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Diffstat (limited to 'libm/ldouble/ndtril.c')
| -rw-r--r-- | libm/ldouble/ndtril.c | 416 |
1 files changed, 416 insertions, 0 deletions
diff --git a/libm/ldouble/ndtril.c b/libm/ldouble/ndtril.c new file mode 100644 index 000000000..b1a15cedf --- /dev/null +++ b/libm/ldouble/ndtril.c @@ -0,0 +1,416 @@ +/* ndtril.c + * + * Inverse of Normal distribution function + * + * + * + * SYNOPSIS: + * + * long double x, y, ndtril(); + * + * x = ndtril( y ); + * + * + * + * DESCRIPTION: + * + * Returns the argument, x, for which the area under the + * Gaussian probability density function (integrated from + * minus infinity to x) is equal to y. + * + * + * For small arguments 0 < y < exp(-2), the program computes + * z = sqrt( -2 log(y) ); then the approximation is + * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) . + * For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) , + * where w = y - 0.5 . + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * Arguments uniformly distributed: + * IEEE 0, 1 5000 7.8e-19 9.9e-20 + * Arguments exponentially distributed: + * IEEE exp(-11355),-1 30000 1.7e-19 4.3e-20 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * ndtril domain x <= 0 -MAXNUML + * ndtril domain x >= 1 MAXNUML + * + */ + + +/* +Cephes Math Library Release 2.3: January, 1995 +Copyright 1984, 1995 by Stephen L. Moshier +*/ + +#include <math.h> +extern long double MAXNUML; + +/* ndtri(y+0.5)/sqrt(2 pi) = y + y^3 R(y^2) + 0 <= y <= 3/8 + Peak relative error 6.8e-21. */ +#if UNK +/* sqrt(2pi) */ +static long double s2pi = 2.506628274631000502416E0L; +static long double P0[8] = { + 8.779679420055069160496E-3L, +-7.649544967784380691785E-1L, + 2.971493676711545292135E0L, +-4.144980036933753828858E0L, + 2.765359913000830285937E0L, +-9.570456817794268907847E-1L, + 1.659219375097958322098E-1L, +-1.140013969885358273307E-2L, +}; +static long double Q0[7] = { +/* 1.000000000000000000000E0L, */ +-5.303846964603721860329E0L, + 9.908875375256718220854E0L, +-9.031318655459381388888E0L, + 4.496118508523213950686E0L, +-1.250016921424819972516E0L, + 1.823840725000038842075E-1L, +-1.088633151006419263153E-2L, +}; +#endif +#if IBMPC +static unsigned short s2p[] = { +0x2cb3,0xb138,0x98ff,0xa06c,0x4000, XPD +}; +#define s2pi *(long double *)s2p +static short P0[] = { +0xb006,0x9fc1,0xa4fe,0x8fd8,0x3ff8, XPD +0x6f8a,0x976e,0x0ed2,0xc3d4,0xbffe, XPD +0xf1f1,0x6fcc,0xf3d0,0xbe2c,0x4000, XPD +0xccfb,0xa681,0xad2c,0x84a3,0xc001, XPD +0x9a0d,0x0082,0xa825,0xb0fb,0x4000, XPD +0x13d1,0x054a,0xf220,0xf500,0xbffe, XPD +0xcee9,0x2c92,0x70bd,0xa9e7,0x3ffc, XPD +0x5fee,0x4a42,0xa6cb,0xbac7,0xbff8, XPD +}; +static short Q0[] = { +/* 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD */ +0x841e,0xfec7,0x1d44,0xa9b9,0xc001, XPD +0x97e6,0xcde0,0xc0e7,0x9e8a,0x4002, XPD +0x66f9,0x8f3e,0x47fd,0x9080,0xc002, XPD +0x212f,0x2185,0x33ec,0x8fe0,0x4001, XPD +0x8e73,0x7bac,0x8df2,0xa000,0xbfff, XPD +0xc143,0xcb94,0xe3ea,0xbac2,0x3ffc, XPD +0x25d9,0xc8f3,0x9573,0xb25c,0xbff8, XPD +}; +#endif +#if MIEEE +static unsigned long s2p[] = { +0x40000000,0xa06c98ff,0xb1382cb3, +}; +#define s2pi *(long double *)s2p +static long P0[24] = { +0x3ff80000,0x8fd8a4fe,0x9fc1b006, +0xbffe0000,0xc3d40ed2,0x976e6f8a, +0x40000000,0xbe2cf3d0,0x6fccf1f1, +0xc0010000,0x84a3ad2c,0xa681ccfb, +0x40000000,0xb0fba825,0x00829a0d, +0xbffe0000,0xf500f220,0x054a13d1, +0x3ffc0000,0xa9e770bd,0x2c92cee9, +0xbff80000,0xbac7a6cb,0x4a425fee, +}; +static long Q0[21] = { +/* 0x3fff0000,0x80000000,0x00000000, */ +0xc0010000,0xa9b91d44,0xfec7841e, +0x40020000,0x9e8ac0e7,0xcde097e6, +0xc0020000,0x908047fd,0x8f3e66f9, +0x40010000,0x8fe033ec,0x2185212f, +0xbfff0000,0xa0008df2,0x7bac8e73, +0x3ffc0000,0xbac2e3ea,0xcb94c143, +0xbff80000,0xb25c9573,0xc8f325d9, +}; +#endif + +/* Approximation for interval z = sqrt(-2 log y ) between 2 and 8 + */ +/* ndtri(p) = z - ln(z)/z - 1/z P1(1/z)/Q1(1/z) + z = sqrt(-2 ln(p)) + 2 <= z <= 8, i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14. + Peak relative error 5.3e-21 */ +#if UNK +static long double P1[10] = { + 4.302849750435552180717E0L, + 4.360209451837096682600E1L, + 9.454613328844768318162E1L, + 9.336735653151873871756E1L, + 5.305046472191852391737E1L, + 1.775851836288460008093E1L, + 3.640308340137013109859E0L, + 3.691354900171224122390E-1L, + 1.403530274998072987187E-2L, + 1.377145111380960566197E-4L, +}; +static long double Q1[9] = { +/* 1.000000000000000000000E0L, */ + 2.001425109170530136741E1L, + 7.079893963891488254284E1L, + 8.033277265194672063478E1L, + 5.034715121553662712917E1L, + 1.779820137342627204153E1L, + 3.845554944954699547539E0L, + 3.993627390181238962857E-1L, + 1.526870689522191191380E-2L, + 1.498700676286675466900E-4L, +}; +#endif +#if IBMPC +static short P1[] = { +0x6105,0xb71e,0xf1f5,0x89b0,0x4001, XPD +0x461d,0x2604,0x8b77,0xae68,0x4004, XPD +0x8b33,0x4a47,0x9ec8,0xbd17,0x4005, XPD +0xa0b2,0xc1b0,0x1627,0xbabc,0x4005, XPD +0x9901,0x28f7,0xad06,0xd433,0x4004, XPD +0xddcb,0x5009,0x7213,0x8e11,0x4003, XPD +0x2432,0x0fa6,0xcfd5,0xe8fa,0x4000, XPD +0x3e24,0xd53c,0x53b2,0xbcff,0x3ffd, XPD +0x4058,0x3d75,0x5393,0xe5f4,0x3ff8, XPD +0x1789,0xf50a,0x7524,0x9067,0x3ff2, XPD +}; +static short Q1[] = { +/* 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD */ +0xd901,0x2673,0x2fad,0xa01d,0x4003, XPD +0x24f5,0xc93c,0x0e9d,0x8d99,0x4005, XPD +0x8cda,0x523a,0x612d,0xa0aa,0x4005, XPD +0x602c,0xb5fc,0x7b9b,0xc963,0x4004, XPD +0xac72,0xd3e7,0xb766,0x8e62,0x4003, XPD +0x048e,0xe34c,0x927c,0xf61d,0x4000, XPD +0x6d88,0xa5cc,0x45de,0xcc79,0x3ffd, XPD +0xe6d1,0x199a,0x9931,0xfa29,0x3ff8, XPD +0x4c7d,0x3675,0x70a0,0x9d26,0x3ff2, XPD +}; +#endif +#if MIEEE +static long P1[30] = { +0x40010000,0x89b0f1f5,0xb71e6105, +0x40040000,0xae688b77,0x2604461d, +0x40050000,0xbd179ec8,0x4a478b33, +0x40050000,0xbabc1627,0xc1b0a0b2, +0x40040000,0xd433ad06,0x28f79901, +0x40030000,0x8e117213,0x5009ddcb, +0x40000000,0xe8facfd5,0x0fa62432, +0x3ffd0000,0xbcff53b2,0xd53c3e24, +0x3ff80000,0xe5f45393,0x3d754058, +0x3ff20000,0x90677524,0xf50a1789, +}; +static