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

Percentage Accurate: 99.5% → 99.8%

Time: 11.4s

Alternatives: 9

Speedup: 1.0×

• 43.5% 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

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt(((2.0 * z) * pow(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(((2.0d0 * z) * (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(((2.0 * z) * Math.pow(Math.exp(t), t)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[N[Exp[t], $MachinePrecision], t], $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. associate-*l/ 99.8%

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

   3. exp-prod 99.8%

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

   4. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. sqrt-pow2 99.8%

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

   2. sqrt-pow1 99.8%

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

   3. exp-prod 99.8%

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

   4. expm1-log1p-u 98.4%

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

   5. expm1-udef 79.5%

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

   6. sqrt-unprod 79.5%

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

   7. associate-*l* 79.5%

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

   8. exp-prod 79.5%

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

5. Applied egg-rr 79.5%

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

   1. expm1-def 98.5%

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

   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. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt(((2.0 * z) * 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(((2.0d0 * z) * 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(((2.0 * z) * Math.exp((t * t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * 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. associate-*l/ 99.8%

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

   3. exp-prod 99.8%

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

   4. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. sqrt-pow2 99.8%

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

   2. sqrt-pow1 99.8%

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

   3. exp-prod 99.8%

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

   4. expm1-log1p-u 98.4%

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

   5. expm1-udef 79.5%

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

   6. sqrt-unprod 79.5%

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

   7. associate-*l* 79.5%

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

   8. exp-prod 79.5%

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

5. Applied egg-rr 79.5%

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

   1. expm1-def 98.5%

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

   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. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

   1. pow-exp 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 86.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * ((1.0 + (0.5 * (t * t))) * sqrt((2.0 * 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) * ((1.0d0 + (0.5d0 * (t * t))) * sqrt((2.0d0 * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $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. Taylor expanded in t around 0 83.2%

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

   1. unpow2 83.2%

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

6. Simplified 83.2%

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

7. Final simplification 83.2%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\left(1 + 0.5 \cdot \left(t \cdot t\right)\right) \cdot \sqrt{2 \cdot z}\right) \]

Alternative 4: 60.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 4.8 \cdot 10^{+25}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+201}:\\ \;\;\;\;t_1 \cdot {\left(4 \cdot \left(z \cdot z\right)\right)}^{0.25}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+266}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot x\right) \cdot \left(0.5 \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 4.8e+25)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= t 7.2e+201)
       (* t_1 (pow (* 4.0 (* z z)) 0.25))
       (if (<= t 8.5e+266)
         (sqrt (* z (* 2.0 (* y y))))
         (sqrt (* (* x x) (* 0.5 z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 4.8e+25) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if (t <= 7.2e+201) {
		tmp = t_1 * pow((4.0 * (z * z)), 0.25);
	} else if (t <= 8.5e+266) {
		tmp = sqrt((z * (2.0 * (y * y))));
	} else {
		tmp = sqrt(((x * x) * (0.5 * z)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 4.8d+25) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else if (t <= 7.2d+201) then
        tmp = t_1 * ((4.0d0 * (z * z)) ** 0.25d0)
    else if (t <= 8.5d+266) then
        tmp = sqrt((z * (2.0d0 * (y * y))))
    else
        tmp = sqrt(((x * x) * (0.5d0 * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 4.8e+25) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else if (t <= 7.2e+201) {
		tmp = t_1 * Math.pow((4.0 * (z * z)), 0.25);
	} else if (t <= 8.5e+266) {
		tmp = Math.sqrt((z * (2.0 * (y * y))));
	} else {
		tmp = Math.sqrt(((x * x) * (0.5 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 4.8e+25:
		tmp = t_1 * math.sqrt((2.0 * z))
	elif t <= 7.2e+201:
		tmp = t_1 * math.pow((4.0 * (z * z)), 0.25)
	elif t <= 8.5e+266:
		tmp = math.sqrt((z * (2.0 * (y * y))))
	else:
		tmp = math.sqrt(((x * x) * (0.5 * z)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 4.8e+25)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (t <= 7.2e+201)
		tmp = Float64(t_1 * (Float64(4.0 * Float64(z * z)) ^ 0.25));
	elseif (t <= 8.5e+266)
		tmp = sqrt(Float64(z * Float64(2.0 * Float64(y * y))));
	else
		tmp = sqrt(Float64(Float64(x * x) * Float64(0.5 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 4.8e+25)
		tmp = t_1 * sqrt((2.0 * z));
	elseif (t <= 7.2e+201)
		tmp = t_1 * ((4.0 * (z * z)) ^ 0.25);
	elseif (t <= 8.5e+266)
		tmp = sqrt((z * (2.0 * (y * y))));
	else
		tmp = sqrt(((x * x) * (0.5 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 4.8e+25], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e+201], N[(t$95$1 * N[Power[N[(4.0 * N[(z * z), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e+266], N[Sqrt[N[(z * N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(0.5 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.79999999999999992e25

