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Merged
syclik merged 9 commits into from
Dec 5, 2013
Merged

Bugfix/issue 376 phi tails #430

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e04daa0
Merge branch 'develop' of https://github.com/stan-dev/stan into develop
Oct 25, 2013
8420773
Merge branch 'develop' of https://github.com/stan-dev/stan into develop
Oct 29, 2013
9874cc9
Merge branch 'develop' of https://github.com/stan-dev/stan into develop
Nov 4, 2013
7169dac
Merge branch 'develop' of https://github.com/stan-dev/stan into develop
Nov 11, 2013
3839df4
Merge branch 'develop' of https://github.com/stan-dev/stan into develop
Nov 16, 2013
a2c73a2
Merge branch 'develop' of https://github.com/stan-dev/stan into develop
Nov 20, 2013
811b78c
Merge branch 'develop' of https://github.com/stan-dev/stan into develop
Nov 28, 2013
856788a
added switch to math::Phi to use erfc to extend negative domain, no c…
Dec 4, 2013
2b4936f
added tests for derivative overflow/underflow and doc about overflow/…
Dec 4, 2013
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6 changes: 5 additions & 1 deletion src/docs/stan-reference/functions.tex
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Expand Up @@ -616,7 +616,11 @@ \section{Probability-Related Functions}\label{Phi-function.section}
The complementary error function of \farg{x}}
%
\fitemindexsort{real}{Phi}{real \farg{x}}{
The cumulative unit normal density function of \farg{x}}{Phi}{phi}
The cumulative unit normal density function of \farg{x};
\code{Phi(\farg{x})} will underflow to 0 for \farg{x} below -37.5 and overflow
to 1 for \farg{x} above 8.25; derivatives will underflow to 0 below
-27.5 and overflow to 1 above 27.5.
}{Phi}{phi}
%
\fitemindexsort{real}{Phi\_approx}{real \farg{x}}{
Fast approximation of the cumulative unit normal density
Expand Down
13 changes: 12 additions & 1 deletion src/stan/math/functions/Phi.hpp
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Expand Up @@ -20,13 +20,24 @@ namespace stan {
* This function can be used to implement the inverse link function
* for probit regression.
*
* Phi will underflow to 0 below -37.5 and overflow to 1 above 8
*
* @param x Argument.
* @return Probability random sample is less than or equal to argument.
*/
template <typename T>
inline typename boost::math::tools::promote_args<T>::type
Phi(const T x) {
return 0.5 * (1.0 + boost::math::erf(INV_SQRT_2 * x));
// overridden in fvar and var, so can hard-code boost versions
// here for scalars only
if (x < -37.5)
return 0;
else if (x < -5.0)
return 0.5 * boost::math::erfc(-INV_SQRT_2 * x);
else if (x > 8.25)
return 1;
else
return 0.5 * (1.0 + boost::math::erf(INV_SQRT_2 * x));
}

}
Expand Down
131 changes: 131 additions & 0 deletions src/test/agrad/fwd/Phi_test.cpp
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Expand Up @@ -17,6 +17,18 @@ TEST(AgradFwdPhi,Fvar) {
EXPECT_FLOAT_EQ(exp(stan::prob::normal_log<false>(1.0,0.0,1.0)),
Phi_x.d_);
}
TEST(AgradFwdPhi, FvarDerivUnderOverFlow) {
using stan::agrad::fvar;
fvar<double> x = -27.5;
x.d_ = 1.0;
fvar<double> Phi_x = Phi(x);
EXPECT_FLOAT_EQ(0, Phi_x.d_);

fvar<double> y = 27.5;
y.d_ = 1.0;
fvar<double> Phi_y = Phi(y);
EXPECT_FLOAT_EQ(0, Phi_y.d_);
}
TEST(AgradFwdPhi, FvarVar_1stDeriv) {
using stan::agrad::fvar;
using stan::agrad::var;
Expand Down Expand Up @@ -154,3 +166,122 @@ TEST(AgradFwdPhi, FvarFvarVar_3rdDeriv) {
a.d_.d_.grad(p,g);
EXPECT_FLOAT_EQ(0, g[0]);
}

double fun_value_of(double x) { return x; }
// inefficient calls by value because neither const& nor & compile
double fun_value_of(stan::agrad::fvar<double> x) {
return x.val();
}
double fun_value_of(stan::agrad::fvar<stan::agrad::fvar<double> > x) {
return x.val().val();
}

