Class BoostErf
This code has been adapted from the Boost
c++
implementation <boost/math/special_functions/erf.hpp>
.
The erf/erfc functions and their inverses are copyright John Maddock 2006 and subject to
the Boost Software License.
Additions made to support the erfcx function are original work under the Apache software license.
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Field Summary
FieldsModifier and TypeFieldDescriptionprivate static final double
Threshold for the erf implementation for |x| where the computation useserf(x)
; otherwiseerfc(x)
is computed.private static final double
Threshold for the scaled complementary error function erfcx where the approximation(1 / sqrt(pi)) / x
can be used.private static final double
Threshold for the scaled complementary error function erfcx for negative x where2 * exp(x*x)
will overflow.private static final double
Threshold for the scaled complementary error function erfcx for x whereexp(x*x) == 1; x <= t
.private static final double
The multiplier used to split the double value into high and low parts.private static final double
1 / sqrt(pi). -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescription(package private) static double
erf
(double x) Returns the error function.(package private) static double
erfc
(double x) Returns the complementary error function.(package private) static double
erfcInv
(double z) Returns the inverse complementary error function.(package private) static double
erfcx
(double x) Returns the scaled complementary error function.private static double
erfImp
(double z, boolean invert, boolean scaled) 53-bit implementation for the error function.(package private) static double
erfInv
(double z) Returns the inverse error function.private static double
erfInvImp
(double p, double q) Common implementation for inverse erf and erfc functions.(package private) static double
expmxx
(double x) Computeexp(-x*x)
with high accuracy.(package private) static double
expxx
(double x) Computeexp(x*x)
with high accuracy.private static double
expxx
(double a, double b) Computeexp(a+b)
with high accuracy assuminga+b = a
.private static double
highPartUnscaled
(double value) Implement Dekker's method to split a value into two parts.private static double
squareLow
(double hx, double lx, double xx) Compute the low part of the double length number(z,zz)
for the exact square ofx
using Dekker's mult12 algorithm.private static double
squareLowUnscaled
(double x, double xx) Compute the low part of the double length number(z,zz)
for the exact square ofx
using Dekker's mult12 algorithm.
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Field Details
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MULTIPLIER
private static final double MULTIPLIERThe multiplier used to split the double value into high and low parts. From Dekker (1971): "The constant should be chosen equal to 2^(p - p/2) + 1, where p is the number of binary digits in the mantissa". Here p is 53 and the multiplier is2^27 + 1
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ONE_OVER_ROOT_PI
private static final double ONE_OVER_ROOT_PI1 / sqrt(pi). Used for the scaled complementary error function erfcx.- See Also:
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ERFCX_APPROX
private static final double ERFCX_APPROXThreshold for the scaled complementary error function erfcx where the approximation(1 / sqrt(pi)) / x
can be used.- See Also:
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COMPUTE_ERF
private static final double COMPUTE_ERFThreshold for the erf implementation for |x| where the computation useserf(x)
; otherwiseerfc(x)
is computed. The final result is achieved by suitable application of symmetry.- See Also:
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ERFCX_NEG_X_MAX
private static final double ERFCX_NEG_X_MAXThreshold for the scaled complementary error function erfcx for negative x where2 * exp(x*x)
will overflow. Value is 26.62873571375149. -
EXP_XX_1
private static final double EXP_XX_1Threshold for the scaled complementary error function erfcx for x whereexp(x*x) == 1; x <= t
. Value is (1 + 5/16) * 2^-27 = 9.778887033462524E-9.Note: This is used for performance. If set to 0 then the result is computed using expm1(x*x) with the same final result.
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Constructor Details
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BoostErf
private BoostErf()Private constructor.
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Method Details
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erfc
static double erfc(double x) Returns the complementary error function.- Parameters:
x
- the value.- Returns:
- the complementary error function.
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erf
static double erf(double x) Returns the error function.- Parameters:
x
- the value.- Returns:
- the error function.
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erfImp
private static double erfImp(double z, boolean invert, boolean scaled) 53-bit implementation for the error function.Note: The
scaled
flag only applies whenz >= 0.5
andinvert == true
. This functionality is used to compute erfcx(z) for positive z.- Parameters:
z
- Point to evaluateinvert
- true to invert the result (for the complementary error function)scaled
- true to compute the scaled complementary error function- Returns:
- the error function result
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erfcx
static double erfcx(double x) Returns the scaled complementary error function.erfcx(x) = exp(x^2) * erfc(x)
- Parameters:
x
- the value.- Returns:
- the scaled complementary error function.
