diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/lib/main.js b/lib/node_modules/@stdlib/math/base/special/covercos/lib/main.js index c5dffde43265..9403790b20eb 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/lib/main.js +++ b/lib/node_modules/@stdlib/math/base/special/covercos/lib/main.js @@ -21,6 +21,7 @@ // MODULES // var sin = require( '@stdlib/math/base/special/sin' ); +var cos = require( '@stdlib/math/base/special/cos' ); // MAIN // @@ -48,7 +49,16 @@ var sin = require( '@stdlib/math/base/special/sin' ); * // returns NaN */ function covercos( x ) { - return 1.0 + sin( x ); + var s; + var c; + + s = sin( x ); + if ( s < 0.0 ) { + // Where `sin(x)` is close to minus one, `1 + sin(x)` cancels. Use `1 + sin(x) = cos(x)^2 / (1 - sin(x))` where `sin(x) < 0`, since `cos(x)` is computed with full relative accuracy near its zeros and the denominator does not cancel: + c = cos( x ); + return ( c * c ) / ( 1.0 - s ); + } + return 1.0 + s; } diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/manifest.json b/lib/node_modules/@stdlib/math/base/special/covercos/manifest.json index 083d8fc81535..498bd6570c7c 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/manifest.json +++ b/lib/node_modules/@stdlib/math/base/special/covercos/manifest.json @@ -37,7 +37,8 @@ "libpath": [], "dependencies": [ "@stdlib/math/base/napi/unary", - "@stdlib/math/base/special/sin" + "@stdlib/math/base/special/sin", + "@stdlib/math/base/special/cos" ] }, { @@ -51,7 +52,8 @@ "libraries": [], "libpath": [], "dependencies": [ - "@stdlib/math/base/special/sin" + "@stdlib/math/base/special/sin", + "@stdlib/math/base/special/cos" ] }, { @@ -65,7 +67,8 @@ "libraries": [], "libpath": [], "dependencies": [ - "@stdlib/math/base/special/sin" + "@stdlib/math/base/special/sin", + "@stdlib/math/base/special/cos" ] } ] diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/src/main.c b/lib/node_modules/@stdlib/math/base/special/covercos/src/main.c index a6160de09d17..6a714fd811b1 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/src/main.c +++ b/lib/node_modules/@stdlib/math/base/special/covercos/src/main.c @@ -18,6 +18,7 @@ #include "stdlib/math/base/special/covercos.h" #include "stdlib/math/base/special/sin.h" +#include "stdlib/math/base/special/cos.h" /** * Computes the coversed cosine. @@ -30,5 +31,14 @@ * // returns 2.0 */ double stdlib_base_covercos( const double x ) { - return 1.0 + stdlib_base_sin( x ); + double s; + double c; + + s = stdlib_base_sin( x ); + if ( s < 0.0 ) { + // Where `sin(x)` is close to minus one, `1 + sin(x)` cancels. Use `1 + sin(x) = cos(x)^2 / (1 - sin(x))` where `sin(x) < 0`, since `cos(x)` is computed with full relative accuracy near its zeros and the denominator does not cancel: + c = stdlib_base_cos( x ); + return ( c * c ) / ( 1.0 - s ); + } + return 1.0 + s; } diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/huge_negative.json b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/huge_negative.json index 53688016f41f..6e89df1576b6 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/huge_negative.json +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/huge_negative.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/huge_positive.json b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/huge_positive.json index 95a6515aca8b..ec33ecce36aa 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/huge_positive.json +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/huge_positive.json @@ -1 +1 @@ -{"expected":[0.004025114534427887,1.9973897635197664,1.144034403935851,0.42668109926594255,0.7149349775545103,1.0323966809718823,0.06052259869500343,0.2169061593337317,0.4535256877576612,0.7461370948499386,1.0647593512449607,1.9247319000324112,0.3562496488838053,1.5261909117677754,0.02598230690010339,1.0970540358255594,0.08468190426394229,0.3317964672358098,0.5089071802166185,1.9824392195464893,1.129246831092774,0.09820953295106483,0.29605871946333606,1.455178503900705,1.9852413403473896,1.1447637052683828,0.10508978915193501,0.2850195201859026,1.441175326956526,1.9902695484196156,1.1931917095112863,0.12808974913228033,0.26274486218046933,1.4118702898502005,1.9942581512503326,1.2248766620203488,0.14441096776381546,0.24124418970940875,1.3666127242873716,1.9972029615226474,1.2563255344066786,0.16163040127422856,0.22054007461877267,1.3519942650148058,1.9999600968585507,1.2714348707876209,0.1894179313184169,0.21082288879