// tests calculating using R 3.0.2 Snow Leopard build (6558)
template <typename T>
void test_tails() {
using stan::agrad::Phi;

EXPECT_EQ(0, fun_value_of(Phi(T(-40))));

EXPECT_FLOAT_EQ(1, 4.60535300958196e-308 / fun_value_of(Phi(T(-37.5))));
EXPECT_FLOAT_EQ(1, 5.72557122252458e-300 / fun_value_of(Phi(T(-37))));
EXPECT_FLOAT_EQ(1, 5.54472571307484e-292 / fun_value_of(Phi(T(-36.5))));
EXPECT_FLOAT_EQ(1, 4.18262406579728e-284 / fun_value_of(Phi(T(-36))));
EXPECT_FLOAT_EQ(1, 2.45769154066194e-276 / fun_value_of(Phi(T(-35.5))));
EXPECT_FLOAT_EQ(1, 1.12491070647241e-268 / fun_value_of(Phi(T(-35))));
EXPECT_FLOAT_EQ(1, 4.01072896657726e-261 / fun_value_of(Phi(T(-34.5))));
EXPECT_FLOAT_EQ(1, 1.11389878557438e-253 / fun_value_of(Phi(T(-34))));
EXPECT_FLOAT_EQ(1, 2.40983869512039e-246 / fun_value_of(Phi(T(-33.5))));
EXPECT_FLOAT_EQ(1, 4.06118562091586e-239 / fun_value_of(Phi(T(-33))));
EXPECT_FLOAT_EQ(1, 5.33142435967881e-232 / fun_value_of(Phi(T(-32.5))));
EXPECT_FLOAT_EQ(1, 5.4520806035124e-225 / fun_value_of(Phi(T(-32))));
EXPECT_FLOAT_EQ(1, 4.34323260103177e-218 / fun_value_of(Phi(T(-31.5))));
EXPECT_FLOAT_EQ(1, 2.6952500812005e-211 / fun_value_of(Phi(T(-31))));
EXPECT_FLOAT_EQ(1, 1.30293791317808e-204 / fun_value_of(Phi(T(-30.5))));
EXPECT_FLOAT_EQ(1, 4.90671392714819e-198 / fun_value_of(Phi(T(-30))));
EXPECT_FLOAT_EQ(1, 1.43947455222918e-191 / fun_value_of(Phi(T(-29.5))));
EXPECT_FLOAT_EQ(1, 3.28978526670438e-185 / fun_value_of(Phi(T(-29))));
EXPECT_FLOAT_EQ(1, 5.85714125380634e-179 / fun_value_of(Phi(T(-28.5))));
EXPECT_FLOAT_EQ(1, 8.12386946965943e-173 / fun_value_of(Phi(T(-28))));
EXPECT_FLOAT_EQ(1, 8.77817055687808e-167 / fun_value_of(Phi(T(-27.5))));
EXPECT_FLOAT_EQ(1, 7.38948100688502e-161 / fun_value_of(Phi(T(-27))));
EXPECT_FLOAT_EQ(1, 4.84616266030332e-155 / fun_value_of(Phi(T(-26.5))));
EXPECT_FLOAT_EQ(1, 2.47606331550339e-149 / fun_value_of(Phi(T(-26))));
EXPECT_FLOAT_EQ(1, 9.85623651896393e-144 / fun_value_of(Phi(T(-25.5))));
EXPECT_FLOAT_EQ(1, 3.05669670638256e-138 / fun_value_of(Phi(T(-25))));
EXPECT_FLOAT_EQ(1, 7.38570686148941e-133 / fun_value_of(Phi(T(-24.5))));
EXPECT_FLOAT_EQ(1, 1.3903921185497e-127 / fun_value_of(Phi(T(-24))));
EXPECT_FLOAT_EQ(1, 2.03936756324998e-122 / fun_value_of(Phi(T(-23.5))));
EXPECT_FLOAT_EQ(1, 2.33063700622065e-117 / fun_value_of(Phi(T(-23))));
EXPECT_FLOAT_EQ(1, 2.07531079906636e-112 / fun_value_of(Phi(T(-22.5))));