long Q1[27] = { +/* 0x3fff0000,0x80000000,0x00000000, */ +0x40030000,0xa01d2fad,0x2673d901, +0x40050000,0x8d990e9d,0xc93c24f5, +0x40050000,0xa0aa612d,0x523a8cda, +0x40040000,0xc9637b9b,0xb5fc602c, +0x40030000,0x8e62b766,0xd3e7ac72, +0x40000000,0xf61d927c,0xe34c048e, +0x3ffd0000,0xcc7945de,0xa5cc6d88, +0x3ff80000,0xfa299931,0x199ae6d1, +0x3ff20000,0x9d2670a0,0x36754c7d, +}; +#endif + +/* ndtri(x) = z - ln(z)/z - 1/z P2(1/z)/Q2(1/z) + z = sqrt(-2 ln(y)) + 8 <= z <= 32 + i.e., y between exp(-32) = 1.27e-14 and exp(-512) = 4.38e-223 + Peak relative error 1.0e-21 */ +#if UNK +static long double P2[8] = { + 3.244525725312906932464E0L, + 6.856256488128415760904E0L, + 3.765479340423144482796E0L, + 1.240893301734538935324E0L, + 1.740282292791367834724E-1L, + 9.082834200993107441750E-3L, + 1.617870121822776093899E-4L, + 7.377405643054504178605E-7L, +}; +static long double Q2[7] = { +/* 1.000000000000000000000E0L, */ + 6.021509481727510630722E0L, + 3.528463857156936773982E0L, + 1.289185315656302878699E0L, + 1.874290142615703609510E-1L, + 9.867655920899636109122E-3L, + 1.760452434084258930442E-4L, + 8.028288500688538331773E-7L, +}; +#endif +#if IBMPC +static short P2[] = { +0xafb1,0x4ff9,0x4f3a,0xcfa6,0x4000, XPD +0xbd81,0xaffa,0x7401,0xdb66,0x4001, XPD +0x3a32,0x3863,0x9d0f,0xf0fd,0x4000, XPD +0x300e,0x633d,0x977a,0x9ed5,0x3fff, XPD +0xea3a,0x56b6,0x74c5,0xb234,0x3ffc, XPD +0x38c6,0x49d2,0x2af6,0x94d0,0x3ff8, XPD +0xc85d,0xe17d,0x5ed1,0xa9a5,0x3ff2, XPD +0x536c,0x808b,0x2542,0xc609,0x3fea, XPD +}; +static short Q2[] = { +/* 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD */ +0xaabd,0x125a,0x34a7,0xc0b0,0x4001, XPD +0x0ded,0xe6da,0x5a11,0xe1d2,0x4000, XPD +0xc742,0x9d16,0x0640,0xa504,0x3fff, XPD +0xea1e,0x4cc2,0x643a,0xbfed,0x3ffc, XPD +0x7a9b,0xfaff,0xf2dd,0xa1ab,0x3ff8, XPD +0xfd90,0x4688,0xc902,0xb898,0x3ff2, XPD +0xf003,0x032a,0xfa7e,0xd781,0x3fea, XPD +}; +#endif +#if MIEEE +static long P2[24] = { +0x40000000,0xcfa64f3a,0x4ff9afb1, +0x40010000,0xdb667401,0xaffabd81, +0x40000000,0xf0fd9d0f,0x38633a32, +0x3fff0000,0x9ed5977a,0x633d300e, +0x3ffc0000,0xb23474c5,0x56b6ea3a, +0x3ff80000,0x94d02af6,0x49d238c6, +0x3ff20000,0xa9a55ed1,0xe17dc85d, +0x3fea0000,0xc6092542,0x808b536c, +}; +static long Q2[21] = { +/* 0x3fff0000,0x80000000,0x00000000, */ +0x40010000,0xc0b034a7,0x125aaabd, +0x40000000,0xe1d25a11,0xe6da0ded, +0x3fff0000,0xa5040640,0x9d16c742, +0x3ffc0000,0xbfed643a,0x4cc2ea1e, +0x3ff80000,0xa1abf2dd,0xfaff7a9b, +0x3ff20000,0xb898c902,0x4688fd90, +0x3fea0000,0xd781fa7e,0x032af003, +}; +#endif + +/* ndtri(x) = z - ln(z)/z - 1/z P3(1/z)/Q3(1/z) + 32 < z < 2048/13 + Peak relative error 1.4e-20 */ +#if UNK +static long double P3[8] = { + 2.020331091302772535752E0L, + 2.133020661587413053144E0L, + 