Skip to content

Commit

Permalink
Merge pull request #2987 from stevengj/erfinv
Browse files Browse the repository at this point in the history 
add inverse error functions erfinv and erfcinv
  • Loading branch information
@JeffBezanson
JeffBezanson committed May 1, 2013
2 parents 6b53b1b + 9c24795 commit ccceddc
Show file tree
Hide file tree
Showing 5 changed files with 253 additions and 1 deletion.
2 changes: 2 additions & 0 deletions base/exports.jl
View file
Original file line number Diff line number Diff line change
Expand Up @@ -318,6 +318,8 @@ export
erfc,
erfcx,
erfi,
erfinv,
erfcinv,
exp,
exp2,
expm1,
Expand Down
213 changes: 212 additions & 1 deletion base/math.jl
View file
Original file line number Diff line number Diff line change
Expand Up @@ -14,7 +14,8 @@ export sin, cos, tan, sinh, cosh, tanh, asin, acos, atan,
airy, airyai, airyprime, airyaiprime, airybi, airybiprime,
besselj0, besselj1, besselj, bessely0, bessely1, bessely,
hankelh1, hankelh2, besseli, besselk, besselh,
beta, lbeta, eta, zeta, digamma
beta, lbeta, eta, zeta, digamma,
erfinv, erfcinv

import Base.log, Base.exp, Base.sin, Base.cos, Base.tan, Base.sinh, Base.cosh,
Base.tanh, Base.asin, Base.acos, Base.atan, Base.asinh, Base.acosh,
Expand Down Expand Up @@ -695,4 +696,214 @@ for f in (:erfcx, :erfi, :Dawson)
end
end

# evaluate p[1] + x * (p[2] + x * (....)), i.e. a polynomial via Horner's rule
macro horner(x, p...)
ex = p[end]
for i = length(p)-1:-1:1
ex = :($(p[i]) + $x * $ex)
end
ex
end

# Compute the inverse of the error function: erf(erfinv(x)) == x,
# using the rational approximants tabulated in:
# J. M. Blair, C. A. Edwards, and J. H. Johnson, "Rational Chebyshev
# approximations for the inverse of the error function," Math. Comp. 30,
# pp. 827--830 (1976).
# http://dx.doi.org/10.1090/S0025-5718-1976-0421040-7
# http://www.jstor.org/stable/2005402
function erfinv(x::Float64)
a = abs(x)
if a >= 1.0
if x == 1.0
return inf(Float64)
elseif x == -1.0
return -inf(Float64)
end
throw(DomainError())
elseif a <= 0.75 # Table 17 in Blair et al.
t = x*x - 0.5625
return x * @horner(t, 0.16030_49558_44066_229311e2,
-0.90784_95926_29603_26650e2,
0.18644_91486_16209_87391e3,
-0.16900_14273_46423_82420e3,
0.65454_66284_79448_7048e2,
-0.86421_30115_87247_794e1,
0.17605_87821_39059_0) /
@horner(t, 0.14780_64707_15138_316110e2,
-0.91374_16702_42603_13936e2,
0.21015_79048_62053_17714e3,
-0.22210_25412_18551_32366e3,
0.10760_45391_60551_23830e3,
-0.20601_07303_28265_443e2,
0.1e1)
elseif a <= 0.9375 # Table 37 in Blair et al.
t = x*x - 0.87890625
return x * @horner(t, -0.15238_92634_40726_128e-1,
0.34445_56924_13612_5216,
-0.29344_39867_25424_78687e1,
0.11763_50570_52178_27302e2,
-0.22655_29282_31011_04193e2,
0.19121_33439_65803_30163e2,
-0.54789_27619_59831_8769e1,
0.23751_66890_24448) /
@horner(t, -0.10846_51696_02059_954e-1,
0.26106_28885_84307_8511,
-0.24068_31810_43937_57995e1,
0.10695_12997_33870_14469e2,
-0.23716_71552_15965_81025e2,
0.24640_15894_39172_84883e2,
-0.10014_37634_97830_70835e2,
0.1e1)
else # Table 57 in Blair et al.
t = 1.0 / sqrt(-log(1.0 - a))
return @horner(t, 0.10501_31152_37334_38116e-3,
0.10532_61131_42333_38164_25e-1,
0.26987_80273_62432_83544_516,
0.23268_69578_89196_90806_414e1,
0.71678_54794_91079_96810_001e1,
0.85475_61182_21678_27825_185e1,
0.68738_08807_35438_39802_913e1,
0.36270_02483_09587_08930_02e1,
0.88606_27392_96515_46814_9) /
(copysign(t, x) *
@horner(t, 0.10501_26668_70303_37690e-3,
0.10532_86230_09333_27531_11e-1,
0.27019_86237_37515_54845_553,
0.23501_43639_79702_53259_123e1,
0.76078_02878_58012_77064_351e1,
0.11181_58610_40569_07827_3451e2,
0.11948_78791_84353_96667_8438e2,
0.81922_40974_72699_07893_913e1,
0.40993_87907_63680_15361_45e1,
0.1e1))
end
end

