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[v1.9.x] modify erfinv implementation based on scipy (#20517) (#20550)
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* modify erfinv implementation based on scipy

* fix lint

* fix lint

* fix host/device gpu error

* fix flag
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@mseth10
mseth10 authored Aug 25, 2021
1 parent 695ba2e commit da38e28
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359 changes: 286 additions & 73 deletions src/operator/contrib/erfinv-inl.h
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@@ -1,49 +1,49 @@
/*
* Copyright (c) 2014 Indiana University
* Copyright (c) 2001-2002 Enthought, Inc. 2003-2019, SciPy Developers.
* All rights reserved.
* Written by Prof. Gary L. Pavlis, Dept. of Geol. Sci.,
* Indiana University, Bloomington, IN
* This software is licensed under the New BSD license:
* Redistribution and use in source and binary forms,
* with or without modification, are permitted provided
* that the following conditions are met:
* Redistributions of source code must retain the above
* copyright notice, this list of conditions and the
* following disclaimer.
* Redistributions in binary form must reproduce the
* above copyright notice, this list of conditions and
* the following disclaimer in the documentation and/or
* other materials provided with the distribution.
* Neither the name of Indiana University nor
* the names of its contributors may be used to endorse
* or promote products derived from this software without
* specific prior written permission.
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND
* CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED
* WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
* WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
* PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL
* THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY
* DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
* PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
* USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER
* IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
* NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
* USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
* POSSIBILITY OF SUCH DAMAGE.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are
* met:
*
* * Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* * Redistributions in binary form must reproduce the above
* copyright notice, this list of conditions and the following
* disclaimer in the documentation and/or other materials provided
* with the distribution.
*
* * Neither the name of the copyright holder nor the names of its
* contributors may be used to endorse or promote products derived
* from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/

/*
* The next function is taken from
* https://github.com/antelopeusersgroup/antelope_contrib/blob/master/lib/location/libgenloc/erfinv.c.
* Output was modified to be inf or -inf when input is 1 or -1.
* The functions in this file are taken from
* https://github.com/scipy/scipy/blob/master/scipy/special/cephes/polevl.h
* https://github.com/scipy/scipy/blob/master/scipy/special/cephes/ndtri.c
* https://github.com/scipy/scipy/blob/master/scipy/special/cephes/erfinv.c
*/

#ifndef MXNET_OPERATOR_CONTRIB_ERFINV_INL_H_
#define MXNET_OPERATOR_CONTRIB_ERFINV_INL_H_

#define _USE_MATH_DEFINES

#include <assert.h>
#include <mxnet/base.h>
#include <limits>
#include "math.h"
Expand All @@ -52,49 +52,262 @@ namespace mxnet {
namespace op {
namespace mshadow_op {

/*! \brief inverse gauss error function */

/*
* Evaluate polynomial
*
*
*
* SYNOPSIS:
*
* int N;
* double x, y, coef[N+1], polevl[];
*
* y = polevl( x, coef, N );
*
*
*
* DESCRIPTION:
*
* Evaluates polynomial of degree N:
*
* 2 N
* y = C + C x + C x +...+ C x
* 0 1 2 N
*
* Coefficients are stored in reverse order:
*
* coef[0] = C , ..., coef[N] = C .
* N 0
*
* The function p1evl() assumes that coef[N] = 1.0 and is
* omitted from the array. Its calling arguments are
* otherwise the same as polevl().
*
*
* SPEED:
*
* In the interest of speed, there are no checks for out
* of bounds arithmetic. This routine is used by most of
* the functions in the library. Depending on available
* equipment features, the user may wish to rewrite the
* program in microcode or assembly language.
*
*/

MSHADOW_XINLINE static double polevl(double x, const double coef[], int N) {
const double *p;
double ans;
int i;

p = coef;
ans = *p++;
i = N;

do {
ans = ans * x + *p++;
} while (--i);

return (ans);
}

MSHADOW_XINLINE static double p1evl(double x, const double coef[], int N) {
const double *p;
double ans;
int i;

p = coef;
ans = x + *p++;
i = N - 1;

do {
ans = ans * x + *p++;
} while (--i);

return (ans);
}


/* Inverse of Normal distribution function
*
* SYNOPSIS:
*
* double x, y, ndtri();
*
* x = ndtri( y );
*
* domain: 0 < y < 1
*
*
*
* 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( -2.0 * log(y) ); then the approximation is
* x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
* There are two rational functions P/Q, one for 0 < y < exp(-32)
* and the other for y up to exp(-2). For larger arguments,
* w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0.125, 1 20000 7.2e-16 1.3e-16
* IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17
*
*/

MSHADOW_XINLINE static double ndtri(double y0) {
assert(y0 > 0 && y0 < 1);

/* sqrt(2pi) */
double s2pi = 2.50662827463100050242E0;

/* approximation for 0 <= |y - 0.5| <= 3/8 */
double P0[5] = {
-5.99633501014107895267E1,
9.80010754185999661536E1,
-5.66762857469070293439E1,
1.39312609387279679503E1,
-1.23916583867381258016E0,
};
double Q0[8] = {
/* 1.00000000000000000000E0, */
1.95448858338141759834E0,
4.67627912898881538453E0,
8.63602421390890590575E1,
-2.25462687854119370527E2,
2.00260212380060660359E2,
-8.20372256168333339912E1,
1.59056225126211695515E1,
-1.18331621121330003142E0,
};