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 84.0%

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

      1. unpow2 84.0%

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

   6. Simplified 84.0%

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

   7. Taylor expanded in t around 0 65.9%

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

      1. unpow1/2 65.9%

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

      2. metadata-eval 65.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2} \cdot {z}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \]

      3. pow-sqr 65.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)}\right) \]

      4. unpow1/2 65.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{{2}^{0.5}} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      5. metadata-eval 65.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({2}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      6. pow-sqr 65.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      7. swap-sqr 65.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\left({2}^{0.25} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)} \]

      8. exp-to-pow 65.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      9. exp-to-pow 63.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      10. exp-sum 63.1%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      11. distribute-rgt-in 63.1%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      12. log-prod 63.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      13. *-commutative 63.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\color{blue}{\log \left(2 \cdot z\right) \cdot 0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      14. exp-to-pow 65.8%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{{\left(2 \cdot z\right)}^{0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      15. exp-to-pow 65.8%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right)\right) \]

      16. exp-to-pow 63.4%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right)\right) \]

      17. exp-sum 63.1%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}}\right) \]

      18. distribute-rgt-in 63.1%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}}\right) \]

      19. log-prod 63.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}}\right) \]

      20. *-commutative 63.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot e^{\color{blue}{\log \left(2 \cdot z\right) \cdot 0.25}}\right) \]

      21. exp-to-pow 65.8%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{{\left(2 \cdot z\right)}^{0.25}}\right) \]

   9. Simplified 66.1%

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

   if 4.79999999999999992e25 < t < 7.19999999999999951e201

   1. Initial program 96.4%

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

      1. associate-*l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. exp-prod 100.0%

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

      4. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 27.9%

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

      1. *-commutative 27.9%

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

      2. sqrt-prod 27.9%

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

      3. pow1/2 27.9%

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

      4. metadata-eval 27.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(z \cdot 2\right)}^{\color{blue}{\left(2 \cdot 0.25\right)}} \]

      5. pow-sqr 27.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left({\left(z \cdot 2\right)}^{0.25} \cdot {\left(z \cdot 2\right)}^{0.25}\right)} \]

      6. pow-prod-down 37.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\left(z \cdot 2\right) \cdot \left(z \cdot 2\right)\right)}^{0.25}} \]

      7. *-commutative 37.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\color{blue}{\left(2 \cdot z\right)} \cdot \left(z \cdot 2\right)\right)}^{0.25} \]

      8. *-commutative 37.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\left(2 \cdot z\right) \cdot \color{blue}{\left(2 \cdot z\right)}\right)}^{0.25} \]

      9. swap-sqr 37.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left(\left(2 \cdot 2\right) \cdot \left(z \cdot z\right)\right)}}^{0.25} \]

      10. metadata-eval 37.7%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\color{blue}{4} \cdot \left(z \cdot z\right)\right)}^{0.25} \]

   6. Applied egg-rr 37.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(4 \cdot \left(z \cdot z\right)\right)}^{0.25}} \]

   if 7.19999999999999951e201 < t < 8.49999999999999955e266

   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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. exp-prod 100.0%

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

      4. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 10.0%

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

      1. add-sqr-sqrt 2.8%

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

      2. sqrt-unprod 30.5%

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

      3. *-commutative 30.5%

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

      4. *-commutative 30.5%

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

      5. sqrt-prod 30.5%

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

      6. *-commutative 30.5%

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

      7. *-commutative 30.5%

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

      8. sqrt-prod 30.5%

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

      9. swap-sqr 30.4%

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

      10. add-sqr-sqrt 30.4%

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

      11. *-commutative 30.4%

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

      12. pow2 30.4%

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

   6. Applied egg-rr 30.4%

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

   7. Taylor expanded in x around 0 24.9%

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

      1. associate-*r* 24.9%

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

      2. unpow2 24.9%

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

   9. Simplified 24.9%

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

   if 8.49999999999999955e266 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. exp-prod 100.0%

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

      4. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 4.7%

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

      1. add-sqr-sqrt 3.2%

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

      2. sqrt-unprod 30.2%

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

      3. *-commutative 30.2%

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

      4. *-commutative 30.2%

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

      5. sqrt-prod 30.2%

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

      6. *-commutative 30.2%

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

      7. *-commutative 30.2%

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

      8. sqrt-prod 30.2%

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

      9. swap-sqr 43.8%

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

      10. add-sqr-sqrt 43.8%

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

      11. *-commutative 43.8%

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

      12. pow2 43.8%

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

   6. Applied egg-rr 43.8%

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

   7. Taylor expanded in x around inf 43.7%

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

      1. *-commutative 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-*l* 43.7%

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

      4. unpow2 43.7%

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

   9. Simplified 43.7%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(z \cdot 0.5\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{+25}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+201}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot {\left(4 \cdot \left(z \cdot z\right)\right)}^{0.25}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+266}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot x\right) \cdot \left(0.5 \cdot z\right)}\\ \end{array} \]

