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

Percentage Accurate: 99.6% → 99.8%

Time: 15.1s

Alternatives: 10

Speedup: 1.0×

• 44.4% 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: 99.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-*l* 99.4%

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

   2. exp-sqrt 99.4%

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

   3. exp-prod 99.4%

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

3. Simplified 99.4%

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

   1. pow1 99.4%

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

   2. sqrt-unprod 99.4%

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

   3. associate-*l* 99.4%

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

   4. pow-exp 99.4%

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

   5. pow2 99.4%

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

6. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 2: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 2.8 \cdot 10^{+54}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+174} \lor \neg \left(t \leq 1.6 \cdot 10^{+223}\right):\\ \;\;\;\;t\_1 \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {t\_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 2.8e+54)
     (* t_1 (sqrt (* 2.0 z)))
     (if (or (<= t 8.5e+174) (not (<= t 1.6e+223)))
       (* t_1 (cbrt (pow (* 2.0 z) 1.5)))
       (sqrt (* (* 2.0 z) (pow t_1 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 2.8e+54) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if ((t <= 8.5e+174) || !(t <= 1.6e+223)) {
		tmp = t_1 * cbrt(pow((2.0 * z), 1.5));
	} else {
		tmp = sqrt(((2.0 * z) * pow(t_1, 2.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 2.8e+54) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else if ((t <= 8.5e+174) || !(t <= 1.6e+223)) {
		tmp = t_1 * Math.cbrt(Math.pow((2.0 * z), 1.5));
	} else {
		tmp = Math.sqrt(((2.0 * z) * Math.pow(t_1, 2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 2.8e+54)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif ((t <= 8.5e+174) || !(t <= 1.6e+223))
		tmp = Float64(t_1 * cbrt((Float64(2.0 * z) ^ 1.5)));
	else
		tmp = sqrt(Float64(Float64(2.0 * z) * (t_1 ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 2.8e+54], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 8.5e+174], N[Not[LessEqual[t, 1.6e+223]], $MachinePrecision]], N[(t$95$1 * N[Power[N[Power[N[(2.0 * z), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.80000000000000015e54

   1. Initial program 98.8%

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

      1. associate-*l* 99.3%

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

      2. exp-sqrt 99.3%

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

      3. exp-prod 99.3%

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

   3. Simplified 99.3%

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

      1. pow1 99.3%

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

      2. sqrt-unprod 99.3%

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

      3. associate-*l* 99.3%

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

      4. pow-exp 99.3%

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

      5. pow2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. unpow1 99.3%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in t around 0 70.1%

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

       1. *-commutative 70.1%

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

   11. Simplified 70.1%

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

   if 2.80000000000000015e54 < t < 8.5000000000000007e174 or 1.6000000000000001e223 < 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. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 22.3%

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

      1. sqrt-prod 22.3%

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

      2. add-cbrt-cube 24.8%

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

      3. pow1/3 24.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{z \cdot 2}\right)}^{0.3333333333333333}} \]

      4. pow3 24.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left({\left(\sqrt{z \cdot 2}\right)}^{3}\right)}}^{0.3333333333333333} \]

      5. sqrt-pow2 24.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left({\left(z \cdot 2\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} \]

      6. *-commutative 24.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\color{blue}{\left(2 \cdot z\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} \]

      7. metadata-eval 24.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\left(2 \cdot z\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 24.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left({\left(2 \cdot z\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 24.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}} \]

      2. *-commutative 24.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\color{blue}{\left(z \cdot 2\right)}}^{1.5}} \]