185063,1.3372894335479857,1.9999773449553957,1.2864776026021898,0.19869067969021936,0.20129935088912843,1.2906405049832141,1.9997492190616029,1.333193183208699,0.20816005298222817,0.1543681136048194,1.2756169148559409,1.99744239197578,1.3791043527107791,0.23911499866980412,0.14611088365093416,1.2605256939742042,1.9962004234802282,1.3935526907545341,0.24937258812131857,0.13806318062393186,1.2127873068126458,1.9947140077733292,1.4382338985705112,0.2598143661891694,0.11419894736726943,1.164535198239283,1.9884693994520892,1.4522599933627078,0.31782637031767036,0.10703845571325565,1.1490643794064126,1.985976252185617,1.4955236597506354,0.329363513710096,0.08600576312704611,1.1666512537500784,1.976588603205723,1.5090686272241176,0.34106521758155506,0.07976254387709703,1.0178659812883804,1.9730991910600504,1.5224886796670738,0.35292861057322067,0.05060565612685053,1.002202174512031,1.9693710002730396,1.5357805240730782,0.3649507816510623,0.04580224881190342,0.9865378273670917,1.9654049456664677,1.548940898895335,0.43086201904534893,0.04123298211148929,0.9708767835640113,1.961202000429028,1.6160448894695572,0.4438115401509005,0.036898977230213736,0.9552228860026664,1.9351605436586692,1.6283081034485942,0.456897538530291,0.03280129764424289,0.939579975829392,1.90274000191312,1.6404171434681198,0.4701168031471301,0.02894094884034859,0.8574156231202033,1.8958907529945015,1.7016114028595513,0.4834660902642187,0.01254360822763445,0.8419290475575677,1.8888216708063121,1.7126869406043936,0.4969421242394949,0.010191521055589292,0.8264812593740667,1.8815344899567563,1.7235875995486574,0.5699770614901846,0.00808231264241932,0.811076049141009,1.8740309985709902,1.778078478523156,0.5841716652355675,0.006216500544735215,0.7957171969819449,1.8663130378523507,1.7878226976049827,0.6606382550255955,0.004594542594669293,0.7804084716451014,1.8221388572483606,1.7973736011080095,0.6754144340790179,0.0032168367875331505,0.70065042598136,1.8131206120618129,1.8067288454358357,0.6902702597924526,0.002402626202952174,0.6857413395428076,1.803902843705607,1.8158861350028206,0.7052020868500751,0.0036102041087143766,0.6709093657528549,1.7944878140315916,1.860818192310607,0.7202062512868856,0.005062275693940799,0.533874870767257,1.7416645868367535,1.8998782820357414,0.6713584273224574,0.0012188310164593652,0.579841209919886,1.7750752595901051,1.8763371388705956,0.7504168485909205,0.008698414761780904,0.506390338328669,1.7202891428333067,1.9130985386010657,0.964273740579678,0.0032550596282758804,0.5516220609447543,1.754902001343973,1.8909959973456623,0.9130902944129033,0.013307474688739829,0.47939026303536747,1.5963840453355937,1.9254226273120607,0.9955956616555559,0.0062695515235872135,0.5238429760794472,1.6367487959246598,1.9047803806820647,0.9443388043236637,0.01888493187613627,0.45290114432691964,1.5709473213598553,1.9368384525934195,1.0269219054044827,0.08334528049950596,0.4965312193286985,1.612282989184219,1.9176767600866713,0.9756419433595291,0.06406166048545636,0.42694898016047067,1.5449502365749734,1.9473348102922325,1.0582217264343046,0.09631484113116406,0.4697135960343314,1.587216252413269,1.9705307302295867,1.006968988804651,0.07555227112673502,0.4015592414948417,1.5184183060274692,1.987106291725655,1.0894644052856557,0.11017133078042107,0.33748707848600223,1.561573187560841,1.9776029314456658,1.0382891944800439,0.08795018837579016,0.27793416754800504,1.4913775696925096,1.9916358636071547,1.1206192785818183,0.12490114988482826,0.22330674168754538,1.417830078936441,1.9837156564558125,1.069571820919669,0.1012432441977058,0.2566175329900032,1.463854566916765,1.9951921865493603,1.282255084970381,0.1404898417457674,0.2039573138206937,1.3891660435753372,1.9998801330486875,1.2327052157706668,0.11541839201666393,0.2360304970922148,1.3118228709565636,1.997771770170713,1.3121672918260432,0.1569221067170926,0.18538916840679664,1.232352636193725,1.9998744545510965,1.2630554683408877,0.1304617195206096,0.2161932651654661,1.2819072902262438,1.9993720827182626,1.341773119599623,0.256860056897929,0.16762052930048887,1.2017715257457795,1.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diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/large_negative.json b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/large_negative.json index a3b3f2df6d3d..fa29b950b889 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/large_negative.json +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/large_negative.