EXPECT_FLOAT_EQ(1, 1.43989243514508e-107 / fun_value_of(Phi(T(-22))));
EXPECT_FLOAT_EQ(1, 7.78439707718263e-103 / fun_value_of(Phi(T(-21.5))));
EXPECT_FLOAT_EQ(1, 3.27927801897904e-98 / fun_value_of(Phi(T(-21))));
EXPECT_FLOAT_EQ(1, 1.0764673258791e-93 / fun_value_of(Phi(T(-20.5))));
EXPECT_FLOAT_EQ(1, 2.75362411860623e-89 / fun_value_of(Phi(T(-20))));
EXPECT_FLOAT_EQ(1, 5.48911547566041e-85 / fun_value_of(Phi(T(-19.5))));
EXPECT_FLOAT_EQ(1, 8.52722395263098e-81 / fun_value_of(Phi(T(-19))));
EXPECT_FLOAT_EQ(1, 1.03236986895633e-76 / fun_value_of(Phi(T(-18.5))));
EXPECT_FLOAT_EQ(1, 9.74094891893715e-73 / fun_value_of(Phi(T(-18))));
EXPECT_FLOAT_EQ(1, 7.16345876623504e-69 / fun_value_of(Phi(T(-17.5))));
EXPECT_FLOAT_EQ(1, 4.10599620209891e-65 / fun_value_of(Phi(T(-17))));
EXPECT_FLOAT_EQ(1, 1.83446300316473e-61 / fun_value_of(Phi(T(-16.5))));
EXPECT_FLOAT_EQ(1, 6.38875440053809e-58 / fun_value_of(Phi(T(-16))));
EXPECT_FLOAT_EQ(1, 1.73446079179387e-54 / fun_value_of(Phi(T(-15.5))));
EXPECT_FLOAT_EQ(1, 3.67096619931275e-51 / fun_value_of(Phi(T(-15))));
EXPECT_FLOAT_EQ(1, 6.05749476441522e-48 / fun_value_of(Phi(T(-14.5))));
EXPECT_FLOAT_EQ(1, 7.7935368191928e-45 / fun_value_of(Phi(T(-14))));
EXPECT_FLOAT_EQ(1, 7.81880730565789e-42 / fun_value_of(Phi(T(-13.5))));
EXPECT_FLOAT_EQ(1, 6.11716439954988e-39 / fun_value_of(Phi(T(-13))));
EXPECT_FLOAT_EQ(1, 3.73256429887771e-36 / fun_value_of(Phi(T(-12.5))));
EXPECT_FLOAT_EQ(1, 1.77648211207768e-33 / fun_value_of(Phi(T(-12))));
EXPECT_FLOAT_EQ(1, 6.59577144611367e-31 / fun_value_of(Phi(T(-11.5))));
EXPECT_FLOAT_EQ(1, 1.91065957449868e-28 / fun_value_of(Phi(T(-11))));
EXPECT_FLOAT_EQ(1, 4.31900631780923e-26 / fun_value_of(Phi(T(-10.5))));
EXPECT_FLOAT_EQ(1, 7.61985302416053e-24 / fun_value_of(Phi(T(-10))));
EXPECT_FLOAT_EQ(1, 1.04945150753626e-21 / fun_value_of(Phi(T(-9.5))));
EXPECT_FLOAT_EQ(1, 1.12858840595384e-19 / fun_value_of(Phi(T(-9))));
EXPECT_FLOAT_EQ(1, 9.47953482220332e-18 / fun_value_of(Phi(T(-8.5))));
EXPECT_FLOAT_EQ(1, 6.22096057427178e-16 / fun_value_of(Phi(T(-8))));
EXPECT_FLOAT_EQ(1, 3.1908916729109e-14 / fun_value_of(Phi(T(-7.5))));
EXPECT_FLOAT_EQ(1, 1.27981254388584e-12 / fun_value_of(Phi(T(-7))));
EXPECT_FLOAT_EQ(1, 4.01600058385912e-11 / fun_value_of(Phi(T(-6.5))));
EXPECT_FLOAT_EQ(1, 9.86587645037698e-10 / fun_value_of(Phi(T(-6))));
EXPECT_FLOAT_EQ(1, 1.89895624658877e-08 / fun_value_of(Phi(T(-5.5))));