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erfcInv
static double erfcInv(double z) Returns the inverse complementary error function.- Parameters:
z
- Value (in[0, 2]
).- Returns:
- t such that
z = erfc(t)
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erfInv
static double erfInv(double z) Returns the inverse error function.- Parameters:
z
- Value (in[-1, 1]
).- Returns:
- t such that
z = erf(t)
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erfInvImp
private static double erfInvImp(double p, double q) Common implementation for inverse erf and erfc functions.- Parameters:
p
- P-valueq
- Q-value (1-p)- Returns:
- the inverse
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expxx
static double expxx(double x) Computeexp(x*x)
with high accuracy. This is performed using information in the round-off fromx*x
.This is accurate at large x to 1 ulp.
At small x the accuracy cannot be improved over using exp(x*x). This occurs at
x <= 1
.Warning: This has no checks for overflow. The method is never called when
x*x > log(MAX_VALUE/2)
.- Parameters:
x
- Value- Returns:
- exp(x*x)
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expmxx
static double expmxx(double x) Computeexp(-x*x)
with high accuracy. This is performed using information in the round-off fromx*x
.This is accurate at large x to 1 ulp until exp(-x*x) is close to sub-normal. For very small exp(-x*x) the adjustment is sub-normal and bits can be lost in the adjustment for a max observed error of
< 2
ulp.At small x the accuracy cannot be improved over using exp(-x*x). This occurs at
x <= 1
.- Parameters:
x
- Value- Returns:
- exp(-x*x)
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expxx
private static double expxx(double a, double b) Computeexp(a+b)
with high accuracy assuminga+b = a
.This is accurate at large positive a to 1 ulp. If a is negative and exp(a) is close to sub-normal a bit of precision may be lost when adjusting result as the adjustment is sub-normal (max observed error
< 2
ulp). For the use case of multiplication of a number less than 1 by exp(-x*x), a = -x*x, the result will be sub-normal and the rounding error is lost.At small |a| the accuracy cannot be improved over using exp(a) as the round-off is too small to create terms that can adjust the standard result by more than 0.5 ulp. This occurs at
|a| <= 1
.- Parameters:
a
- High bits of a split numberb
- Low bits of a split number- Returns:
- exp(a+b)
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squareLowUnscaled
private static double squareLowUnscaled(double x, double xx) Compute the low part of the double length number(z,zz)
for the exact square ofx
using Dekker's mult12 algorithm. The standard precision productx*x
must be provided. The numberx
is split into high and low parts using Dekker's algorithm.Warning: This method does not perform scaling in Dekker's split and large finite numbers can create NaN results.
- Parameters:
x
- Number to squarexx
- Standard precision productx*x
- Returns:
- the low part of the square double length number
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highPartUnscaled
private static double highPartUnscaled(double value) Implement Dekker's method to split a value into two parts. Multiplying by (2^s + 1) creates a big value from which to derive the two split parts.c = (2^s + 1) * a a_big = c - a a_hi = c - a_big a_lo = a - a_hi a = a_hi + a_lo
The multiplicand allows a p-bit value to be split into (p-s)-bit value
a_hi
and a non-overlapping (s-1)-bit valuea_lo
. Combined they have (p-1) bits of significand but the sign bit ofa_lo
contains a bit of information. The constant is chosen so that s is ceil(p/2) where the precision p for a double is 53-bits (1-bit of the mantissa is assumed to be 1 for a non sub-normal number) and s is 27.This conversion does not use scaling and the result of overflow is NaN. Overflow may occur when the exponent of the input value is above 996.
Splitting a NaN or infinite value will return NaN.
- Parameters:
value
- Value.- Returns:
- the high part of the value.
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squareLow
private static double squareLow(double hx, double lx, double xx) Compute the low part of the double length number(z,zz)
for the exact square ofx
using Dekker's mult12 algorithm. The standard precision productx*x
must be provided. The numberx
should already be split into low and high parts.Note: This uses the high part of the result
(z,zz)
asx * x
and nothx * hx + hx * lx + lx * hx
as specified in Dekker's original paper. See Shewchuk (1997) for working examples.- Parameters:
hx
- High part of factor.lx
- Low part of factor.xx
- Square of the factor.- Returns:
lx * ly - (((xy - hx * hy) - lx * hy) - hx * ly)
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