (z - y[0]) + THREE_PIO2_1T; + return -3; + } + // Case: |x| ~= 4pi/2 + if ( ix === TWO_PI_HIGH_WORD ) { + return remPio2Medium( x, ix, y ); + } + if ( x > 0.0 ) { + z = x - ( 4.0*PIO2_1 ); + y[ 0 ] = z - FOUR_PIO2_1T; + y[ 1 ] = (z - y[0]) - FOUR_PIO2_1T; + return 4; + } + z = x + ( 4.0*PIO2_1 ); + y[ 0 ] = z + FOUR_PIO2_1T; + y[ 1 ] = (z - y[0]) + FOUR_PIO2_1T; + return -4; + } + // Case: |x| ~< 2^20*pi/2 (medium size) + if ( ix < MEDIUM ) { + return remPio2Medium( x, ix, y ); + } + // Case: x is NaN or infinity + if ( ix >= EXPONENT_MASK ) { + y[ 0 ] = NaN; + y[ 1 ] = NaN; + return 0.0; + } + // Set z = scalbn(|x|, ilogb(x)-23)... + low = getLowWord( x ); + e0 = (ix >> 20) - 1046; // e0 = ilogb(z) - 23 => unbiased exponent minus 23 + z = fromWords( ix - ((e0 << 20)|0), low ); + for ( i = 0; i < 2; i++ ) { + TX[ i ] = z|0; + z = (z - TX[i]) * TWO24; + } + TX[ 2 ] = z; + nx = 3; + while ( TX[ nx-1 ] === ZERO ) { + // Skip zero term... + nx -= 1; + } + n = remPio2Kernel( TX, TY, e0, nx, 1 ); + if ( x < 0.0 ) { + y[ 0 ] = -TY[ 0 ]; + y[ 1 ] = -TY[ 1 ]; + return -n; + } + y[ 0 ] = TY[ 0 ]; + y[ 1 ] = TY[ 1 ]; + return n; +} // end FUNCTION remPio2() + + +// EXPORTS // + +module.exports = remPio2; diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/lib/rem_pio2_medium.js b/lib/node_modules/@stdlib/math/base/special/sincos/lib/rem_pio2_medium.js new file mode 100644 index 000000000000..be7892fedcff --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/lib/rem_pio2_medium.js @@ -0,0 +1,109 @@ +'use strict'; + +/* +* 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?view=co}. +* +* The implementation follows the original, but has been modified for JavaScript. +*/ + +/* +* ==================================================== +* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. +* +* Developed at SunSoft, a Sun Microsystems, Inc. business. +* Permission to use, copy, modify, and distribute this +* software is freely granted, provided that this notice +* is preserved. +* ==================================================== +*/ + +// MODULES // + +var round = require( '@stdlib/math/base/special/round' ); +var getHighWord = require( '@stdlib/math/base/utils/float64-get-high-word' ); + + +// VARIABLES // + +// 53 bits of 2/PI: +var INVPIO2 = 6.36619772367581382433e-01; // 0x3FE45F30, 0x6DC9C883 + +// First 33 bits of PI/2: +var PIO2_1 = 1.57079632673412561417e+00; // 0x3FF921FB, 0x54400000 + +// PIO2_1T = PI/2 - PIO2_1: +var PIO2_1T = 6.07710050650619224932e-11; // 0x3DD0B461, 0x1A626331 + +// Another 33 bits of PI/2: +var PIO2_2 = 6.07710050630396597660e-11; // 0x3DD0B461, 0x1A600000 + +// PIO2_2T = PI/2 - ( PIO2_1 + PIO2_2 ): +var PIO2_2T = 2.02226624879595063154e-21; // 0x3BA3198A, 0x2E037073 + +// Another 33 bits of PI/2: +var PIO2_3 = 2.02226624871116645580e-21; // 0x3BA3198A, 0x2E000000 + +// PIO2_3T = PI/2 - ( PIO2_1 + PIO2_2 + PIO2_3 ): +var PIO2_3T = 8.47842766036889956997e-32; // 0x397B839A, 0x252049C1 + +// Exponent mask (2047 => 0x7ff): +var EXPONENT_MASK = 0x7ff; + + +// MAIN // + +/** +* Computes `x - n*pi/2 = r` for medium-sized inputs. +* +* @private +* @param {number} x - input value +* @param {uint32} ix - high word of `x` +* @param {Array} y - remainder elements +* @returns {integer} factor of `pi/2` +*/ +function remPio2( x, ix, y ) { + var high; + var n; + var t; + var r; + var w; + var i; + var j; + + n = round( x * INVPIO2 ); + r = x - ( n * PIO2_1 ); + w = n * PIO2_1T; + + // First rounding (good to 85 bits)... + j = ix >> 20; + y[ 0 ] = r - w; + high = getHighWord( y[0] ); + i = j - ( (high >> 20) & EXPONENT_MASK ); + + // Check if a second iteration is needed (good to 118 bits)... + if ( i > 16 ) { + t = r; + w = n * PIO2_2; + r = t - w; + w = (n * PIO2_2T) - ((t-r) - w); + y[ 0 ] = r - w; + high = getHighWord( y[0] ); + i = j - ( (high >> 20) & EXPONENT_MASK ); + + // Check if a third iteration is needed (151 bits accumulated)... + if ( i > 49 ) { + t = r; + w = n * PIO2_3; + r = t - w; + w = (n * PIO2_3T) - ((t-r) - w); + y[ 0 ] = r - w; + } + } + y[ 1 ] = (r - y[0]) - w; + return n; +} // end FUNCTION remPio2() + + +// EXPORTS // + +module.exports = remPio2; diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/lib/sincos.js b/lib/node_modules/@stdlib/math/base/special/sincos/lib/sincos.js new file mode 100644 index 000000000000..cfb8b4a0c21c --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/lib/sincos.js @@ -0,0 +1,139 @@ +'use strict'; + +/* +* The following copyright, license, and long comment 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_sin.c?view=log}. +* +* The implementation follows the original, but has been modified for JavaScript. +*/ + +/* +* ==================================================== +* 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. +* ==================================================== +*/ + +// MODULES // + +var getHighWord = require( '@stdlib/math/base/utils/float64-get-high-word' ); +var kernelSincos = require( './kernel_sincos.js' ); +var remPio2 = require( './rem_pio2.js' ); + + +// VARIABLES // + +// Absolute value mask: 0x7fffffff = 2147483647 => 01111111111111111111111111111111 +var ABS_MASK = 0x7fffffff; + +// Exponent mask: 0x7ff00000 = 2146435072 => 01111111111100000000000000000000 +var EXPONENT_MASK = 0x7ff00000; + +// High word for PI/4: 0x3fe921fb = 1072243195 => 00111111111010010010000111111011 +var PIO4_HIGH_WORD = 0x3fe921fb; + +// The smaller of the two cutoffs for the sine and cosine kernels. +// 2^-27 = 0x3e400000 +var SMALL_HIGH_WORD = 0x3e400000; + +// Array for storing remainder elements: +var Y = [ 0.0, 0.0 ]; + + +// MAIN // + +/** +* Simultaneously computes the sine and cosine of a number. +* +* ## Method +* +* * Let `S`, `C`, and `T` denote the `sin`, `cos` and `tan`, respectively, on `[-PI/4, +PI/4]`. +* * Reduce the argument `x` to `y1+y2 = x-k*pi/2` in `[-pi/4 , +pi/4]`, and let `n = k mod 4`. We have +* +* | n | sin(x) | cos(x) | tan(x) | +* |:---:|:--------:|:--------:|:--------:| +* | 0 | S | C | T | +* | 1 | C | -S | -1/T | +* | 2 | -S | -C | T | +* | 3 | -C | S | -1/T | +* +* +* @private +* @param {(Array|TypedArray|Object)} out - destination array +* @param {number} x - input value +* @returns {(Array|TypedArray|Object)} sine and cosine (in radians) +* +* @example +* var v = sincos( [ 0.0, 0.0 ], 0.0 ); +* // returns [ ~0.0, ~1.0 ] +* +* @example +* var v = sincos( [ 0.0, 0.0 ], Math.PI/2.0 ); +* // returns [ ~1.0, ~0.0 ] +* +* @example +* var v = sincos( [ 0.0, 0.0 ], -Math.PI/6.0 ); +* // returns [ ~-0.5, ~0.866 ] +* +* @example +* var v = sincos( [ 0.0, 0.0 ], NaN ); +* // returns [ NaN, NaN ] +*/ +function sincos( out, x ) { + var ix; + var n; + + ix = getHighWord( x ); + + // Case: |x| ~< pi/4 + ix &= ABS_MASK; + if ( ix <= PIO4_HIGH_WORD ) { + // Case: |x| ~< 2^-26 + if ( ix < SMALL_HIGH_WORD ) { + if ( (x|0) === 0 ) { + out[0] = x; + out[1] = 0.0; + } + } + return kernelSincos( out, x, 0.0, 0 ); + } + // Case: x is NaN or infinity + if ( ix >= EXPONENT_MASK ) { + out[0] = NaN; + out[1] = NaN; + return out; + } + // Argument reduction... + n = remPio2( x, Y ); + + // Compute the sine and cosine together: + kernelSincos( out, Y[0], Y[1], 1 ); + + switch ( n & 3 ) { + case 1: + ix = out[1]; + out[1] = -out[0]; + out[0] = ix; + return out; + case 2: + out[0] *= -1; + out[1] *= -1; + return out; + case 3: + // Passing + ix = -out[1]; + out[1] = out[0]; + out[0] = ix; + return out; + default: + return out; + } +} // end FUNCTION sincos() + + +// EXPORTS // + +module.exports = sincos; diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/package.json b/lib/node_modules/@stdlib/math/base/special/sincos/package.json new file mode 100644 index 000000000000..5a4270f4fc5a --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/package.json @@ -0,0 +1,49 @@ +{ + "name": "@stdlib/math/base/special/sincos", + "version": "0.0.0", + "description": "Simultaneously compute the sine and cosine of a number.", + "author": { + "name": "The Stdlib Authors", + "url": "https://github.com/stdlib-js/stdlib/graphs/contributors" + }, + "contributors": [ + { + "name": "The Stdlib Authors", + "url": "https://github.com/stdlib-js/stdlib/graphs/contributors" + } + ], + "scripts": {}, + "main": "./lib", + "repository": { + "type": "git", + "url": "git://github.com/stdlib-js/stdlib.git" + }, + "homepage": "https://github.com/stdlib-js/stdlib", + "keywords": [ + "stdlib", + "stdmath", + "mathematics", + "math", + "math.sin", + "math.cos", + "sin", + "cos", + "sincos", + "sine", + "cosine", + "trig", + "trigonometry", + "radians", + "angle" + ], + "bugs": { + "url": "https://github.com/stdlib-js/stdlib/issues" + }, + "dependencies": {}, + "devDependencies": {}, + "engines": { + "node": ">=0.10.0", + "npm": ">2.7.0" + }, + "license": "Apache-2.0" +} diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/REQUIRE b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/REQUIRE new file mode 100644 index 000000000000..becfcba22e9e --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/REQUIRE @@ -0,0 +1,2 @@ +julia 0.5 +JSON 0.5 diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/huge_negative.json b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/huge_negative.json new file mode 100644 index 000000000000..647898cda8ec --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/huge_negative.json @@ -0,0 +1 @@ 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\ No newline at end of file diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/large_negative.json b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/large_negative.json new file mode 100644 index 000000000000..d4571a0bd856 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/large_negative.json @@ -0,0 +1 @@ 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\ No newline at end of file diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/medium_negative.json b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/medium_negative.json new file mode 100644 index 000000000000..4128f3e17647 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/medium_negative.json @@ -0,0 +1 @@ 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\ No newline at end of file diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/medium_positive.json b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/medium_positive.json new file mode 100644 index 000000000000..67fc3fa4dcc5 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/medium_positive.json @@ -0,0 +1 @@ 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\ No newline at end of file diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/runner.jl new file mode 100644 index 000000000000..69081a1642b3 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/test/fixtures/julia/runner.jl @@ -0,0 +1,69 @@ +#!