Alternative 5: 84.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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(((2.0d0 * z) * ((t * t) + 1.0d0)))
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) * ((t * t) + 1.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $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. 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. associate-*l/ 99.8%

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

   3. exp-prod 99.8%

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

   4. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. sqrt-pow2 99.8%

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

   2. sqrt-pow1 99.8%

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

   3. exp-prod 99.8%

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

   4. expm1-log1p-u 98.4%

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

   5. expm1-udef 79.5%

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

   6. sqrt-unprod 79.5%

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

   7. associate-*l* 79.5%

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

   8. exp-prod 79.5%

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

5. Applied egg-rr 79.5%

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

   1. expm1-def 98.5%

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

   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. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 82.7%

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

   1. unpow2 82.7%

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

10. Simplified 82.7%

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

11. Final simplification 82.7%

    \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)} \]

Alternative 6: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{+195}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.3e+195)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (sqrt (* z (* 2.0 (* y y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.3e+195) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt((z * (2.0 * (y * y))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= 2.3d+195) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt((z * (2.0d0 * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.3e+195) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt((z * (2.0 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 2.3e+195:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt((z * (2.0 * (y * y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.3e+195)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(z * Float64(2.0 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.3e+195)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt((z * (2.0 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.3e+195], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(z * N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.3000000000000001e195

   1. Initial program 99.3%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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 81.4%

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

      1. unpow2 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in t around 0 61.6%

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

      1. unpow1/2 61.6%

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

      2. metadata-eval 61.6%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2} \cdot {z}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \]

      3. pow-sqr 61.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)}\right) \]

      4. unpow1/2 61.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{{2}^{0.5}} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      5. metadata-eval 61.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({2}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      6. pow-sqr 61.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      7. swap-sqr 61.3%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\left({2}^{0.25} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)} \]

      8. exp-to-pow 61.3%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      9. exp-to-pow 59.3%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      10. exp-sum 59.1%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      11. distribute-rgt-in 59.1%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      12. log-prod 59.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      13. *-commutative 59.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\color{blue}{\log \left(2 \cdot z\right) \cdot 0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      14. exp-to-pow 61.5%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{{\left(2 \cdot z\right)}^{0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      15. exp-to-pow 61.5%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right)\right) \]

      16. exp-to-pow 59.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right)\right) \]

      17. exp-sum 59.1%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}}\right) \]

      18. distribute-rgt-in 59.1%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}}\right) \]

      19. log-prod 59.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}}\right) \]

      20. *-commutative 59.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot e^{\color{blue}{\log \left(2 \cdot z\right) \cdot 0.25}}\right) \]

      21. exp-to-pow 61.5%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{{\left(2 \cdot z\right)}^{0.25}}\right) \]

   9. Simplified 61.7%

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

   if 2.3000000000000001e195 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. associate-*l/ 100.0%

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

      3. exp-prod 100.0%

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

      4. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 8.2%

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

      1. add-sqr-sqrt 2.8%

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

      2. sqrt-unprod 29.2%

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

      3. *-commutative 29.2%

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

      4. *-commutative 29.2%

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

      5. sqrt-prod 29.2%

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

      6. *-commutative 29.2%

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

      7. *-commutative 29.2%

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

      8. sqrt-prod 29.2%

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

      9. swap-sqr 33.0%

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

      10. add-sqr-sqrt 33.0%

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

      11. *-commutative 33.0%

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

      12. pow2 33.0%

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

   6. Applied egg-rr 33.0%

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

   7. Taylor expanded in x around 0 17.6%

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

      1. associate-*r* 17.6%

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

      2. unpow2 17.6%

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

   9. Simplified 17.6%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{+195}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}\\ \end{array} \]