EXPECT_FLOAT_EQ(1, 2.86651571879194e-07 / fun_value_of(Phi(T(-5))));
EXPECT_FLOAT_EQ(1, 3.39767312473006e-06 / fun_value_of(Phi(T(-4.5))));
EXPECT_FLOAT_EQ(1, 3.16712418331199e-05 / fun_value_of(Phi(T(-4))));
EXPECT_FLOAT_EQ(1, 0.000232629079035525 / fun_value_of(Phi(T(-3.5))));
EXPECT_FLOAT_EQ(1, 0.00134989803163009 / fun_value_of(Phi(T(-3))));
EXPECT_FLOAT_EQ(1, 0.00620966532577613 / fun_value_of(Phi(T(-2.5))));
EXPECT_FLOAT_EQ(1, 0.0227501319481792 / fun_value_of(Phi(T(-2))));
EXPECT_FLOAT_EQ(1, 0.0668072012688581 / fun_value_of(Phi(T(-1.5))));
EXPECT_FLOAT_EQ(1, 0.158655253931457 / fun_value_of(Phi(T(-1))));
EXPECT_FLOAT_EQ(1, 0.308537538725987 / fun_value_of(Phi(T(-0.5))));
EXPECT_FLOAT_EQ(1, 0.5 / fun_value_of(Phi(T(0))));
EXPECT_FLOAT_EQ(1, 0.691462461274013 / fun_value_of(Phi(T(0.5))));
EXPECT_FLOAT_EQ(1, 0.841344746068543 / fun_value_of(Phi(T(1))));
EXPECT_FLOAT_EQ(1, 0.933192798731142 / fun_value_of(Phi(T(1.5))));
EXPECT_FLOAT_EQ(1, 0.977249868051821 / fun_value_of(Phi(T(2))));
EXPECT_FLOAT_EQ(1, 0.993790334674224 / fun_value_of(Phi(T(2.5))));
EXPECT_FLOAT_EQ(1, 0.99865010196837 / fun_value_of(Phi(T(3))));
EXPECT_FLOAT_EQ(1, 0.999767370920964 / fun_value_of(Phi(T(3.5))));
EXPECT_FLOAT_EQ(1, 0.999968328758167 / fun_value_of(Phi(T(4))));
EXPECT_FLOAT_EQ(1, 0.999996602326875 / fun_value_of(Phi(T(4.5))));
EXPECT_FLOAT_EQ(1, 0.999999713348428 / fun_value_of(Phi(T(5))));
EXPECT_FLOAT_EQ(1, 0.999999981010438 / fun_value_of(Phi(T(5.5))));
EXPECT_FLOAT_EQ(1, 0.999999999013412 / fun_value_of(Phi(T(6))));
EXPECT_FLOAT_EQ(1, 0.99999999995984 / fun_value_of(Phi(T(6.5))));
EXPECT_FLOAT_EQ(1, 0.99999999999872 / fun_value_of(Phi(T(7))));
EXPECT_FLOAT_EQ(1, 0.999999999999968 / fun_value_of(Phi(T(7.5))));
EXPECT_FLOAT_EQ(1, 0.999999999999999 / fun_value_of(Phi(T(8))));
EXPECT_FLOAT_EQ(1, 1 / fun_value_of(Phi(T(8.5))));
EXPECT_FLOAT_EQ(1, 1 / fun_value_of(Phi(T(9))));
EXPECT_FLOAT_EQ(1, 1 / fun_value_of(Phi(T(9.5))));
EXPECT_FLOAT_EQ(1, 1 / fun_value_of(Phi(T(10))));
}
TEST(AgradRev, PhiTails) {
using stan::agrad::fvar;
test_tails<fvar<double> >();
test_tails<fvar<fvar<double> > >();
}
109 changes: 109 additions & 0 deletions src/test/agrad/rev/Phi_test.cpp
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Expand Up @@ -10,12 +10,16 @@ TEST(AgradRev, Phi) {
y_values.push_back(0.0);
y_values.push_back(0.9);
y_values.push_back(-5.0);
y_values.push_back(-27.5);
y_values.push_back(27.5);