/usr/bin/env julia + +import JSON + +""" + gen( x, filepath ) + +Generate fixture data and write to file. + +# Arguments + +* `x`: domain +* `filepath::AbstractString`: filepath of the output file + +# Examples + +``` julia +julia> x = linspace( -1000.0, 1000.0, 2001 ); +julia> gen( x, \"./data.json\" ); +``` +""" +function gen( x, filepath ) + s = sin( x ); + c = cos( x ); + data = Dict([ + ("x", x), + ("sine", s), + ("cosine", c) + ]); + outfile = open( filepath, "w" ); + write( outfile, JSON.json(data) ); + close( outfile ); +end + +# Get the filename: +file = @__FILE__; + +# Extract the directory in which this file resides: +dir = dirname( file ); + +# Negative medium sized values: +x = linspace( -256.0*pi, 0.0, 4000 ); +out = joinpath( dir, "medium_negative.json" ); +gen( x, out ); + +# Positive medium sized values: +x = linspace( 0.0, 256.0*pi, 4000 ); +out = joinpath( dir, "medium_positive.json" ); +gen( x, out ); + +# Negative large values: +x = linspace( -2.0^20*(pi/2.0), -2.0^60*(pi/2.0), 4000 ); +out = joinpath( dir, "large_negative.json" ); +gen( x, out ); + +# Positive large values: +x = linspace( 2.0^20*(pi/2.0), 2.0^60*(pi/2.0), 4000 ); +out = joinpath( dir, "large_positive.json" ); +gen( x, out ); + +# Negative huge values: +x = linspace( -2.0^60*(pi/2.0), -2.0^1000*(pi/2.0), 4000 ); +out = joinpath( dir, "huge_negative.json" ); +gen( x, out ); + +# Positive huge values: +x = linspace( 2.0^60*(pi/2.0), 2.0^1000*(pi/2.0), 4000 ); +out = joinpath( dir, "huge_positive.json" ); +gen( x, out ); diff --git a/lib/node_modules/@stdlib/math/base/special/sincos/test/test.js b/lib/node_modules/@stdlib/math/base/special/sincos/test/test.js new file mode 100644 index 000000000000..15f62f6754bb --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/sincos/test/test.js @@ -0,0 +1,285 @@ +'use strict'; + +// MODULES // + +var tape = require( 'tape' ); +var isnan = require( '@stdlib/math/base/assert/is-nan' ); +var PINF = require( '@stdlib/math/constants/float64-pinf' ); +var NINF = require( '@stdlib/math/constants/float64-ninf' ); +var EPS = require( '@stdlib/math/constants/float64-eps' ); +var abs = require( '@stdlib/math/base/special/abs' ); +var sincos = require( './../lib' ); + + +// FIXTURES // + +var mediumNegative = require( './fixtures/julia/medium_negative.json' ); +var mediumPositive = require( './fixtures/julia/medium_positive.json' ); +var largeNegative = require( './fixtures/julia/large_negative.json' ); +var largePositive = require( './fixtures/julia/large_positive.json' ); +var hugeNegative = require( './fixtures/julia/huge_negative.json' ); +var hugePositive = require( './fixtures/julia/huge_positive.json' ); + + +// TESTS // + +tape( 'main export is a function', function test( t ) { + t.ok( true, __filename ); + t.equal( typeof sincos, 'function', 'main export is a function' ); + t.end(); +}); + +tape( 'the function computes in-place the sine and cosine (for -256*pi < x < 0)', function test( t ) { + var cosine; + var delta; + var sine; + var tol; + var x; + var y; + var z; + var i; + + z = [ 0.0, 0.0 ]; + x = mediumNegative.x; + sine = mediumNegative.sine; + cosine = mediumNegative.cosine; + + for ( i = 0; i < x.length; i++ ) { + y = sincos( z, x[i] ); + t.equal( y, z, 'returns a reference to out' ); + if ( y[0] === sine[ i ] ) { + t.equal( y[0], sine[ i ], 'x: '+x[i]+'. Expected: '+sine[i] ); + } else { + delta = abs( y[0] - sine[i] ); + tol = EPS * abs( sine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[0]+'. Expected: '+sine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + if ( y[1] === cosine[ i ] ) { + t.equal( y[1], cosine[ i ], 'x: '+x[i]+'. Expected: '+cosine[i] ); + } else { + delta = abs( y[1] - cosine[i] ); + tol = EPS * abs( cosine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[1]+'. Expected: '+cosine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + } + t.end(); +}); + +tape( 'the function computes the sine and cosine (for -256*pi < x < 0)', function test( t ) { + var cosine; + var delta; + var sine; + var tol; + var x; + var y; + var i; + + x = mediumNegative.x; + sine = mediumNegative.sine; + cosine = mediumNegative.cosine; + + for ( i = 0; i < x.length; i++ ) { + y = sincos( x[i] ); + if ( y[0] === sine[ i ] ) { + t.equal( y[0], sine[ i ], 'x: '+x[i]+'. Expected: '+sine[i] ); + } else { + delta = abs( y[0] - sine[i] ); + tol = EPS * abs( sine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[0]+'. Expected: '+sine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + if ( y[1] === cosine[ i ] ) { + t.equal( y[1], cosine[ i ], 'x: '+x[i]+'. Expected: '+cosine[i] ); + } else { + delta = abs( y[1] - cosine[i] ); + tol = EPS * abs( cosine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[1]+'. Expected: '+cosine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + } + t.end(); +}); + +tape( 'the function computes the sine and cosine (for 0 < x < 256*pi)', function test( t ) { + var cosine; + var delta; + var sine; + var tol; + var x; + var y; + var i; + + x = mediumPositive.x; + sine = mediumPositive.sine; + cosine = mediumPositive.cosine; + + for ( i = 0; i < x.length; i++ ) { + y = sincos( x[i] ); + if ( y[0] === sine[ i ] ) { + t.equal( y[0], sine[ i ], 'x: '+x[i]+'. Expected: '+sine[i] ); + } else { + delta = abs( y[0] - sine[i] ); + tol = EPS * abs( sine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[0]+'. Expected: '+sine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + if ( y[1] === cosine[ i ] ) { + t.equal( y[1], cosine[ i ], 'x: '+x[i]+'. Expected: '+cosine[i] ); + } else { + delta = abs( y[1] - cosine[i] ); + tol = EPS * abs( cosine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[1]+'. Expected: '+cosine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + } + t.end(); +}); + +tape( 'the function computes the sine and cosine (for -2**60 (pi/2) < x < -2**20 (pi/2))', function test( t ) { + var cosine; + var delta; + var sine; + var tol; + var x; + var y; + var i; + + x = largeNegative.x; + sine = largeNegative.sine; + cosine = largeNegative.cosine; + + for ( i = 0; i < x.length; i++ ) { + y = sincos( x[i] ); + if ( y[0] === sine[ i ] ) { + t.equal( y[0], sine[ i ], 'x: '+x[i]+'. Expected: '+sine[i] ); + } else { + delta = abs( y[0] - sine[i] ); + tol = EPS * abs( sine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[0]+'. Expected: '+sine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + if ( y[1] === cosine[ i ] ) { + t.equal( y[1], cosine[ i ], 'x: '+x[i]+'. Expected: '+cosine[i] ); + } else { + delta = abs( y[1] - cosine[i] ); + tol = EPS * abs( cosine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[1]+'. Expected: '+cosine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + } + t.end(); +}); + +tape( 