   9. Simplified 24.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}} \]

   if 8.5000000000000007e174 < t < 1.6000000000000001e223

   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. Add Preprocessing

   3. Taylor expanded in t around 0 4.6%

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

      1. *-commutative 4.6%

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

      2. associate-*l* 4.6%

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

   5. Simplified 4.6%

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

      1. pow1 4.6%

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

      2. *-commutative 4.6%

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

      3. *-commutative 4.6%

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

      4. associate-*r* 4.6%

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

      5. sqrt-prod 4.6%

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

      6. metadata-eval 4.6%

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

      7. pow-sqr 2.9%

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

      8. pow-prod-down 29.6%

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

      9. *-commutative 29.6%

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

      10. *-commutative 29.6%

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

      11. swap-sqr 43.1%

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

      12. add-sqr-sqrt 43.1%

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

      13. *-commutative 43.1%

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

      14. pow2 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. unpow1/2 43.1%

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

      2. *-commutative 43.1%

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

      3. *-commutative 43.1%

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

   9. Simplified 43.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+54}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+174} \lor \neg \left(t \leq 1.6 \cdot 10^{+223}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 18.5:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z + z \cdot {t}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot e^{0.5 \cdot {t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 18.5)
     (* t_1 (sqrt (* 2.0 (+ z (* z (pow t 2.0))))))
     (* t_1 (exp (* 0.5 (pow t 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 18.5) {
		tmp = t_1 * sqrt((2.0 * (z + (z * pow(t, 2.0)))));
	} else {
		tmp = t_1 * exp((0.5 * pow(t, 2.0)));
	}
	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 <= 18.5d0) then
        tmp = t_1 * sqrt((2.0d0 * (z + (z * (t ** 2.0d0)))))
    else
        tmp = t_1 * exp((0.5d0 * (t ** 2.0d0)))
    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 <= 18.5) {
		tmp = t_1 * Math.sqrt((2.0 * (z + (z * Math.pow(t, 2.0)))));
	} else {
		tmp = t_1 * Math.exp((0.5 * Math.pow(t, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 18.5:
		tmp = t_1 * math.sqrt((2.0 * (z + (z * math.pow(t, 2.0)))))
	else:
		tmp = t_1 * math.exp((0.5 * math.pow(t, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 18.5)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z + Float64(z * (t ^ 2.0))))));
	else
		tmp = Float64(t_1 * exp(Float64(0.5 * (t ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 18.5)
		tmp = t_1 * sqrt((2.0 * (z + (z * (t ^ 2.0)))));
	else
		tmp = t_1 * exp((0.5 * (t ^ 2.0)));
	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, 18.5], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z + N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(0.5 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 18.5

   1. Initial program 98.8%

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

      1. associate-*l* 99.3%

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

      2. exp-sqrt 99.3%

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

      3. exp-prod 99.3%

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

   3. Simplified 99.3%

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

      1. pow1 99.3%

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

      2. sqrt-unprod 99.3%

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

      3. associate-*l* 99.3%

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

      4. pow-exp 99.3%

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

      5. pow2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. unpow1 99.3%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in t around 0 91.3%

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

       1. distribute-lft-out 91.3%

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

       2. *-commutative 91.3%

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

   11. Simplified 91.3%

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

   if 18.5 < 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. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. pow-to-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. pow2 100.0%

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

      6. log-prod 100.0%

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

      7. add-log-exp 100.0%

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

      8. pow2 100.0%

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

      9. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in t around inf 100.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 18.5:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z + z \cdot {t}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot e^{0.5 \cdot {t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 17:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot e^{0.5 \cdot {t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 17.0)
     (* t_1 (sqrt (* 2.0 z)))
     (* t_1 (exp (* 0.5 (pow t 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 17.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = t_1 * exp((0.5 * pow(t, 2.0)));
	}
	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 <= 17.0d0) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = t_1 * exp((0.5d0 * (t ** 2.0d0)))
    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 <= 17.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = t_1 * Math.exp((0.5 * Math.pow(t, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 17.0:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = t_1 * math.exp((0.5 * math.pow(t, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 17.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(t_1 * exp(Float64(0.5 * (t ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 17.0)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = t_1 * exp((0.5 * (t ^ 2.0)));
	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, 17.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(0.5 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 17

   1. Initial program 98.8%

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

      1. associate-*l* 99.3%

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

      2. exp-sqrt 99.3%

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

      3. exp-prod 99.3%

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

   3. Simplified 99.3%

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

      1. pow1 99.3%

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

      2. sqrt-unprod 99.3%

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

      3. associate-*l* 99.3%

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

      4. pow-exp 99.3%

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

      5. pow2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. unpow1 99.3%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in t around 0 72.4%

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

       1. *-commutative 72.4%

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

   11. Simplified 72.4%

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

   if 17 < 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. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. pow-to-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. pow2 100.0%

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

      6. log-prod 100.0%

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

      7. add-log-exp 100.0%

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

      8. pow2 100.0%

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

      9. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in t around inf 100.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 17:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot e^{0.5 \cdot {t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1.5 \cdot 10^{+42}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {t\_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.5e+42)
     (* t_1 (sqrt (* 2.0 z)))
     (sqrt (* (* 2.0 z) (pow t_1 2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.5e+42) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = sqrt(((2.0 * z) * pow(t_1, 2.0)));
	}
	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 <= 1.5d+42) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = sqrt(((2.0d0 * z) * (t_1 ** 2.0d0)))
    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 <= 1.5e+42) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(((2.0 * z) * Math.pow(t_1, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 1.5e+42:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(((2.0 * z) * math.pow(t_1, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.5e+42)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(Float64(2.0 * z) * (t_1 ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 1.5e+42)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = sqrt(((2.0 * z) * (t_1 ^ 2.0)));
	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, 1.5e+42], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.50000000000000014e42

   1. Initial program 98.8%

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

      1. associate-*l* 99.3%

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

      2. exp-sqrt 99.3%

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

      3. exp-prod 99.3%

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

   3. Simplified 99.3%

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

      1. pow1 99.3%

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

      2. sqrt-unprod 99.3%

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

      3. associate-*l* 99.3%

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

      4. pow-exp 99.3%

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

      5. pow2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. unpow1 99.3%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in t around 0 70.3%

