Error Function Inverses

Synopsis

#include <boost/math/special_functions/erf.hpp>

namespace boost{ namespace math{

template <class T>
calculated-result-type erf_inv(T p);

template <class T, class Policy>
calculated-result-type erf_inv(T p, const Policy&);

template <class T>
calculated-result-type erfc_inv(T p);

template <class T, class Policy>
calculated-result-type erfc_inv(T p, const Policy&);

}} // namespaces

The return type of these functions is computed using the result type calculation rules: the return type is double if T is an integer type, and T otherwise.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

Description

template <class T>
calculated-result-type erf_inv(T z);

template <class T, class Policy>
calculated-result-type erf_inv(T z, const Policy&);

Returns the inverse error function of z, that is a value x such that:

p = erf(x);

template <class T>
calculated-result-type erfc_inv(T z);

template <class T, class Policy>
calculated-result-type erfc_inv(T z, const Policy&);

Returns the inverse of the complement of the error function of z, that is a value x such that:

p = erfc(x);

Accuracy

For types up to and including 80-bit long doubles the approximations used are accurate to less than ~ 2 epsilon. For higher precision types these functions have the same accuracy as the forward error functions.

Table 8.30. Error rates for erf_inv

	GNU C++ version 7.1.0 linux double	GNU C++ version 7.1.0 linux long double	Sun compiler version 0x5150 Sun Solaris long double	Microsoft Visual C++ version 14.1 Win32 double
Inverse Erf Function	Max = 0ε (Mean = 0ε)	Max = 0.996ε (Mean = 0.389ε)	Max = 1.08ε (Mean = 0.395ε)	Max = 1.09ε (Mean = 0.502ε)

Table 8.31. Error rates for erfc_inv

	GNU C++ version 7.1.0 linux double	GNU C++ version 7.1.0 linux long double	Sun compiler version 0x5150 Sun Solaris long double	Microsoft Visual C++ version 14.1 Win32 double
Inverse Erfc Function	Max = 0ε (Mean = 0ε)	Max = 0.996ε (Mean = 0.397ε)	Max = 1.08ε (Mean = 0.403ε)	Max = 1ε (Mean = 0.491ε)
Inverse Erfc Function: extreme values		Max = 1.62ε (Mean = 0.383ε)	Max = 1.62ε (Mean = 0.383ε)

The following error plot are based on an exhaustive search of the functions domain, MSVC-15.5 at double precision, and GCC-7.1/Ubuntu for long double and __float128.

Testing

There are two sets of tests:

Basic sanity checks attempt to "round-trip" from x to p and back again. These tests have quite generous tolerances: in general both the error functions and their inverses change so rapidly in some places that round tripping to more than a couple of significant digits isn't possible. This is especially true when p is very near one: in this case there isn't enough "information content" in the input to the inverse function to get back where you started.
Accuracy checks using high-precision test values. These measure the accuracy of the result, given exact input values.

Implementation

These functions use a rational approximation devised by JM to calculate an initial approximation to the result that is accurate to ~10^-19, then only if that has insufficient accuracy compared to the epsilon for T, do we clean up the result using Halley iteration.

Constructing rational approximations to the erf/erfc functions is actually surprisingly hard, especially at high precision. For this reason no attempt has been made to achieve 10^-34 accuracy suitable for use with 128-bit reals.

In the following discussion, p is the value passed to erf_inv, and q is the value passed to erfc_inv, so that p = 1 - q and q = 1 - p and in both cases we want to solve for the same result x.

For p < 0.5 the inverse erf function is reasonably smooth and the approximation:

x = p(p + 10)(Y + R(p))

Gives a good result for a constant Y, and R(p) optimised for low absolute error compared to |Y|.

For q < 0.5 things get trickier, over the interval 0.5 > q > 0.25 the following approximation works well:

x = sqrt(-2log(q)) / (Y + R(q))

While for q < 0.25, let

z = sqrt(-log(q))

Then the result is given by:

x = z(Y + R(z - B))

As before Y is a constant and the rational function R is optimised for low absolute error compared to |Y|. B is also a constant: it is the smallest value of z for which each approximation is valid. There are several approximations of this form each of which reaches a little further into the tail of the erfc function (at long double precision the extended exponent range compared to double means that the tail goes on for a very long way indeed).