'the function computes the sine and cosine (for 2**20 (pi/2) < x < 2**60 (pi/2))', function test( t ) { + var cosine; + var delta; + var sine; + var tol; + var x; + var y; + var i; + + x = largePositive.x; + sine = largePositive.sine; + cosine = largePositive.cosine; + + for ( i = 0; i < x.length; i++ ) { + y = sincos( x[i] ); + if ( y[0] === sine[ i ] ) { + t.equal( y[0], sine[ i ], 'x: '+x[i]+'. Expected: '+sine[i] ); + } else { + delta = abs( y[0] - sine[i] ); + tol = EPS * abs( sine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[0]+'. Expected: '+sine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + if ( y[1] === cosine[ i ] ) { + t.equal( y[1], cosine[ i ], 'x: '+x[i]+'. Expected: '+cosine[i] ); + } else { + delta = abs( y[1] - cosine[i] ); + tol = EPS * abs( cosine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[1]+'. Expected: '+cosine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + } + t.end(); +}); + +tape( 'the function computes the sine and cosine (for x <= -2**60 (PI/2))', function test( t ) { + var cosine; + var delta; + var sine; + var tol; + var x; + var y; + var i; + + x = hugeNegative.x; + sine = hugeNegative.sine; + cosine = hugeNegative.cosine; + + for ( i = 0; i < x.length; i++ ) { + y = sincos( x[i] ); + if ( y[0] === sine[ i ] ) { + t.equal( y[0], sine[ i ], 'x: '+x[i]+'. Expected: '+sine[i] ); + } else { + delta = abs( y[0] - sine[i] ); + tol = EPS * abs( sine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[0]+'. Expected: '+sine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + if ( y[1] === cosine[ i ] ) { + t.equal( y[1], cosine[ i ], 'x: '+x[i]+'. Expected: '+cosine[i] ); + } else { + delta = abs( y[1] - cosine[i] ); + tol = EPS * abs( cosine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[1]+'. Expected: '+cosine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + } + t.end(); +}); + +tape( 'the function computes the sine and cosine (for x >= 2**60 (PI/2))', function test( t ) { + var cosine; + var delta; + var sine; + var tol; + var x; + var y; + var i; + + x = hugePositive.x; + sine = hugePositive.sine; + cosine = hugePositive.cosine; + + for ( i = 0; i < x.length; i++ ) { + y = sincos( x[i] ); + if ( y[0] === sine[ i ] ) { + t.equal( y[0], sine[ i ], 'x: '+x[i]+'. Expected: '+sine[i] ); + } else { + delta = abs( y[0] - sine[i] ); + tol = EPS * abs( sine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[0]+'. Expected: '+sine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + if ( y[1] === cosine[ i ] ) { + t.equal( y[1], cosine[ i ], 'x: '+x[i]+'. Expected: '+cosine[i] ); + } else { + delta = abs( y[1] - cosine[i] ); + tol = EPS * abs( cosine[i] ); + t.ok( delta <= tol, 'within tolerance. x: '+x[i]+'. Value: '+y[1]+'. Expected: '+cosine[i]+'. tol: '+tol+'. delta: '+delta+'.' ); + } + } + t.end(); +}); + +tape( 'the function returns `NaN` if provided a `NaN`', function test( t ) { + var v = sincos( NaN ); + t.equal( isnan( v[0] ), true, 'returns NaN' ); + t.equal( isnan( v[1] ), true, 'returns NaN' ); + t.end(); +}); + +tape( 'the function returns `NaN` if provided `+infinity`', function test( t ) { + var v = sincos( PINF ); + t.equal( isnan( v[0] ), true, 'returns NaN' ); + t.equal( isnan( v[1] ), true, 'returns NaN' ); + t.end(); +}); + +tape( 'the function returns `NaN` if provided `-infinity`', function test( t ) { + var v = sincos( NINF ); + t.equal( isnan( v[0] ), true, 'returns NaN' ); + t.equal( isnan( v[1] ), true, 'returns NaN' ); + t.end(); +});