function erfinv(x::Float32)
a = abs(x)
if a >= 1.0f0
if x == 1.0f0
return inf(Float64)
elseif x == -1.0f0
return -inf(Float64)
end
throw(DomainError())
elseif a <= 0.75f0 # Table 10 in Blair et al.
t = x*x - 0.5625f0
return x * @horner(t, -0.13095_99674_22f2,
0.26785_22576_0f2,
-0.92890_57365f1) /
@horner(t, -0.12074_94262_97f2,
0.30960_61452_9f2,
-0.17149_97799_1f2,
0.1f1)
elseif a <= 0.9375f0 # Table 29 in Blair et al.
t = x*x - 0.87890625f0
return x * @horner(t, -0.12402_56522_1f0,
0.10688_05957_4f1,
-0.19594_55607_8f1,
0.42305_81357f0) /
@horner(t, -0.88276_97997f-1,
0.89007_43359f0,
-0.21757_03119_6f1,
0.1f1)
else # Table 50 in Blair et al.
t = 1.0f0 / sqrt(-log(1.0f0 - a))
return @horner(t, 0.15504_70003_116f0,
0.13827_19649_631f1,
0.69096_93488_87f0,
-0.11280_81391_617f1,
0.68054_42468_25f0,
-0.16444_15679_1f0) /
(copysign(t, x) *
@horner(t, 0.15502_48498_22f0,
0.13852_28141_995f1,
0.1f1))
end
end

erfinv(x::Integer) = erfinv(float(x))
@vectorize_1arg Real erfinv

# Inverse complementary error function: use Blair tables for y = 1-x,
# exploiting the greater accuracy of y (vs. x) when y is small.
function erfcinv(y::Float64)
if y > 0.0625
return erfinv(1.0 - y)
elseif y <= 0.0
if y == 0.0
return inf(Float64)
end
throw(DomainError())
elseif y >= 1e-100 # Table 57 in Blair et al.
t = 1.0 / sqrt(-log(y))
return @horner(t, 0.10501_31152_37334_38116e-3,
0.10532_61131_42333_38164_25e-1,
0.26987_80273_62432_83544_516,
0.23268_69578_89196_90806_414e1,
0.71678_54794_91079_96810_001e1,
0.85475_61182_21678_27825_185e1,
0.68738_08807_35438_39802_913e1,
0.36270_02483_09587_08930_02e1,
0.88606_27392_96515_46814_9) /
(t *
@horner(t, 0.10501_26668_70303_37690e-3,
0.10532_86230_09333_27531_11e-1,
0.27019_86237_37515_54845_553,
0.23501_43639_79702_53259_123e1,
0.76078_02878_58012_77064_351e1,
0.11181_58610_40569_07827_3451e2,
0.11948_78791_84353_96667_8438e2,
0.81922_40974_72699_07893_913e1,
0.40993_87907_63680_15361_45e1,
0.1e1))
else # Table 80 in Blair et al.
t = 1.0 / sqrt(-log(y))
return @horner(t, 0.34654_29858_80863_50177e-9,
0.25084_67920_24075_70520_55e-6,
0.47378_13196_37286_02986_534e-4,
0.31312_60375_97786_96408_3388e-2,
0.77948_76454_41435_36994_854e-1,
0.70045_68123_35816_43868_271e0,
0.18710_42034_21679_31668_683e1,
0.71452_54774_31351_45428_3e0) /
(t * @horner(t, 0.34654_29567_31595_11156e-9,
0.25084_69079_75880_27114_87e-6,
0.47379_53129_59749_13536_339e-4,
0.31320_63536_46177_68848_0813e-2,
0.78073_48906_27648_97214_733e-1,
0.70715_04479_95337_58619_993e0,
0.19998_51543_49112_15105_214e1,
0.15072_90269_27316_80008_56e1,
0.1e1))
end
end