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

       1. *-commutative 70.3%

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

   11. Simplified 70.3%

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

   if 1.50000000000000014e42 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 20.8%

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

      1. *-commutative 20.8%

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

      2. associate-*l* 20.8%

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

   5. Simplified 20.8%

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

      1. pow1 20.8%

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

      2. *-commutative 20.8%

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

      3. *-commutative 20.8%

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

      4. associate-*r* 20.8%

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

      5. sqrt-prod 20.8%

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

      6. metadata-eval 20.8%

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

      7. pow-sqr 11.6%

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

      8. pow-prod-down 25.6%

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

      9. *-commutative 25.6%

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

      10. *-commutative 25.6%

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

      11. swap-sqr 34.9%

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

      12. add-sqr-sqrt 34.9%

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

      13. *-commutative 34.9%

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

      14. pow2 34.9%

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

   7. Applied egg-rr 34.9%

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

      1. unpow1/2 34.9%

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

      2. *-commutative 34.9%

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

      3. *-commutative 34.9%

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

   9. Simplified 34.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+42}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((2.0 * z))) * 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[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 7: 42.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+135} \lor \neg \left(y \leq 2.9 \cdot 10^{+47}\right):\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (or (<= y -1.3e+135) (not (<= y 2.9e+47)))
     (* y (- t_1))
     (* 0.5 (* x t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((y <= -1.3e+135) || !(y <= 2.9e+47)) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	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 = sqrt((2.0d0 * z))
    if ((y <= (-1.3d+135)) .or. (.not. (y <= 2.9d+47))) then
        tmp = y * -t_1
    else
        tmp = 0.5d0 * (x * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if ((y <= -1.3e+135) || !(y <= 2.9e+47)) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (y <= -1.3e+135) or not (y <= 2.9e+47):
		tmp = y * -t_1
	else:
		tmp = 0.5 * (x * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if ((y <= -1.3e+135) || !(y <= 2.9e+47))
		tmp = Float64(y * Float64(-t_1));
	else
		tmp = Float64(0.5 * Float64(x * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((y <= -1.3e+135) || ~((y <= 2.9e+47)))
		tmp = y * -t_1;
	else
		tmp = 0.5 * (x * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -1.3e+135], N[Not[LessEqual[y, 2.9e+47]], $MachinePrecision]], N[(y * (-t$95$1)), $MachinePrecision], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e135 or 2.8999999999999998e47 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 67.5%

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

      1. *-commutative 67.5%

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

      2. associate-*l* 67.5%

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

   5. Simplified 67.5%

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

   6. Taylor expanded in x around 0 57.4%

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

      1. mul-1-neg 57.4%

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

      2. distribute-lft-neg-out 57.4%

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

      3. *-commutative 57.4%

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

   8. Simplified 57.4%

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

      1. associate-*r* 57.4%

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

      2. sqrt-prod 57.4%

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

      3. distribute-rgt-neg-out 57.4%

         \[\leadsto \color{blue}{-\sqrt{2 \cdot z} \cdot y} \]

      4. sqrt-prod 57.4%

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

      5. add-sqr-sqrt 30.7%

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

      6. sqrt-unprod 26.2%

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

      7. sqr-neg 26.2%

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

      8. sqrt-unprod 0.2%

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

      9. add-sqr-sqrt 1.7%

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

      10. associate-*r* 1.7%

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

      11. *-commutative 1.7%

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

      12. *-commutative 1.7%

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

      13. associate-*l* 1.7%

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

      14. sqrt-prod 1.7%

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

      15. *-commutative 1.7%

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

      16. add-sqr-sqrt 0.2%

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

      17. sqrt-unprod 26.2%

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

      18. sqr-neg 26.2%

          \[\leadsto -\sqrt{\color{blue}{y \cdot y}} \cdot \sqrt{2 \cdot z} \]

      19. sqrt-unprod 30.7%

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

      20. add-sqr-sqrt 57.4%

          \[\leadsto -\color{blue}{y} \cdot \sqrt{2 \cdot z} \]

   10. Applied egg-rr 57.4%

       \[\leadsto \color{blue}{-y \cdot \sqrt{2 \cdot z}} \]
   11. Step-by-step derivation

       1. distribute-rgt-neg-in 57.4%

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

       2. *-commutative 57.4%

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

   12. Simplified 57.4%

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

   if -1.3e135 < y < 2.8999999999999998e47

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l* 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in x around inf 46.3%

