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Add Fresnel integral
* Add Fresnel integral * Add equation to readme * Add additional Fresnel integral examples * Add equation to readme * Fix incorrect value in example output * Remove unneeded blank line * Remove unused fixture generation code * Fix labels in tests * Fix test labels and comment format * Remove equation file * Fix wording * Add Cephes method and copyright * Fix incorrect values in repl docs * Add a note on scaling * Fix PR review issues * Align equations * Align equations * Fix formatting issues * Move scipy benchmark
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‎docs/links/database.json‎

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"frechet"
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]
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},
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"https://en.wikipedia.org/wiki/Fresnel_integral": {
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"id": "fresnel-integral",
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"description": "Wikipedia entry for Fresnel integral.",
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"short_url": "",
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"keywords": [
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"math",
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"special",
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"fresnel",
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"integral"
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]
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},
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"https://en.wikipedia.org/wiki/Function_composition_%28computer_science%29": {
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"id": "function-composition",
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"description": "Wikipedia entry for function composition.",

‎lib/node_modules/@stdlib/math/base/special/fresnel/README.md‎

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# fresnel
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> Compute the [Fresnel integrals][fresnel-integral] S(x) and C(x).
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<section class="intro">
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The [Fresnel integrals][fresnel-integral] are defined as
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<!-- <equation class="equation" label="eq:fresnel_integrals" align="center" raw="\begin{align} S(x) &= \int_0^x \sin\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t, \\ C(x) &= \int_0^x \cos\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t. \end{align}" alt="Fresnel integral"> -->
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<!-- </equation> -->
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Some sources define the Fresnel integrals using t<sup>2</sup> for the argument of the sine and cosine. To get these functions, multiply the computed integrals by `√(π/2)` and multiply the argument `x` by `√(2/π)`.
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var fresnel = require( '@stdlib/math/base/special/fresnel' );
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```
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#### fresnel( \[out,] x )
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Simultaneously computes the [Fresnel integrals][fresnel-integral] S(x) and C(x).
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```javascript
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var v = fresnel( 0.0 );
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// returns [ ~0.0, ~0.0 ]
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v = fresnel( 1.0 );
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// returns [ ~0.438, ~0.780 ]
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v = fresnel( Infinity );
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// returns [ ~0.5, ~0.5 ]
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v = fresnel( -Infinity );
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// returns [ ~-0.5, ~-0.5 ]
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v = fresnel( NaN );
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// returns [ NaN, NaN ]
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```
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By default, the function returns the S(x) and C(x) as a two-element `array`. To avoid extra memory allocation, the function supports providing an output (destination) object.
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```javascript
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var out = new Float64Array( 2 );
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var v = fresnel( out, 0.0 );
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// return <Float64Array>[ ~0.0, ~0.0 ]
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var bool = ( v === out );
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// returns true
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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```javascript
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var linspace = require( '@stdlib/math/utils/linspace' );
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var fresnel = require( '@stdlib/math/base/special/fresnel' );
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var x = linspace( 0.0, 10.0, 100 );
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var i;
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for ( i = 0; i < x.length; i++ ) {
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console.log( fresnel( x[ i ] ) );
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}
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```
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</section>
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<!-- /.examples -->
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<section class="links">
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[fresnel-integral]: https://en.wikipedia.org/wiki/Fresnel_integral
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</section>
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<!-- /.links -->

‎lib/node_modules/@stdlib/math/base/special/fresnel/benchmark/benchmark.js‎

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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var randu = require( '@stdlib/math/base/random/randu' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var pkg = require( './../package.json' ).name;
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var fresnel = require( './../lib' );
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// MAIN //
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bench( pkg, function benchmark( b ) {
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var x;
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var y;
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var i;
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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x = ( randu()*20.0 ) - 10.0;
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y = fresnel( x );
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if ( isnan( y[0] ) || isnan( y[1] ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( y[0] ) || isnan( y[1] ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});
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bench( pkg+'::in-place', function benchmark( b ) {
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var x;
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var y;
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var i;
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y = [ 0.0, 0.0 ];
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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x = ( randu()*20.0 ) - 10.0;
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fresnel( y, x );
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if ( isnan( y[0] ) || isnan( y[1] ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( y[0] ) || isnan( y[1] ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});

