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fix: avoid cancellation near one in math/base/special/asech #16110

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fix: avoid cancellation near one in math/base/special/asech #16110

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8 changes: 8 additions & 0 deletions lib/node_modules/@stdlib/math/base/special/asech/lib/main.js
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Expand Up @@ -22,6 +22,8 @@
// MODULES //

var acosh = require( '@stdlib/math/base/special/acosh' );
var log1p = require( '@stdlib/math/base/special/log1p' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );


// MAIN //
Expand All @@ -32,7 +34,7 @@
* ## Method
*
* ```tex
* \operatorname{asech}(x) = \operatorname{acosh}(\frac{1}{x})

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Unknown word: "asech"
* ```
*
* @param {number} x - input value
Expand All @@ -51,6 +53,12 @@
* // returns NaN
*/
function asech( x ) {
var d;
if ( x >= 0.5 && x <= 1.0 ) {
// `acosh( 1/x )` loses the low bits of `1/x - 1` to the rounding of `1/x`, which near one is most of the result, so there use `asech(x) = ln( (1+sqrt(1-x^2))/x ) = log1p( ((1-x) + sqrt((1-x)*(1+x)))/x )`, where `1-x` is exact:
d = 1.0 - x;
return log1p( ( d + sqrt( d * ( 1.0 + x ) ) ) / x );
}
return acosh( 1.0 / x );
}

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12 changes: 9 additions & 3 deletions lib/node_modules/@stdlib/math/base/special/asech/manifest.json
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Expand Up @@ -37,7 +37,9 @@
"libpath": [],
"dependencies": [
"@stdlib/math/base/napi/unary",
"@stdlib/math/base/special/acosh"
"@stdlib/math/base/special/acosh",
"@stdlib/math/base/special/log1p",
"@stdlib/math/base/special/sqrt"
]
},
{
Expand All @@ -51,7 +53,9 @@
"libraries": [],
"libpath": [],
"dependencies": [
"@stdlib/math/base/special/acosh"
"@stdlib/math/base/special/acosh",
"@stdlib/math/base/special/log1p",
"@stdlib/math/base/special/sqrt"
]
},
{
Expand All @@ -65,7 +69,9 @@
"libraries": [],
"libpath": [],
"dependencies": [
"@stdlib/math/base/special/acosh"
"@stdlib/math/base/special/acosh",
"@stdlib/math/base/special/log1p",
"@stdlib/math/base/special/sqrt"
]
}
]
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8 changes: 8 additions & 0 deletions lib/node_modules/@stdlib/math/base/special/asech/src/main.c
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Expand Up @@ -18,6 +18,8 @@

#include "stdlib/math/base/special/asech.h"
#include "stdlib/math/base/special/acosh.h"
#include "stdlib/math/base/special/log1p.h"
#include "stdlib/math/base/special/sqrt.h"

/**
* Computes the hyperbolic arcsecant of a number.
Expand All @@ -30,5 +32,11 @@
* // returns 0.0
*/
double stdlib_base_asech( const double x ) {
double d;
if ( x >= 0.5 && x <= 1.0 ) {
// `acosh( 1/x )` loses the low bits of `1/x - 1` to the rounding of `1/x`, which near one is most of the result, so there use `asech(x) = ln( (1+sqrt(1-x^2))/x ) = log1p( ((1-x) + sqrt((1-x)*(1+x)))/x )`, where `1-x` is exact:
d = 1.0 - x;
return stdlib_base_log1p( ( d + stdlib_base_sqrt( d * ( 1.0 + x ) ) ) / x );
}
return stdlib_base_acosh( 1.0 / x );
}
2 changes: 1 addition & 1 deletion lib/node_modules/@stdlib/math/base/special/asech/test/fixtures/julia/data.json
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1 change: 1 addition & 0 deletions lib/node_modules/@stdlib/math/base/special/asech/test/fixtures/julia/near_one.json
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19 changes: 18 additions & 1 deletion lib/node_modules/@stdlib/math/base/special/asech/test/fixtures/julia/runner.jl
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Expand Up @@ -19,7 +19,7 @@
import JSON

"""
gen( domain, name )

Check failure on line 22 in lib/node_modules/@stdlib/math/base/special/asech/test/fixtures/julia/runner.jl

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wrong indentation type (spaces instead of tabs)

Generate fixture data and write to file.

Expand All @@ -37,7 +37,9 @@
"""
function gen( domain, name )
x = collect( domain );
y = asech.( x );
y = setprecision( BigFloat, 256 ) do
Float64.( asech.( BigFloat.( x ) ) );
end

# Store data to be written to file as a collection:
data = Dict([
Expand All @@ -64,3 +66,18 @@
# Generate values over the function domain:
x = range( 0.00001, stop = 1.0, length = 2003 );
gen( x, "data.json" );

# Values close to one, evaluated in extended precision (in `Float64`, `asech` is `acosh(1/x)` and loses the low bits of `1/x - 1`):
x = 1.0 .- exp10.( range( -16.0, stop = -1.0, length = 1000 ) );
y = setprecision( BigFloat, 256 ) do
Float64.( asech.( BigFloat.( x ) ) );
end
data = Dict([
("x", x),
("expected", y)
]);
outfile = open( joinpath( dir, "near_one.json" ), "w" );
write( outfile, JSON.json(data) );
write( outfile, "\n" );
close( outfile );

19 changes: 18 additions & 1 deletion lib/node_modules/@stdlib/math/base/special/asech/test/test.js
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Expand Up @@ -32,6 +32,7 @@ var asech = require( './../lib' );
// FIXTURES //

var data = require( './fixtures/julia/data.json' );
var nearOne = require( './fixtures/julia/near_one.json' );


// TESTS //
Expand All @@ -53,7 +54,23 @@ tape( 'the function computes the hyperbolic arcsecant on the interval (0, 1]', f

for ( i = 0; i < x.length; i++ ) {
y = asech( x[i] );
t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' );
t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
}
t.end();
});

tape( 'the function computes the hyperbolic arcsecant close to one', function test( t ) {
var expected;
var x;
var y;
var i;

x = nearOne.x;
expected = nearOne.expected;

for ( i = 0; i < x.length; i++ ) {
y = asech( x[i] );
t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
}
t.end();
});
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19 changes: 18 additions & 1 deletion lib/node_modules/@stdlib/math/base/special/asech/test/test.native.js
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Expand Up @@ -41,6 +41,7 @@ var opts = {
// FIXTURES //

var data = require( './fixtures/julia/data.json' );
var nearOne = require( './fixtures/julia/near_one.json' );


// TESTS //
Expand All @@ -62,7 +63,23 @@ tape( 'the function computes the hyperbolic arcsecant on the interval (0, 1]', o

for ( i = 0; i < x.length; i++ ) {
y = asech( x[i] );
t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' );
t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
}
t.end();
});

tape( 'the function computes the hyperbolic arcsecant close to one', opts, function test( t ) {
var expected;
var x;
var y;
var i;

x = nearOne.x;
expected = nearOne.expected;

for ( i = 0; i < x.length; i++ ) {
y = asech( x[i] );
t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
}
t.end();
});
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