2.114822217898707063183E-1L, +-6.500909615246067985872E-3L, +-7.279315200737344309241E-4L, +-1.275404675610280787619E-5L, +-6.433966387613344714022E-8L, +-7.772828380948163386917E-11L, +}; +static long double Q3[7] = { +/* 1.000000000000000000000E0L, */ + 2.278210997153449199574E0L, + 2.345321838870438196534E-1L, +-6.916708899719964982855E-3L, +-7.908542088737858288849E-4L, +-1.387652389480217178984E-5L, +-7.001476867559193780666E-8L, +-8.458494263787680376729E-11L, +}; +#endif +#if IBMPC +static short P3[] = { +0x87b2,0x0f31,0x1ac7,0x814d,0x4000, XPD +0x491c,0xcd74,0x6917,0x8883,0x4000, XPD +0x935e,0x1776,0xcba9,0xd88e,0x3ffc, XPD +0xbafd,0x8abb,0x9518,0xd505,0xbff7, XPD +0xc87e,0x2ed3,0xa84a,0xbed2,0xbff4, XPD +0x0094,0xa402,0x36b5,0xd5fa,0xbfee, XPD +0xbc53,0x0fc3,0x1ab2,0x8a2b,0xbfe7, XPD +0x30b4,0x71c0,0x223d,0xaaed,0xbfdd, XPD +}; +static short Q3[] = { +/* 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD */ +0xdfc1,0x8a57,0x357f,0x91ce,0x4000, XPD +0xcc4f,0x9e03,0x346e,0xf029,0x3ffc, XPD +0x38b1,0x9788,0x8f42,0xe2a5,0xbff7, XPD +0xb281,0x2117,0x53da,0xcf51,0xbff4, XPD +0xf2ab,0x1d42,0x3760,0xe8cf,0xbfee, XPD +0x741b,0xf14f,0x06b0,0x965b,0xbfe7, XPD +0x37c2,0xa91f,0x16ea,0xba01,0xbfdd, XPD +}; +#endif +#if MIEEE +static long P3[24] = { +0x40000000,0x814d1ac7,0x0f3187b2, +0x40000000,0x88836917,0xcd74491c, +0x3ffc0000,0xd88ecba9,0x1776935e, +0xbff70000,0xd5059518,0x8abbbafd, +0xbff40000,0xbed2a84a,0x2ed3c87e, +0xbfee0000,0xd5fa36b5,0xa4020094, +0xbfe70000,0x8a2b1ab2,0x0fc3bc53, +0xbfdd0000,0xaaed223d,0x71c030b4, +}; +static long Q3[21] = { +/* 0x3fff0000,0x80000000,0x00000000, */ +0x40000000,0x91ce357f,0x8a57dfc1, +0x3ffc0000,0xf029346e,0x9e03cc4f, +0xbff70000,0xe2a58f42,0x978838b1, +0xbff40000,0xcf5153da,0x2117b281, +0xbfee0000,0xe8cf3760,0x1d42f2ab, +0xbfe70000,0x965b06b0,0xf14f741b, +0xbfdd0000,0xba0116ea,0xa91f37c2, +}; +#endif +#ifdef ANSIPROT +extern long double polevll ( long double, void *, int ); +extern long double p1evll ( long double, void *, int ); +extern long double logl ( long double ); +extern long double sqrtl ( long double ); +#else +long double polevll(), p1evll(), logl(), sqrtl(); +#endif + +long double ndtril(y0) +long double y0; +{ +long double x, y, z, y2, x0, x1; +int code; + +if( y0 <= 0.0L ) + { + mtherr( "ndtril", DOMAIN ); + return( -MAXNUML ); + } +if( y0 >= 1.0L ) + { + mtherr( "ndtri", DOMAIN ); + return( MAXNUML ); + } +code = 1; +y = y0; +if( y > (1.0L - 0.13533528323661269189L) ) /* 0.135... = exp(-2) */ + { + y = 1.0L - y; + code = 0; + } + +if( y > 0.13533528323661269189L ) + { + y = y - 0.5L; + y2 = y * y; + x = y + y * (y2 * polevll( y2, P0, 7 )/p1evll( y2, Q0, 7 )); + x = x * s2pi; + return(x); + } + +x = sqrtl( -2.0L * logl(y) ); +x0 = x - logl(x)/x; +z = 1.0L/x; +if( x < 8.0L ) + x1 = z * polevll( z, P1, 9 )/p1evll( z, Q1, 9 ); +else if( x < 32.0L ) + x1 = z * polevll( z, P2, 7 )/p1evll( z, Q2, 7 ); +else + x1 = z * polevll( z, P3, 7 )/p1evll( z, Q3, 7 ); +x = x0 - x1; +if( code != 0 ) + x = -x; +return( x ); +} |