‎lib/node_modules/@stdlib/math/base/special/fresnel/benchmark/c/cephes/Makefile‎

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# VARIABLES #
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ifndef VERBOSE
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QUIET := @
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endif
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# Specify the path to Cephes:
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CEPHES ?=
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# Specify a list of Cephes source files:
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CEPHES_SRC ?=
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# Determine the OS:
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#
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# [1]: https://en.wikipedia.org/wiki/Uname#Examples
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# [2]: http://stackoverflow.com/a/27776822/2225624
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OS ?= $(shell uname)
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ifneq (, $(findstring MINGW,$(OS)))
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OS := WINNT
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else
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ifneq (, $(findstring MSYS,$(OS)))
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OS := WINNT
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else
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ifneq (, $(findstring CYGWIN,$(OS)))
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OS := WINNT
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endif
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endif
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endif
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# Define the program used for compiling C source files:
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ifdef C_COMPILER
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CC := $(C_COMPILER)
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else
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CC := gcc
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endif
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# Define the command-line options when compiling C files:
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CFLAGS ?= \
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-std=c99 \
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-O3 \
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-Wall \
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-pedantic
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# Determine whether to generate [position independent code][1]:
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#
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# [1]: https://gcc.gnu.org/onlinedocs/gcc/Code-Gen-Options.html#Code-Gen-Options
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# [2]: http://stackoverflow.com/questions/5311515/gcc-fpic-option
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ifeq ($(OS), WINNT)
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fPIC ?=
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else
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fPIC ?= -fPIC
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endif
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# List of C targets:
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c_targets := benchmark.out
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# TARGETS #
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# Default target.
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#
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# This target is the default target.
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all: $(c_targets)
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.PHONY: all
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# Compile C source.
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#
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# This target compiles C source files.
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$(c_targets): %.out: %.c
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$(QUIET) $(CC) $(CFLAGS) $(fPIC) -o $@ $(CEPHES_SRC) $< -lm
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# Run a benchmark.
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#
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# This target runs a benchmark.
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run: $(c_targets)
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$(QUIET) ./$<
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.PHONY: run
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# Perform clean-up.
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#
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# This target removes generated files.
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clean:
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$(QUIET) -rm -f *.o *.out
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.PHONY: clean

‎lib/node_modules/@stdlib/math/base/special/fresnel/benchmark/c/cephes/benchmark.c‎

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/**
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* Benchmark Cephes `fresnl`.
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*/
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#include <stdlib.h>
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#include <stdio.h>
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#include <math.h>
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#include <sys/time.h>
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#define NAME "fresnel"
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#define ITERATIONS 1000000
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#define REPEATS 3
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/**
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* Define prototypes for external functions.
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*/
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extern int fresnl( double x, double *S, double *C );
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/**
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* Prints the TAP version.
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*/
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void print_version() {
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printf( "TAP version 13\n" );
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}
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/**
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* Prints the TAP summary.
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*
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* @param total total number of tests
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* @param passing total number of passing tests
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*/
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void print_summary( int total, int passing ) {
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printf( "#\n" );
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printf( "1..%d\n", total ); // TAP plan
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printf( "# total %d\n", total );
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printf( "# pass %d\n", passing );
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printf( "#\n" );
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printf( "# ok\n" );
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}
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/**
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* Prints benchmarks results.
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*
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* @param elapsed elapsed time in seconds
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*/
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void print_results( double elapsed ) {
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double rate = (double)ITERATIONS / elapsed;
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printf( " ---\n" );
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printf( " iterations: %d\n", ITERATIONS );
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printf( " elapsed: %0.9f\n", elapsed );
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printf( " rate: %0.9f\n", rate );
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printf( " ...\n" );
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}
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/**
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* Returns a clock time.
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*
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* @return clock time
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*/
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double tic() {
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struct timeval now;
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gettimeofday( &now, NULL );
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return (double)now.tv_sec + (double)now.tv_usec/1.0e6;
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}
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/**
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* Generates a random double on the interval [0,1].
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*
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* @return random double
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*/
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double rand_double() {
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int r = rand();
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return (double)r / ( (double)RAND_MAX + 1.0 );
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}
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/**
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* Runs a benchmark.
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*
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* @return elapsed time in seconds
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*/
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double benchmark() {
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double elapsed;
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double x;
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double S;
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double C;
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double t;
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int i;
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t = tic();
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for ( i = 0; i < ITERATIONS; i++ ) {
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x = ( 20.0*rand_double() ) - 10.0;
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fresnl( x, &S, &C );
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if ( S != S || C != C ) {
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printf( "should not return NaN\n" );
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break;
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}
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}
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elapsed = tic() - t;
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if ( S != S || C != C ) {
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printf( "should not return NaN\n" );
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}
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return elapsed;
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}
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/**
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* Main execution sequence.
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*/
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int main( void ) {
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double elapsed;
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int i;
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// Use the current time to seed the random number generator:
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srand( time( NULL ) );
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print_version();
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for ( i = 0; i < REPEATS; i++ ) {
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printf( "# c::cephes::%s\n", NAME );
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elapsed = benchmark();
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print_results( elapsed );
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printf( "ok %d benchmark finished\n", i+1 );
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}
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print_summary( REPEATS, REPEATS );
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}

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