// d/dy = exp(normal_log(value_of(y), 0.0, 1.0))
std::vector<double> dy_values;
dy_values.push_back(0.3989423);
dy_values.push_back(0.2660852);
dy_values.push_back(1.4867195e-06);
dy_values.push_back(0);
dy_values.push_back(0);

for (size_t i = 0; i < y_values.size(); i++) {
var y, phi_y;
Expand All @@ -30,3 +34,108 @@ TEST(AgradRev, Phi) {
<< "y = " << y;
}
}

// tests calculating using R 3.0.2 Snow Leopard build (6558)
TEST(AgradRev, PhiTails) {
using stan::agrad::Phi;
using stan::agrad::var;

EXPECT_EQ(0, Phi(var(-40)).val());

EXPECT_FLOAT_EQ(1, 4.60535300958196e-308 / Phi(var(-37.5)).val());
EXPECT_FLOAT_EQ(1, 5.72557122252458e-300 / Phi(var(-37)).val());
EXPECT_FLOAT_EQ(1, 5.54472571307484e-292 / Phi(var(-36.5)).val());
EXPECT_FLOAT_EQ(1, 4.18262406579728e-284 / Phi(var(-36)).val());
EXPECT_FLOAT_EQ(1, 2.45769154066194e-276 / Phi(var(-35.5)).val());
EXPECT_FLOAT_EQ(1, 1.12491070647241e-268 / Phi(var(-35)).val());
EXPECT_FLOAT_EQ(1, 4.01072896657726e-261 / Phi(var(-34.5)).val());
EXPECT_FLOAT_EQ(1, 1.11389878557438e-253 / Phi(var(-34)).val());
EXPECT_FLOAT_EQ(1, 2.40983869512039e-246 / Phi(var(-33.5)).val());
EXPECT_FLOAT_EQ(1, 4.06118562091586e-239 / Phi(var(-33)).val());
EXPECT_FLOAT_EQ(1, 5.33142435967881e-232 / Phi(var(-32.5)).val());
EXPECT_FLOAT_EQ(1, 5.4520806035124e-225 / Phi(var(-32)).val());
EXPECT_FLOAT_EQ(1, 4.34323260103177e-218 / Phi(var(-31.5)).val());
EXPECT_FLOAT_EQ(1, 2.6952500812005e-211 / Phi(var(-31)).val());
EXPECT_FLOAT_EQ(1, 1.30293791317808e-204 / Phi(var(-30.5)).val());
EXPECT_FLOAT_EQ(1, 4.90671392714819e-198 / Phi(var(-30)).val());
EXPECT_FLOAT_EQ(1, 1.43947455222918e-191 / Phi(var(-29.5)).val());
EXPECT_FLOAT_EQ(1, 3.28978526670438e-185 / Phi(var(-29)).val());
EXPECT_FLOAT_EQ(1, 5.85714125380634e-179 / Phi(var(-28.5)).val());
EXPECT_FLOAT_EQ(1, 8.12386946965943e-173 / Phi(var(-28)).val());
EXPECT_FLOAT_EQ(1, 8.77817055687808e-167 / Phi(var(-27.5)).val());
EXPECT_FLOAT_EQ(1, 7.38948100688502e-161 / Phi(var(-27)).val());
EXPECT_FLOAT_EQ(1, 4.84616266030332e-155 / Phi(var(-26.5)).val());
EXPECT_FLOAT_EQ(1, 2.47606331550339e-149 / Phi(var(-26)).val());
EXPECT_FLOAT_EQ(1, 9.85623651896393e-144 / Phi(var(-25.5)).val());