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/large_positive.json b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/large_positive.json index bfda28ca64fc..232cdd39aeb9 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/large_positive.json +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/large_positive.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/medium_negative.json b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/medium_negative.json index 838a0a698921..fd1b5b39d1c0 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/medium_negative.json +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/medium_negative.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/medium_positive.json b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/medium_positive.json index 07e8dca964be..845f964cd141 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/medium_positive.json +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/medium_positive.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/near_extrema.json b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/near_extrema.json new file mode 100644 index 000000000000..248cd2490269 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/near_extrema.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/runner.jl index 63fbfea9f564..de6728d7eea8 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/runner.jl +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/fixtures/julia/runner.jl @@ -37,7 +37,9 @@ julia> gen( x, \"./data.json\" ); """ function gen( domain, filepath ) x = collect( domain ); - y = 1.0 .+ sin.( x ); + y = setprecision( BigFloat, 256 ) do + Float64.( 1 .+ sin.( BigFloat.( x ) ) ); + end data = Dict([ ("x", x), ("expected", y) @@ -83,3 +85,14 @@ gen( x, out ); x = range( 2.0^60*(pi/2.0), stop = 2.0^1000*(pi/2.0), length = 4000 ) out = joinpath( dir, "huge_positive.json" ); gen( x, out ); + +# Values close to where `sin(x)` is one or minus one, where the textbook formulas cancel: +x = Float64[]; +for k in -10:10 + for d in exp10.( range( -8.0, stop = -1.0, length = 50 ) ) + push!( x, (2*k+0.5)*pi + d, (2*k+0.5)*pi - d, (2*k-0.5)*pi + d, (2*k-0.5)*pi - d ); + end +end +out = joinpath( dir, "near_extrema.json" ); +gen( x, out ); + diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/test.js b/lib/node_modules/@stdlib/math/base/special/covercos/test/test.js index d867210331fd..146bba7a614f 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/test.js +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/test.js @@ -36,6 +36,7 @@ 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' ); +var nearExtrema = require( './fixtures/julia/near_extrema.json' ); // TESTS // @@ -57,7 +58,7 @@ tape( 'the function computes the coversed cosine (for -256*pi < x < 0)', functio for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 3 ), true, 'returns expected value' ); } t.end(); }); @@ -73,7 +74,7 @@ tape( 'the function computes the coversed cosine (for 0 < x < 256*pi)', function for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); } t.end(); }); @@ -89,7 +90,7 @@ tape( 'the function computes the coversed cosine (for -2**60 (pi/2) < x < -2**20 for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 3 ), true, 'returns expected value' ); } t.end(); }); @@ -105,7 +106,7 @@ tape( 'the function computes the coversed cosine (for 2**20 (pi/2) < x < 2**60 ( for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); } t.end(); }); @@ -121,7 +122,7 @@ tape( 'the function computes the coversed cosine (for x <= -2**60 (PI/2))', func for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 3 ), true, 'returns expected value' ); } t.end(); }); @@ -137,7 +138,23 @@ tape( 'the function computes the coversed cosine (for x >= 2**60 (PI/2))', funct for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); + } + t.end(); +}); + +tape( 'the function computes the coversed cosine close to where `sin(x)` is one or minus one', function test( t ) { + var expected; + var x; + var y; + var i; + + x = nearExtrema.x; + expected = nearExtrema.expected; + + for ( i = 0; i < x.length; i++ ) { + y = covercos( x[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); } t.end(); }); diff --git a/lib/node_modules/@stdlib/math/base/special/covercos/test/test.native.js b/lib/node_modules/@stdlib/math/base/special/covercos/test/test.native.js