Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A

Percentage Accurate: 99.4% → 99.4%

Time: 7.6s

Alternatives: 3

Speedup: 1.0×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (sqrt (* z 2.0)) (- (* x 0.5) y)) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (sqrt((z * 2.0)) * ((x * 0.5) - y)) * exp(((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt((z * 2.0d0)) * ((x * 0.5d0) - y)) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (Math.sqrt((z * 2.0)) * ((x * 0.5) - y)) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (math.sqrt((z * 2.0)) * ((x * 0.5) - y)) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y)) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (sqrt((z * 2.0)) * ((x * 0.5) - y)) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Final simplification 99.9%

   \[\leadsto \left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

Alternative 2: 58.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 37:\\ \;\;\;\;t_1 \cdot \sqrt{z + z}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt[3]{{\left(z + z\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 37.0) (* t_1 (sqrt (+ z z))) (* t_1 (cbrt (pow (+ z z) 1.5))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 37.0) {
		tmp = t_1 * sqrt((z + z));
	} else {
		tmp = t_1 * cbrt(pow((z + z), 1.5));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 37.0) {
		tmp = t_1 * Math.sqrt((z + z));
	} else {
		tmp = t_1 * Math.cbrt(Math.pow((z + z), 1.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 37.0)
		tmp = Float64(t_1 * sqrt(Float64(z + z)));
	else
		tmp = Float64(t_1 * cbrt((Float64(z + z) ^ 1.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 37.0], N[(t$95$1 * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[Power[N[(z + z), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 37

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around inf 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot \sqrt{z}} \]
   3. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \color{blue}{\left(\left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \sqrt{2}\right)} \cdot \sqrt{z} \]

      2. associate-*l* 99.6%

         \[\leadsto \color{blue}{\left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

      3. unpow2 99.6%

         \[\leadsto \left(e^{0.5 \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      4. *-commutative 99.6%

         \[\leadsto \left(e^{\color{blue}{\left(t \cdot t\right) \cdot 0.5}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      5. associate-*r* 99.6%

         \[\leadsto \left(e^{\color{blue}{t \cdot \left(t \cdot 0.5\right)}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      6. *-commutative 99.6%

         \[\leadsto \left(e^{\color{blue}{\left(t \cdot 0.5\right) \cdot t}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      7. exp-prod 99.6%

         \[\leadsto \left(\color{blue}{{\left(e^{t \cdot 0.5}\right)}^{t}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      8. exp-prod 99.6%

         \[\leadsto \left({\color{blue}{\left({\left(e^{t}\right)}^{0.5}\right)}}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      9. unpow1/2 99.6%

         \[\leadsto \left({\color{blue}{\left(\sqrt{e^{t}}\right)}}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

   4. Simplified 99.6%

      \[\leadsto \color{blue}{\left({\left(\sqrt{e^{t}}\right)}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   5. Taylor expanded in t around 0 71.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot x - y\right)} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]
   6. Step-by-step derivation

      1. sqrt-unprod 72.2%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt{2 \cdot z}} \]

      2. count-2 72.2%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt{\color{blue}{z + z}} \]

      3. expm1-log1p-u 70.3%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)} \]

      4. expm1-udef 41.4%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)} \]

   7. Applied egg-rr 41.4%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 70.3%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)} \]

      2. expm1-log1p 72.2%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt{z + z}} \]

   9. Simplified 72.2%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt{z + z}} \]

   if 37 < t

   1. Initial program 100.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around inf 100.0%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot \sqrt{z}} \]
   3. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \sqrt{2}\right)} \cdot \sqrt{z} \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

      3. unpow2 100.0%

         \[\leadsto \left(e^{0.5 \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      4. *-commutative 100.0%

         \[\leadsto \left(e^{\color{blue}{\left(t \cdot t\right) \cdot 0.5}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \left(e^{\color{blue}{t \cdot \left(t \cdot 0.5\right)}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      6. *-commutative 100.0%

         \[\leadsto \left(e^{\color{blue}{\left(t \cdot 0.5\right) \cdot t}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      7. exp-prod 100.0%

         \[\leadsto \left(\color{blue}{{\left(e^{t \cdot 0.5}\right)}^{t}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      8. exp-prod 100.0%

         \[\leadsto \left({\color{blue}{\left({\left(e^{t}\right)}^{0.5}\right)}}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