EXPECT_FLOAT_EQ(1, 3.05669670638256e-138 / Phi(var(-25)).val());
EXPECT_FLOAT_EQ(1, 7.38570686148941e-133 / Phi(var(-24.5)).val());
EXPECT_FLOAT_EQ(1, 1.3903921185497e-127 / Phi(var(-24)).val());
EXPECT_FLOAT_EQ(1, 2.03936756324998e-122 / Phi(var(-23.5)).val());
EXPECT_FLOAT_EQ(1, 2.33063700622065e-117 / Phi(var(-23)).val());
EXPECT_FLOAT_EQ(1, 2.07531079906636e-112 / Phi(var(-22.5)).val());
EXPECT_FLOAT_EQ(1, 1.43989243514508e-107 / Phi(var(-22)).val());
EXPECT_FLOAT_EQ(1, 7.78439707718263e-103 / Phi(var(-21.5)).val());
EXPECT_FLOAT_EQ(1, 3.27927801897904e-98 / Phi(var(-21)).val());
EXPECT_FLOAT_EQ(1, 1.0764673258791e-93 / Phi(var(-20.5)).val());
EXPECT_FLOAT_EQ(1, 2.75362411860623e-89 / Phi(var(-20)).val());
EXPECT_FLOAT_EQ(1, 5.48911547566041e-85 / Phi(var(-19.5)).val());
EXPECT_FLOAT_EQ(1, 8.52722395263098e-81 / Phi(var(-19)).val());
EXPECT_FLOAT_EQ(1, 1.03236986895633e-76 / Phi(var(-18.5)).val());
EXPECT_FLOAT_EQ(1, 9.74094891893715e-73 / Phi(var(-18)).val());
EXPECT_FLOAT_EQ(1, 7.16345876623504e-69 / Phi(var(-17.5)).val());
EXPECT_FLOAT_EQ(1, 4.10599620209891e-65 / Phi(var(-17)).val());
EXPECT_FLOAT_EQ(1, 1.83446300316473e-61 / Phi(var(-16.5)).val());
EXPECT_FLOAT_EQ(1, 6.38875440053809e-58 / Phi(var(-16)).val());
EXPECT_FLOAT_EQ(1, 1.73446079179387e-54 / Phi(var(-15.5)).val());
EXPECT_FLOAT_EQ(1, 3.67096619931275e-51 / Phi(var(-15)).val());
EXPECT_FLOAT_EQ(1, 6.05749476441522e-48 / Phi(var(-14.5)).val());
EXPECT_FLOAT_EQ(1, 7.7935368191928e-45 / Phi(var(-14)).val());
EXPECT_FLOAT_EQ(1, 7.81880730565789e-42 / Phi(var(-13.5)).val());
EXPECT_FLOAT_EQ(1, 6.11716439954988e-39 / Phi(var(-13)).val());
EXPECT_FLOAT_EQ(1, 3.73256429887771e-36 / Phi(var(-12.5)).val());
EXPECT_FLOAT_EQ(1, 1.77648211207768e-33 / Phi(var(-12)).val());
EXPECT_FLOAT_EQ(1, 6.59577144611367e-31 / Phi(var(-11.5)).val());
EXPECT_FLOAT_EQ(1, 1.91065957449868e-28 / Phi(var(-11)).val());
EXPECT_FLOAT_EQ(1, 4.31900631780923e-26 / Phi(var(-10.5)).val());
EXPECT_FLOAT_EQ(1, 7.61985302416053e-24 / Phi(var(-10)).val());
EXPECT_FLOAT_EQ(1, 1.04945150753626e-21 / Phi(var(-9.5)).val());
EXPECT_FLOAT_EQ(1, 1.12858840595384e-19 / Phi(var(-9)).val());
EXPECT_FLOAT_EQ(1, 9.47953482220332e-18 / Phi(var(-8.5)).val());