function erfcinv(y::Float32)
if y > 0.0625f0
return erfinv(1.0f0 - y)
elseif y <= 0.0f0
if y == 0.0f0
return inf(Float32)
end
throw(DomainError())
else # Table 50 in Blair et al.
t = 1.0f0 / sqrt(-log(y))
return @horner(t, 0.15504_70003_116f0,
0.13827_19649_631f1,
0.69096_93488_87f0,
-0.11280_81391_617f1,
0.68054_42468_25f0,
-0.16444_15679_1f0) /
(t *
@horner(t, 0.15502_48498_22f0,
0.13852_28141_995f1,
0.1f1))
end
end

erfcinv(x::Integer) = erfcinv(float(x))
@vectorize_1arg Real erfcinv

end # module
16 changes: 16 additions & 0 deletions doc/helpdb.jl
View file
Original file line number Diff line number Diff line change
Expand Up @@ -2695,6 +2695,22 @@ FloatingPoint
"),

("Mathematical Functions","Base","erfinv","erfinv(x)
Compute the inverse error function of a real \"x\",
so that \\operatorname{erf}(\\operatorname{erfinv}(x)) = x.
"),

("Mathematical Functions","Base","erfcinv","erfcinv(x)
Compute the inverse complementary error function of a real \"x\",
so that \\operatorname{erfc}(\\operatorname{erfcinv}(x)) = x.
"),

("Mathematical Functions","Base","dawson","dawson(x)
Expand Down
10 changes: 10 additions & 0 deletions doc/stdlib/base.rst
View file
Original file line number Diff line number Diff line change
Expand Up @@ -1527,6 +1527,16 @@ Mathematical Functions
Compute the Dawson function (scaled imaginary error function) of ``x``,
defined by :math:`\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)`.

.. function:: erfinv(x)

Compute the inverse error function of a real ``x``,
defined by :math:`\operatorname{erf}(\operatorname{erfinv}(x)) = x`.

.. function:: erfcinv(x)

Compute the inverse error complementary function of a real ``x``,
defined by :math:`\operatorname{erfc}(\operatorname{erfcinv}(x)) = x`.

.. function:: real(z)

Return the real part of the complex number ``z``
Expand Down
13 changes: 13 additions & 0 deletions test/math.jl
View file
Original file line number Diff line number Diff line change
Expand Up @@ -9,6 +9,8 @@
@test_approx_eq erfc(1) 0.15729920705028513066
@test_approx_eq erfcx(1) 0.42758357615580700442
@test_approx_eq erfi(1) 1.6504257587975428760
@test_approx_eq erfinv(0.84270079294971486934) 1
@test_approx_eq erfcinv(0.15729920705028513066) 1
@test_approx_eq dawson(1) 0.53807950691276841914
# TODO: complex versions only supported on 64-bit for now
if WORD_SIZE==64
Expand All @@ -18,6 +20,17 @@ if WORD_SIZE==64
@test_approx_eq erfi(1+2im) -0.011259006028815025076+1.0036063427256517509im
@test_approx_eq dawson(1+2im) -13.388927316482919244-11.828715103889593303im
end
for x in logspace(-200, -0.01)
@test_approx_eq_eps erf(erfinv(x)) x 1e-12*x
@test_approx_eq_eps erf(erfinv(-x)) -x 1e-12*x
@test_approx_eq_eps erfc(erfcinv(2*x)) 2*x 1e-12*x
if x > 1e-20
xf = float32(x)
@test_approx_eq_eps erf(erfinv(xf)) xf 1e-5*xf
@test_approx_eq_eps erf(erfinv(-xf)) -xf 1e-5*xf
@test_approx_eq_eps erfc(erfcinv(2xf)) 2xf 1e-5*xf
end
end

# airy
@test_approx_eq airy(1.8) 0.0470362168668458052247
Expand Down

0 comments on commit ccceddc

Please sign in to comment.