index 2a4e8fca8549..34d47ec82138 100644 --- a/lib/node_modules/@stdlib/math/base/special/covercos/test/test.native.js +++ b/lib/node_modules/@stdlib/math/base/special/covercos/test/test.native.js @@ -45,6 +45,7 @@ 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' ); +var nearExtrema = require( './fixtures/julia/near_extrema.json' ); // TESTS // @@ -66,7 +67,7 @@ tape( 'the function computes the coversed cosine (for -256*pi < x < 0)', opts, f for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 3 ), true, 'returns expected value' ); } t.end(); }); @@ -82,7 +83,7 @@ tape( 'the function computes the coversed cosine (for 0 < x < 256*pi)', opts, fu for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); } t.end(); }); @@ -98,7 +99,7 @@ tape( 'the function computes the coversed cosine (for -2**60 (pi/2) < x < -2**20 for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 3 ), true, 'returns expected value' ); } t.end(); }); @@ -114,7 +115,7 @@ tape( 'the function computes the coversed cosine (for 2**20 (pi/2) < x < 2**60 ( for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); } t.end(); }); @@ -130,7 +131,7 @@ tape( 'the function computes the coversed cosine (for x <= -2**60 (PI/2))', opts for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 3 ), true, 'returns expected value' ); } t.end(); }); @@ -146,7 +147,23 @@ tape( 'the function computes the coversed cosine (for x >= 2**60 (PI/2))', opts, for ( i = 0; i < x.length; i++ ) { y = covercos( x[i] ); - t.strictEqual( isAlmostSameValue( y, expected[ i ], 0 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); + } + t.end(); +}); + +tape( 'the function computes the coversed cosine close to where `sin(x)` is one or minus one', opts, function test( t ) { + var expected; + var x; + var y; + var i; + + x = nearExtrema.x; + expected = nearExtrema.expected; + + for ( i = 0; i < x.length; i++ ) { + y = covercos( x[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); } t.end(); }); diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/lib/main.js b/lib/node_modules/@stdlib/math/base/special/hacovercos/lib/main.js index 66a593122a2b..99164022e367 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/lib/main.js +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/lib/main.js @@ -21,6 +21,7 @@ // MODULES // var sin = require( '@stdlib/math/base/special/sin' ); +var cos = require( '@stdlib/math/base/special/cos' ); // MAIN // @@ -48,7 +49,16 @@ var sin = require( '@stdlib/math/base/special/sin' ); * // returns NaN */ function hacovercos( x ) { - return (1.0 + sin( x )) / 2.0; + var s; + var c; + + s = sin( x ); + if ( s < 0.0 ) { + // Where `sin(x)` is close to minus one, `1 + sin(x)` cancels. Use `1 + sin(x) = cos(x)^2 / (1 - sin(x))` where `sin(x) < 0`, since `cos(x)` is computed with full relative accuracy near its zeros and the denominator does not cancel: + c = cos( x ); + return ( c * c ) / ( 2.0 * ( 1.0 - s ) ); + } + return ( 1.0 + s ) / 2.0; } diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/manifest.json b/lib/node_modules/@stdlib/math/base/special/hacovercos/manifest.json index 083d8fc81535..498bd6570c7c 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/manifest.json +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/manifest.json @@ -37,7 +37,8 @@ "libpath": [], "dependencies": [ "@stdlib/math/base/napi/unary", - "@stdlib/math/base/special/sin" + "@stdlib/math/base/special/sin", + "@stdlib/math/base/special/cos" ] }, { @@ -51,7 +52,8 @@ "libraries": [], "libpath": [], "dependencies": [ - "@stdlib/math/base/special/sin" + "@stdlib/math/base/special/sin", + "@stdlib/math/base/special/cos" ] }, { @@ -65,7 +67,8 @@ "libraries": [], "libpath": [], "dependencies": [ - "@stdlib/math/base/special/sin" + "@stdlib/math/base/special/sin", + "@stdlib/math/base/special/cos" ] } ] diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/src/main.c b/lib/node_modules/@stdlib/math/base/special/hacovercos/src/main.c index 80b9eaaa5d0f..75a352f09782 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/src/main.c +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/src/main.c @@ -18,6 +18,7 @@ #include "stdlib/math/base/special/hacovercos.h" #include "stdlib/math/base/special/sin.h" +#include "stdlib/math/base/special/cos.h" /** * Computes the half-value coversed cosine. @@ -30,5 +31,14 @@ * // returns 1.0 */ double stdlib_base_hacovercos( const double x ) { - return ( 1.0 + stdlib_base_sin( x ) ) / 2.0; + double s; + double c; + + s = stdlib_base_sin( x ); + if ( s < 0.0 ) { + // Where `sin(x)` is close to minus one, `1 + sin(x)` cancels. Use `1 + sin(x) = cos(x)^2 / (1 - sin(x))` where `sin(x) < 0`, since `cos(x)` is computed with full relative accuracy near its zeros and the denominator does not cancel: + c = stdlib_base_cos( x ); + return ( c * c ) / ( 2.0 * ( 1.0 - s ) ); + } + return ( 1.0 + s ) / 2.0; } diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/huge_negative.json b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/huge_negative.json index c598bcef081e..cd372ab07bdc 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/huge_negative.json +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/huge_negative.json @@ -1 +1 @@ -{"expected":[0.9979874427327861,0.0013051182401168093,0.4279827980320745,0.7866594503670288,0.6425325112227449,0.48380165951405885,0.9697387006524982,0.8915469203331341,0.7732371561211694,0.6269314525750307,0.46762032437751966,0.03763404998379433,0.8218751755580973,0.23690454411611228,0.9870088465499482,0.4514729820872203,0.9576590478680289,0.8341017663820951,0.7455464098916907,0.008780390226755364,0.4353765844536131,0.9508952335244676,0.851970640268332,0.2724107480496475,0.007379329826305181,0.4276181473658086,0.9474551054240326,0.8574902399070488,0.279412336521737,0.004865225790192218,0.4034041452443569,0.9359551254338598,0.8686275689097653,0.29406485507489977,0.0028709243748337165,0.3875616689898256,0.9277945161180923,0.8793779051452957,0.3166936378563142,0.001398519238676288,0.3718372327966607,0.9191847993628857,0.8897299626906137,0.3240028674925971,1.9951570724652257e-5,0.36428256460618963,0.9052910343407916,0.8945885556040747,0.3313552832260072,1.1327522302095883e-5,0.3567611986989051,0.9006546601548904,0.8993503245554357,0.35467974750839293,0.00012539046919857189,0.3334034083956505,0.8959199735088859,0.9228159431975903,0.36219154257202957,0.001278804012110002,0.31044782364461043,0.880442500665098,0.9269445581745329,0.36973715301289795,0.001899788259885904,0.303223654622733,0.8753137059393408,0.9309684096880341,0.39360634659367705,0.002642996113335405,0.28088305071474434,0.8700928169054153,0.9429005263163652,0.4177324008803584,0.005765300273955387,0.2738700033186461,0.8410868148411648,0.9464807721433721,0.4254678102967937,0.00701187390719149,0.2522381701246823,0.835318243144952,0.956997118436477,0.4166743731249608,0.011705698397138531,0.24546568638794114,0.8294673912092225,0.9601187280614515,0.4910670093558098,0.013450404469974864,0.23875566016646305,0.8235356947133896,0.9746971719365747,0.49889891274398446,0.015314499863480158,0.23210973796346096,0.8175246091744688,0.9770988755940483,0.5067310863164541,0.017297527166766136,0.22552955055233248,0.7845689904773255,0.9793835089442553,0.5145616082179943,0.019398999785486037,0.19197755526522137,0.7780942299245497,0.9815505113848931,0.5223885569986668,0.03241972817066541,0.1858459482757029,0.7715512307348544,0.9835993511778786,0.5302100120853039,0.04862999904343995,0.17979142826594013,0.7649415984264349,0.9855295255798258,0.5712921884398984,0.05205462350274931,0.14919429857022432,0.7582669548678906,0.9937281958861828,0.5790354762212162,0.05558916459684393,0.14365652969780313,0.7515289378802525,0.9949042394722054,0.5867593703129667,0.059232755021621786,0.13820620022567132,0.7150114692549077,0.9959588436787903,0.5944619754294955,0.06298450071450495,0.11096076073842193,0.7079141673822162,0.9968917497276324,0.6021414015090276,0.06684348107382465,0.10608865119750865,0.6696808724872023,0.9977027287026654,0.6097957641774493,0.08893057137581967,0.1013131994459952,0.662292782960491,0.9983915816062334,0.64967478700932,0.09343969396909357,0.09663557728208211,0.6548648701037737,0.9987986868985239,0.6571293302285962,0.09804857814719653,0.0920569324985897,0.6473989565749625,0.9981948979456428,0.6645453171235726,0.10275609298420413,0.06959090384469646,0.6398968743565572,0.9974688621530297,0.7330625646163715,0.12916770658162324,0.05006085898212931,0.6643207863387712,0.9993905844917703,0.710079395040057,0.11246237020494743,0.061831430564702194,0.6247915757045397,0.9956507926191096,0.7468048308356655,0.1398554285833467,0.0434507306994672,0.517863129710161,0.9983724701858621,0.7241889695276229,0.12254899932801361,0.054502001327168925,0.5434548527935483,0.9933462626556301,0.7603048684823163,0.20180797733220313,0.03728868