      9. unpow1/2 100.0%

         \[\leadsto \left({\color{blue}{\left(\sqrt{e^{t}}\right)}}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\left({\left(\sqrt{e^{t}}\right)}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   5. Taylor expanded in t around 0 15.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot x - y\right)} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]
   6. Step-by-step derivation

      1. add-cbrt-cube 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}} \]

      2. sqrt-unprod 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\left(\color{blue}{\sqrt{2 \cdot z}} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

      3. count-2 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\left(\sqrt{\color{blue}{z + z}} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

      4. sqrt-unprod 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\left(\sqrt{z + z} \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

      5. count-2 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\left(\sqrt{z + z} \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

      6. add-sqr-sqrt 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\color{blue}{\left(z + z\right)} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

      7. sqrt-unprod 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\left(z + z\right) \cdot \color{blue}{\sqrt{2 \cdot z}}} \]

      8. count-2 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\left(z + z\right) \cdot \sqrt{\color{blue}{z + z}}} \]

   7. Applied egg-rr 27.0%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt[3]{\left(z + z\right) \cdot \sqrt{z + z}}} \]
   8. Step-by-step derivation

      1. *-commutative 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\color{blue}{\sqrt{z + z} \cdot \left(z + z\right)}} \]

      2. unpow1/2 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\color{blue}{{\left(z + z\right)}^{0.5}} \cdot \left(z + z\right)} \]

      3. pow-plus 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{\color{blue}{{\left(z + z\right)}^{\left(0.5 + 1\right)}}} \]

      4. metadata-eval 27.0%

         \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt[3]{{\left(z + z\right)}^{\color{blue}{1.5}}} \]

   9. Simplified 27.0%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt[3]{{\left(z + z\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 37:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(z + z\right)}^{1.5}}\\ \end{array} \]

Alternative 3: 56.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (- (* x 0.5) y) (sqrt (+ z z))))

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt((z + z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * 0.5d0) - y) * sqrt((z + z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * Math.sqrt((z + z));
}

Python

def code(x, y, z, t):
	return ((x * 0.5) - y) * math.sqrt((z + z))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z + z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * 0.5) - y) * sqrt((z + z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Taylor expanded in t around inf 99.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot \sqrt{z}} \]
3. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \color{blue}{\left(\left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \sqrt{2}\right)} \cdot \sqrt{z} \]

   2. associate-*l* 99.7%

      \[\leadsto \color{blue}{\left(e^{0.5 \cdot {t}^{2}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   3. unpow2 99.7%

      \[\leadsto \left(e^{0.5 \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

   4. *-commutative 99.7%

      \[\leadsto \left(e^{\color{blue}{\left(t \cdot t\right) \cdot 0.5}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

   5. associate-*r* 99.7%

      \[\leadsto \left(e^{\color{blue}{t \cdot \left(t \cdot 0.5\right)}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

   6. *-commutative 99.7%

      \[\leadsto \left(e^{\color{blue}{\left(t \cdot 0.5\right) \cdot t}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

   7. exp-prod 99.7%

      \[\leadsto \left(\color{blue}{{\left(e^{t \cdot 0.5}\right)}^{t}} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

   8. exp-prod 99.7%

      \[\leadsto \left({\color{blue}{\left({\left(e^{t}\right)}^{0.5}\right)}}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

   9. unpow1/2 99.7%

      \[\leadsto \left({\color{blue}{\left(\sqrt{e^{t}}\right)}}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]

4. Simplified 99.7%

   \[\leadsto \color{blue}{\left({\left(\sqrt{e^{t}}\right)}^{t} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

5. Taylor expanded in t around 0 55.5%

   \[\leadsto \color{blue}{\left(0.5 \cdot x - y\right)} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right) \]
6. Step-by-step derivation

   1. sqrt-unprod 55.7%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   2. count-2 55.7%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt{\color{blue}{z + z}} \]

   3. expm1-log1p-u 54.3%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)} \]

   4. expm1-udef 33.6%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)} \]

7. Applied egg-rr 33.6%

   \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)} \]
8. Step-by-step derivation

   1. expm1-def 54.3%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)} \]

   2. expm1-log1p 55.7%

      \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt{z + z}} \]

9. Simplified 55.7%

   \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt{z + z}} \]

10. Final simplification 55.7%

    \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z} \]

Developer target: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(exp(1.0), ((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (exp(1.0d0) ** ((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.pow(Math.exp(1.0), ((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.exp(1.0), ((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * (exp(1.0) ^ Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(1.0) ^ ((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[1.0], $MachinePrecision], N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))