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

Percentage Accurate: 99.5% → 99.8%

Time: 10.0s

Alternatives: 6

Speedup: 1.0×

• 44.5% 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: 99.5% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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)) * sqrt(exp((t * t))))
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.sqrt(Math.exp((t * t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)))))
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)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)\right)} \]

   2. expm1-udef 73.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} - 1\right)} \]

   3. sqrt-unprod 73.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1\right) \]

   4. associate-*l* 73.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot e^{t \cdot t}\right)}}\right)} - 1\right) \]

   5. exp-prod 73.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot \color{blue}{{\left(e^{t}\right)}^{t}}\right)}\right)} - 1\right) \]

5. Applied egg-rr 73.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 98.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. exp-prod 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 84.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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((2.0d0 * (z + (z * (t * t)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)\right)} \]

   2. expm1-udef 73.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} - 1\right)} \]

   3. sqrt-unprod 73.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1\right) \]

   4. associate-*l* 73.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot e^{t \cdot t}\right)}}\right)} - 1\right) \]

   5. exp-prod 73.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot \color{blue}{{\left(e^{t}\right)}^{t}}\right)}\right)} - 1\right) \]

5. Applied egg-rr 73.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 98.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. exp-prod 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 86.6%

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

   1. distribute-lft-out 86.6%

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

   2. *-commutative 86.6%

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

   3. unpow2 86.6%

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

10. Simplified 86.6%

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

11. Final simplification 86.6%

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

Alternative 4: 56.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 58.5%

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

   1. *-commutative 58.5%

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

   2. sqrt-prod 58.6%

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

6. Applied egg-rr 58.6%

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

7. Final simplification 58.6%

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

Alternative 5: 16.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\sqrt{z \cdot \left(y \cdot \left(y \cdot 2\right)\right)} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return -sqrt((z * (y * (y * 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 * (y * (y * 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0 58.6%

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

   1. associate-*l* 58.5%

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

4. Simplified 58.5%

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

5. Taylor expanded in x around 0 33.2%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 16.5%

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}} \cdot \sqrt{\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}}\right)} \]

   2. sqrt-unprod 18.0%

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\left(\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}\right) \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}\right)}} \]

   3. *-commutative 18.0%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)\right)} \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}\right)} \]

   4. *-commutative 18.0%

      \[\leadsto -1 \cdot \sqrt{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)\right)}} \]

   5. swap-sqr 16.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \left(\sqrt{2} \cdot y\right)\right)}} \]

   6. add-sqr-sqrt 16.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{z} \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \left(\sqrt{2} \cdot y\right)\right)} \]

   7. swap-sqr 16.5%

      \[\leadsto -1 \cdot \sqrt{z \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(y \cdot y\right)\right)}} \]

   8. add-sqr-sqrt 16.5%

      \[\leadsto -1 \cdot \sqrt{z \cdot \left(\color{blue}{2} \cdot \left(y \cdot y\right)\right)} \]

7. Applied egg-rr 16.5%

   \[\leadsto -1 \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}} \]
8. Step-by-step derivation

   1. associate-*r* 16.5%

      \[\leadsto -1 \cdot \sqrt{z \cdot \color{blue}{\left(\left(2 \cdot y\right) \cdot y\right)}} \]

   2. *-commutative 16.5%

      \[\leadsto -1 \cdot \sqrt{z \cdot \color{blue}{\left(y \cdot \left(2 \cdot y\right)\right)}} \]

9. Simplified 16.5%

   \[\leadsto -1 \cdot \color{blue}{\sqrt{z \cdot \left(y \cdot \left(2 \cdot y\right)\right)}} \]

10. Final simplification 16.5%

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

Alternative 6: 16.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\sqrt{\left(y \cdot z\right) \cdot \left(y \cdot 2\right)} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return -sqrt(((y * z) * (y * 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(((y * z) * (y * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0 58.6%

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

   1. associate-*l* 58.5%

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

4. Simplified 58.5%

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

5. Taylor expanded in x around 0 33.2%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 16.5%

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}} \cdot \sqrt{\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}}\right)} \]

   2. sqrt-unprod 18.0%

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\left(\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}\right) \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}\right)}} \]

   3. *-commutative 18.0%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)\right)} \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \sqrt{z}\right)} \]

   4. *-commutative 18.0%

      \[\leadsto -1 \cdot \sqrt{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)\right)}} \]

   5. swap-sqr 16.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \left(\sqrt{2} \cdot y\right)\right)}} \]

   6. add-sqr-sqrt 16.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{z} \cdot \left(\left(\sqrt{2} \cdot y\right) \cdot \left(\sqrt{2} \cdot y\right)\right)} \]

   7. swap-sqr 16.5%

      \[\leadsto -1 \cdot \sqrt{z \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(y \cdot y\right)\right)}} \]

   8. add-sqr-sqrt 16.5%

      \[\leadsto -1 \cdot \sqrt{z \cdot \left(\color{blue}{2} \cdot \left(y \cdot y\right)\right)} \]

7. Applied egg-rr 16.5%

   \[\leadsto -1 \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}} \]
8. Step-by-step derivation

   1. associate-*r* 16.5%

      \[\leadsto -1 \cdot \sqrt{z \cdot \color{blue}{\left(\left(2 \cdot y\right) \cdot y\right)}} \]

   2. *-commutative 16.5%

      \[\leadsto -1 \cdot \sqrt{z \cdot \color{blue}{\left(y \cdot \left(2 \cdot y\right)\right)}} \]

9. Simplified 16.5%

   \[\leadsto -1 \cdot \color{blue}{\sqrt{z \cdot \left(y \cdot \left(2 \cdot y\right)\right)}} \]
10. Step-by-step derivation

    1. sqrt-prod 16.9%

       \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{y \cdot \left(2 \cdot y\right)}\right)} \]

    2. *-commutative 16.9%

       \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{y \cdot \left(2 \cdot y\right)} \cdot \sqrt{z}\right)} \]

    3. expm1-log1p-u 16.7%

       \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{y \cdot \left(2 \cdot y\right)} \cdot \sqrt{z}\right)\right)} \]

    4. expm1-udef 14.0%

       \[\leadsto -1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{y \cdot \left(2 \cdot y\right)} \cdot \sqrt{z}\right)} - 1\right)} \]

    5. *-commutative 14.0%

       \[\leadsto -1 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{z} \cdot \sqrt{y \cdot \left(2 \cdot y\right)}}\right)} - 1\right) \]

    6. sqrt-prod 14.1%

       \[\leadsto -1 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{z \cdot \left(y \cdot \left(2 \cdot y\right)\right)}}\right)} - 1\right) \]

11. Applied egg-rr 14.1%

    \[\leadsto -1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(y \cdot \left(2 \cdot y\right)\right)}\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 16.3%

       \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot \left(y \cdot \left(2 \cdot y\right)\right)}\right)\right)} \]

    2. expm1-log1p 16.5%

       \[\leadsto -1 \cdot \color{blue}{\sqrt{z \cdot \left(y \cdot \left(2 \cdot y\right)\right)}} \]

    3. associate-*r* 18.0%

       \[\leadsto -1 \cdot \sqrt{\color{blue}{\left(z \cdot y\right) \cdot \left(2 \cdot y\right)}} \]

13. Simplified 18.0%

    \[\leadsto -1 \cdot \color{blue}{\sqrt{\left(z \cdot y\right) \cdot \left(2 \cdot y\right)}} \]

14. Final simplification 18.0%

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

Developer target: 99.5% 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 2023238 
(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))))