634396959,0.502202169172222,0.9968652242382063,0.7380785119602764,0.18162560203767014,0.04760980965896766,0.5278305978381682,0.9905575340619319,0.7735494278365402,0.21452633932007237,0.03158077370329021,0.48653904729775865,0.9583273597502471,0.7517343903356508,0.19385850540789057,0.04116161995666434,0.5121790283202354,0.9679691697572719,0.7865255099197647,0.22752488171251328,0.026332594853883673,0.4708891367828478,0.951842579434418,0.7651432019828344,0.20639187379336554,0.014734634885206654,0.4965155055976745,0.9622238644366325,0.7992203792525792,0.24079084698626535,0.0064468541371724974,0.4552677973571722,0.9449143346097895,0.8312564607569989,0.21921340621957952,0.011198534277167105,0.48085540275997807,0.9560249058121049,0.8610329162259975,0.2543112151537452,0.004182068196422639,0.43969036070909084,0.9375494250575859,0.8883466291562273,0.29108496053177946,0.00814217177209381,0.4652140895401655,0.9493783779011471,0.8716912335049984,0.26807271654161746,0.002403906725319871,0.3588724575148095,0.9297550791271163,0.8980213430896531,0.3054169782123314,5.9933475656248625e-5,0.3836473921146666,0.942290803991668,0.8819847514538925,0.34408856452171815,0.001114114914643527,0.3439163540869784,0.9215389466414536,0.9073054157966016,0.3838236819031375,6.277272445170956e-5,0.36847226582955617,0.9347691402396952,0.891903367417267,0.3590463548868781,0.00031395864086875003,0.32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diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/huge_positive.json b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/huge_positive.json index d11389dda521..1289c34cf2f3 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/huge_positive.json +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/huge_positive.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/large_negative.json b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/large_negative.json index 706a322e0d9e..9dfe76ec86d0 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/large_negative.json +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/large_negative.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/large_positive.json b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/large_positive.json index ebbd9b78e54a..184d1fb3a4bf 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/large_positive.json +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/large_positive.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/medium_negative.json b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/medium_negative.json index afdacce58e49..0906b14c82d2 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/medium_negative.json +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/medium_negative.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/medium_positive.json b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/medium_positive.json index 1d9716ed807d..4c5a31f97197 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/medium_positive.json +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/medium_positive.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/near_extrema.json b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/near_extrema.json new file mode 100644 index 000000000000..900e7c9e809e --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/near_extrema.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/runner.jl index b746c58a24df..33b27c9aa4be 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/runner.jl +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/fixtures/julia/runner.jl @@ -36,7 +36,9 @@ julia> gen( x, \"./data.json\" ); ``` """ function gen( x, filepath ) - y = (1.0 .+ sin.( x )) ./ 2.0; + y = setprecision( BigFloat, 256 ) do + Float64.( (1 .+ sin.( BigFloat.( x ) )) ./ 2 ); + end data = Dict([ ("x", x), ("expected", y) @@ -82,3 +84,14 @@ gen( x, out ); x = range( 2.0^60*(pi/2.0), stop = 2.0^1000*(pi/2.0), length = 4000 ) out = joinpath( dir, "huge_positive.json" ); gen( x, out ); + +# Values close to where `sin(x)` is one or minus one, where the textbook formulas cancel: +x = Float64[]; +for k in -10:10 + for d in exp10.( range( -8.0, stop = -1.0, length = 50 ) ) + push!