/* Approximation for interval z = sqrt(-2 log y ) between 2 and 8
* i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
*/
double P1[9] = {
4.05544892305962419923E0,
3.15251094599893866154E1,
5.71628192246421288162E1,
4.40805073893200834700E1,
1.46849561928858024014E1,
2.18663306850790267539E0,
-1.40256079171354495875E-1,
-3.50424626827848203418E-2,
-8.57456785154685413611E-4,
};
double Q1[8] = {
/* 1.00000000000000000000E0, */
1.57799883256466749731E1,
4.53907635128879210584E1,
4.13172038254672030440E1,
1.50425385692907503408E1,
2.50464946208309415979E0,
-1.42182922854787788574E-1,
-3.80806407691578277194E-2,
-9.33259480895457427372E-4,
};

/* Approximation for interval z = sqrt(-2 log y ) between 8 and 64
* i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
*/
double P2[9] = {
3.23774891776946035970E0,
6.91522889068984211695E0,
3.93881025292474443415E0,
1.33303460815807542389E0,
2.01485389549179081538E-1,
1.23716634817820021358E-2,
3.01581553508235416007E-4,
2.65806974686737550832E-6,
6.23974539184983293730E-9,
};
double Q2[8] = {
/* 1.00000000000000000000E0, */
6.02427039364742014255E0,
3.67983563856160859403E0,
1.37702099489081330271E0,
2.16236993594496635890E-1,
1.34204006088543189037E-2,
3.28014464682127739104E-4,
2.89247864745380683936E-6,
6.79019408009981274425E-9,
};

double x, y, z, y2, x0, x1;
bool code = true;
y = y0;
if (y > (1.0 - 0.13533528323661269189)) { /* 0.135... = exp(-2) */
y = 1.0 - y;
code = false;
}

if (y > 0.13533528323661269189) {
y = y - 0.5;
y2 = y * y;
x = y + y * (y2 * polevl(y2, P0, 4) / p1evl(y2, Q0, 8));
x = x * s2pi;
return (x);
}

x = sqrt(-2.0 * log(y));
x0 = x - log(x) / x;

z = 1.0 / x;
if (x < 8.0) { /* y > exp(-32) = 1.2664165549e-14 */
x1 = z * polevl(z, P1, 8) / p1evl(z, Q1, 8);
} else {
x1 = z * polevl(z, P2, 8) / p1evl(z, Q2, 8);
}

x = x0 - x1;
if (code) {
x = -x;
}
return (x);
}


/*! \brief inverse of the error function */
struct erfinv : public mxnet_op::tunable {
template<typename DType>
MSHADOW_XINLINE static DType Map(DType v) {
/* Function to calculate inverse error function. Rational approximation
is used to generate an initial approximation, which is then improved to
full accuracy by two steps of Newton's method. Code is a direct
translation of the erfinv m file in matlab version 2.0.
Author: Gary L. Pavlis, Indiana University
Date: February 1996
*/
const double central_range = 0.7;
/* Inverse of the error function.
* Computes the inverse of the error function on the restricted domain
* -1 < y < 1. This restriction ensures the existence of a unique result
* such that erf(erfinv(y)) = y.
*/
const double domain_lb = -1;
const double domain_ub = 1;

const double thresh = 1e-7;
double y = static_cast<double>(v);
double y_fab = std::fabs(y);
/*working variables */
double x = 0.0;
double z, num, dem;
/* coefficients in rational expansion */
double a[4]={ 0.886226899, -1.645349621, 0.914624893, -0.140543331};
double b[4]={-2.118377725, 1.442710462, -0.329097515, 0.012229801};
double c[4]={-1.970840454, -1.624906493, 3.429567803, 1.641345311};
double d[2]={ 3.543889200, 1.637067800};
if (y_fab > 1.0) {
/* This needs IEEE constant*/
return DType(std::numeric_limits<double>::quiet_NaN());
} else if (y_fab == 1.0) {
return DType((std::copysign(1.0, y))*std::numeric_limits<double>::infinity());
} else if (y_fab <= central_range) {
z = y*y;
num = (((a[3]*z + a[2])*z + a[1])*z + a[0]);
dem = ((((b[3]*z + b[2])*z + b[1])*z +b[0])*z + 1.0);
x = y*num/dem;
} else {
z = std::sqrt(-std::log((1.0-y_fab)/2.0));
num = ((c[3]*z + c[2])*z + c[1])*z + c[0];
dem = (d[1]*z + d[0])*z + 1.0;
x = (std::copysign(1.0, y))*num/dem;

/*
* For small arguments, use the Taylor expansion
* erf(y) = 2/\sqrt{\pi} (y - y^3 / 3 + O(y^5)), y\to 0
* where we only retain the linear term.
* Otherwise, y + 1 loses precision for |y| << 1.
*/
if ((-thresh < y) && (y < thresh)) {
return DType(y / M_2_SQRTPI);
}
/* Two steps of Newton-Raphson correction */
x = x - (std::erf(x) - y)/((2.0/std::sqrt(M_PI))*std::exp(-x*x));
x = x - (std::erf(x) - y)/((2.0/std::sqrt(M_PI))*std::exp(-x*x));

return DType(x);
if ((domain_lb < y) && (y < domain_ub)) {
return DType(ndtri(0.5 * (y+1)) * M_SQRT1_2);
} else if (y == domain_lb || y == domain_ub) {
return DType(std::copysign(1.0, y) * std::numeric_limits<double>::infinity());
} else {
return DType(std::numeric_limits<double>::quiet_NaN());
}
}
};

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