EXPECT_FLOAT_EQ(1, 6.22096057427178e-16 / Phi(var(-8)).val());
EXPECT_FLOAT_EQ(1, 3.1908916729109e-14 / Phi(var(-7.5)).val());
EXPECT_FLOAT_EQ(1, 1.27981254388584e-12 / Phi(var(-7)).val());
EXPECT_FLOAT_EQ(1, 4.01600058385912e-11 / Phi(var(-6.5)).val());
EXPECT_FLOAT_EQ(1, 9.86587645037698e-10 / Phi(var(-6)).val());
EXPECT_FLOAT_EQ(1, 1.89895624658877e-08 / Phi(var(-5.5)).val());
EXPECT_FLOAT_EQ(1, 2.86651571879194e-07 / Phi(var(-5)).val());
EXPECT_FLOAT_EQ(1, 3.39767312473006e-06 / Phi(var(-4.5)).val());
EXPECT_FLOAT_EQ(1, 3.16712418331199e-05 / Phi(var(-4)).val());
EXPECT_FLOAT_EQ(1, 0.000232629079035525 / Phi(var(-3.5)).val());
EXPECT_FLOAT_EQ(1, 0.00134989803163009 / Phi(var(-3)).val());
EXPECT_FLOAT_EQ(1, 0.00620966532577613 / Phi(var(-2.5)).val());
EXPECT_FLOAT_EQ(1, 0.0227501319481792 / Phi(var(-2)).val());
EXPECT_FLOAT_EQ(1, 0.0668072012688581 / Phi(var(-1.5)).val());
EXPECT_FLOAT_EQ(1, 0.158655253931457 / Phi(var(-1)).val());
EXPECT_FLOAT_EQ(1, 0.308537538725987 / Phi(var(-0.5)).val());
EXPECT_FLOAT_EQ(1, 0.5 / Phi(var(0)).val());
EXPECT_FLOAT_EQ(1, 0.691462461274013 / Phi(var(0.5)).val());
EXPECT_FLOAT_EQ(1, 0.841344746068543 / Phi(var(1)).val());
EXPECT_FLOAT_EQ(1, 0.933192798731142 / Phi(var(1.5)).val());
EXPECT_FLOAT_EQ(1, 0.977249868051821 / Phi(var(2)).val());
EXPECT_FLOAT_EQ(1, 0.993790334674224 / Phi(var(2.5)).val());
EXPECT_FLOAT_EQ(1, 0.99865010196837 / Phi(var(3)).val());
EXPECT_FLOAT_EQ(1, 0.999767370920964 / Phi(var(3.5)).val());
EXPECT_FLOAT_EQ(1, 0.999968328758167 / Phi(var(4)).val());
EXPECT_FLOAT_EQ(1, 0.999996602326875 / Phi(var(4.5)).val());
EXPECT_FLOAT_EQ(1, 0.999999713348428 / Phi(var(5)).val());
EXPECT_FLOAT_EQ(1, 0.999999981010438 / Phi(var(5.5)).val());
EXPECT_FLOAT_EQ(1, 0.999999999013412 / Phi(var(6)).val());
EXPECT_FLOAT_EQ(1, 0.99999999995984 / Phi(var(6.5)).val());
EXPECT_FLOAT_EQ(1, 0.99999999999872 / Phi(var(7)).val());
EXPECT_FLOAT_EQ(1, 0.999999999999968 / Phi(var(7.5)).val());
EXPECT_FLOAT_EQ(1, 0.999999999999999 / Phi(var(8)).val());
EXPECT_FLOAT_EQ(1, 1 / Phi(var(8.5)).val());
EXPECT_FLOAT_EQ(1, 1 / Phi(var(9)).val());
EXPECT_FLOAT_EQ(1, 1 / Phi(var(9.5)).val());
EXPECT_FLOAT_EQ(1, 1 / Phi(var(10)).val());
}
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