( x, (2*k+0.5)*pi + d, (2*k+0.5)*pi - d, (2*k-0.5)*pi + d, (2*k-0.5)*pi - d ); + end +end +out = joinpath( dir, "near_extrema.json" ); +gen( x, out ); + diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/test.js b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/test.js index 5ac905b7e4e1..0388f066a02e 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/test.js +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/test.js @@ -36,6 +36,7 @@ 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' ); +var nearExtrema = require( './fixtures/julia/near_extrema.json' ); // TESTS // @@ -59,7 +60,7 @@ tape( 'the function computes the half-value coversed cosine (for -256*pi < x < 0 for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 3 ), true, 'returns expected value' ); } t.end(); }); @@ -77,7 +78,7 @@ tape( 'the function computes the half-value coversed cosine (for 0 < x < 256*pi) for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 2 ), true, 'returns expected value' ); } t.end(); }); @@ -95,7 +96,7 @@ tape( 'the function computes the half-value coversed cosine (for -2**60 (pi/2) < for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 3 ), true, 'returns expected value' ); } t.end(); }); @@ -113,7 +114,7 @@ tape( 'the function computes the half-value coversed cosine (for 2**20 (pi/2) < for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 2 ), true, 'returns expected value' ); } t.end(); }); @@ -131,7 +132,7 @@ tape( 'the function computes the half-value coversed cosine (for x <= -2**60 (PI for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 3 ), true, 'returns expected value' ); } t.end(); }); @@ -149,7 +150,25 @@ tape( 'the function computes the half-value coversed cosine (for x >= 2**60 (PI/ for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 2 ), true, 'returns expected value' ); + } + t.end(); +}); + +tape( 'the function computes the half-value coversed cosine close to where `sin(x)` is one or minus one', function test( t ) { + var expected; + var x; + var y; + var i; + var e; + + x = nearExtrema.x; + expected = nearExtrema.expected; + + for ( i = 0; i < x.length; i++ ) { + y = hacovercos( x[i] ); + e = expected[ i ]; + t.strictEqual( isAlmostSameValue( y, e, 2 ), true, 'returns expected value' ); } t.end(); }); diff --git a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/test.native.js b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/test.native.js index 3da34387752d..142913b0d169 100644 --- a/lib/node_modules/@stdlib/math/base/special/hacovercos/test/test.native.js +++ b/lib/node_modules/@stdlib/math/base/special/hacovercos/test/test.native.js @@ -45,6 +45,7 @@ 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' ); +var nearExtrema = require( './fixtures/julia/near_extrema.json' ); // TESTS // @@ -68,7 +69,7 @@ tape( 'the function computes the half-value coversed cosine (for -256*pi < x < 0 for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 3 ), true, 'returns expected value' ); } t.end(); }); @@ -86,7 +87,7 @@ tape( 'the function computes the half-value coversed cosine (for 0 < x < 256*pi) for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 2 ), true, 'returns expected value' ); } t.end(); }); @@ -104,7 +105,7 @@ tape( 'the function computes the half-value coversed cosine (for -2**60 (pi/2) < for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 3 ), true, 'returns expected value' ); } t.end(); }); @@ -122,7 +123,7 @@ tape( 'the function computes the half-value coversed cosine (for 2**20 (pi/2) < for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 2 ), true, 'returns expected value' ); } t.end(); }); @@ -140,7 +141,7 @@ tape( 'the function computes the half-value coversed cosine (for x <= -2**60 (PI for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 3 ), true, 'returns expected value' ); } t.end(); }); @@ -158,7 +159,25 @@ tape( 'the function computes the half-value coversed cosine (for x >= 2**60 (PI/ for ( i = 0; i < x.length; i++ ) { y = hacovercos( x[i] ); e = expected[ i ]; - t.strictEqual( isAlmostSameValue( y, e, 1 ), true, 'returns expected value' ); + t.strictEqual( isAlmostSameValue( y, e, 2 ), true, 'returns expected value' ); + } + t.end(); +}); + +tape( 'the function computes the half-value coversed cosine close to where `sin(x)` is one or minus one', opts, function test( t ) { + var expected; + var x; + var y; + var i; + var e; + + x = nearExtrema.x; + expected = nearExtrema.expected; + + for ( i = 0; i < x.length; i++ ) { + y = hacovercos( x[i] ); + e = expected[ i ]; + t.strictEqual( isAlmostSameValue( y, e, 2 ), true, 'returns expected value' ); } t.end(); });