Alternative 7: 23.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot x\right) \cdot \left(0.5 \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.12e+29)
   (sqrt (* z (* 2.0 (* y y))))
   (sqrt (* (* x x) (* 0.5 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.12e+29) {
		tmp = sqrt((z * (2.0 * (y * y))));
	} else {
		tmp = sqrt(((x * x) * (0.5 * z)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 1.12d+29) then
        tmp = sqrt((z * (2.0d0 * (y * y))))
    else
        tmp = sqrt(((x * x) * (0.5d0 * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.12e+29) {
		tmp = Math.sqrt((z * (2.0 * (y * y))));
	} else {
		tmp = Math.sqrt(((x * x) * (0.5 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.12e+29:
		tmp = math.sqrt((z * (2.0 * (y * y))))
	else:
		tmp = math.sqrt(((x * x) * (0.5 * z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.12e+29)
		tmp = sqrt(Float64(z * Float64(2.0 * Float64(y * y))));
	else
		tmp = sqrt(Float64(Float64(x * x) * Float64(0.5 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.12e+29)
		tmp = sqrt((z * (2.0 * (y * y))));
	else
		tmp = sqrt(((x * x) * (0.5 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.12e+29], N[Sqrt[N[(z * N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(0.5 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1200000000000001e29

   1. Initial program 99.3%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. associate-*l/ 99.8%

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

      3. exp-prod 99.8%

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

      4. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 57.4%

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

      1. add-sqr-sqrt 19.6%

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

      2. sqrt-unprod 18.7%

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

      3. *-commutative 18.7%

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

      4. *-commutative 18.7%

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

      5. sqrt-prod 18.7%

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

      6. *-commutative 18.7%

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

      7. *-commutative 18.7%

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

      8. sqrt-prod 18.7%

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

      9. swap-sqr 18.7%

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

      10. add-sqr-sqrt 18.7%

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

      11. *-commutative 18.7%

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

      12. pow2 18.7%

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

   6. Applied egg-rr 18.7%

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

   7. Taylor expanded in x around 0 16.9%

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

      1. associate-*r* 16.9%

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

      2. unpow2 16.9%

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

   9. Simplified 16.9%

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

   if 1.1200000000000001e29 < x

   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. Step-by-step derivation

      1. associate-*l* 99.9%

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

      2. associate-*l/ 99.9%

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

      3. exp-prod 99.9%

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

      4. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 53.1%

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

      1. add-sqr-sqrt 47.5%

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

      2. sqrt-unprod 55.7%

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

      3. *-commutative 55.7%

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

      4. *-commutative 55.7%

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

      5. sqrt-prod 55.8%

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

      6. *-commutative 55.8%

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

      7. *-commutative 55.8%

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

      8. sqrt-prod 55.7%

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

      9. swap-sqr 55.8%

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

      10. add-sqr-sqrt 55.8%

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

      11. *-commutative 55.8%

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

      12. pow2 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in x around inf 54.4%

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

      1. *-commutative 54.4%

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

      2. *-commutative 54.4%

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

      3. associate-*l* 54.4%

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

      4. unpow2 54.4%

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

   9. Simplified 54.4%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot x\right) \cdot \left(0.5 \cdot z\right)}\\ \end{array} \]

Alternative 8: 16.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[Sqrt[N[(2.0 * N[(y * N[(y * z), $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. associate-*l/ 99.8%

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

   3. exp-prod 99.8%

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

   4. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 56.4%

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

   1. add-sqr-sqrt 26.2%

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

   2. sqrt-unprod 27.5%

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

   3. *-commutative 27.5%

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

   4. *-commutative 27.5%

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

   5. sqrt-prod 27.6%

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

   6. *-commutative 27.6%

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

   7. *-commutative 27.6%

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

   8. sqrt-prod 27.6%

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

   9. swap-sqr 27.5%

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

   10. add-sqr-sqrt 27.6%

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

   11. *-commutative 27.6%

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

   12. pow2 27.6%

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

6. Applied egg-rr 27.6%

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

7. Taylor expanded in x around 0 17.3%

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

   1. unpow2 17.3%

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

   2. associate-*l* 17.3%

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

   3. *-commutative 17.3%

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

9. Simplified 17.3%

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

10. Final simplification 17.3%

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

Alternative 9: 16.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 * (y * y))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[Sqrt[N[(z * N[(2.0 * N[(y * y), $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. associate-*l/ 99.8%

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

   3. exp-prod 99.8%

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

   4. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 56.4%

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

   1. add-sqr-sqrt 26.2%

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

   2. sqrt-unprod 27.5%

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

   3. *-commutative 27.5%

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

   4. *-commutative 27.5%

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

   5. sqrt-prod 27.6%

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

   6. *-commutative 27.6%

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

   7. *-commutative 27.6%

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

   8. sqrt-prod 27.6%

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

   9. swap-sqr 27.5%

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

   10. add-sqr-sqrt 27.6%

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

   11. *-commutative 27.6%

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

   12. pow2 27.6%

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

6. Applied egg-rr 27.6%

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

7. Taylor expanded in x around 0 17.3%

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

   1. associate-*r* 17.3%

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

   2. unpow2 17.3%

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

9. Simplified 17.3%

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

10. Final simplification 17.3%

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

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 2023208 
(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))))