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

      1. associate-*r* 46.3%

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

      2. *-commutative 46.3%

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

   8. Simplified 46.3%

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

      1. pow1 46.3%

         \[\leadsto \color{blue}{{\left(\left(0.5 \cdot \left(\sqrt{2} \cdot x\right)\right) \cdot \sqrt{z}\right)}^{1}} \]

      2. associate-*l* 46.3%

         \[\leadsto {\color{blue}{\left(0.5 \cdot \left(\left(\sqrt{2} \cdot x\right) \cdot \sqrt{z}\right)\right)}}^{1} \]

      3. *-commutative 46.3%

         \[\leadsto {\left(0.5 \cdot \left(\color{blue}{\left(x \cdot \sqrt{2}\right)} \cdot \sqrt{z}\right)\right)}^{1} \]

      4. associate-*l* 46.3%

         \[\leadsto {\left(0.5 \cdot \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)}\right)}^{1} \]

      5. sqrt-prod 46.4%

         \[\leadsto {\left(0.5 \cdot \left(x \cdot \color{blue}{\sqrt{2 \cdot z}}\right)\right)}^{1} \]

   10. Applied egg-rr 46.4%

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

       1. unpow1 46.4%

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

   12. Simplified 46.4%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+135} \lor \neg \left(y \leq 2.9 \cdot 10^{+47}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-*l* 99.4%

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

   2. exp-sqrt 99.4%

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

   3. exp-prod 99.4%

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

3. Simplified 99.4%

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

   1. pow1 99.4%

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

   2. sqrt-unprod 99.4%

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

   3. associate-*l* 99.4%

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

   4. pow-exp 99.4%

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

   5. pow2 99.4%

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

6. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in t around 0 62.4%

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

    1. *-commutative 62.4%

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

11. Simplified 62.4%

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

12. Final simplification 62.4%

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

Alternative 9: 30.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0 62.2%

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

   1. *-commutative 62.2%

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

   2. associate-*l* 62.2%

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

5. Simplified 62.2%

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

6. Taylor expanded in x around 0 30.2%

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

   1. mul-1-neg 30.2%

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

   2. distribute-lft-neg-out 30.2%

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

   3. *-commutative 30.2%

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

8. Simplified 30.2%

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

   1. associate-*r* 30.2%

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

   2. sqrt-prod 30.2%

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

   3. distribute-rgt-neg-out 30.2%

      \[\leadsto \color{blue}{-\sqrt{2 \cdot z} \cdot y} \]

   4. sqrt-prod 30.2%

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

   5. add-sqr-sqrt 14.6%

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

   6. sqrt-unprod 14.4%

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

   7. sqr-neg 14.4%

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

   8. sqrt-unprod 1.4%

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

   9. add-sqr-sqrt 2.7%

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

   10. associate-*r* 2.7%

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

   11. *-commutative 2.7%

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

   12. *-commutative 2.7%

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

   13. associate-*l* 2.7%

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

   14. sqrt-prod 2.7%

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

   15. *-commutative 2.7%

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

   16. add-sqr-sqrt 1.4%

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

   17. sqrt-unprod 14.4%

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

   18. sqr-neg 14.4%

       \[\leadsto -\sqrt{\color{blue}{y \cdot y}} \cdot \sqrt{2 \cdot z} \]

   19. sqrt-unprod 14.6%

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

   20. add-sqr-sqrt 30.2%

       \[\leadsto -\color{blue}{y} \cdot \sqrt{2 \cdot z} \]

10. Applied egg-rr 30.2%

    \[\leadsto \color{blue}{-y \cdot \sqrt{2 \cdot z}} \]
11. Step-by-step derivation

    1. distribute-rgt-neg-in 30.2%

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

    2. *-commutative 30.2%

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

12. Simplified 30.2%

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

13. Final simplification 30.2%

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

Alternative 10: 2.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \sqrt{2 \cdot z} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0 62.2%

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

   1. *-commutative 62.2%

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

   2. associate-*l* 62.2%

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

5. Simplified 62.2%

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

6. Taylor expanded in x around 0 30.2%

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

   1. mul-1-neg 30.2%

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

   2. distribute-lft-neg-out 30.2%

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

   3. *-commutative 30.2%

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

8. Simplified 30.2%

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

   1. pow1 30.2%

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

   2. *-commutative 30.2%

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

   3. *-commutative 30.2%

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

   4. associate-*l* 30.2%

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

   5. sqrt-prod 30.2%

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

   6. *-commutative 30.2%

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

   7. add-sqr-sqrt 15.5%

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

   8. sqrt-unprod 17.4%

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

   9. sqr-neg 17.4%

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

   10. sqrt-unprod 1.3%

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

   11. add-sqr-sqrt 2.7%

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

10. Applied egg-rr 2.7%

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

    1. unpow1 2.7%

       \[\leadsto \color{blue}{y \cdot \sqrt{2 \cdot z}} \]

    2. *-commutative 2.7%

       \[\leadsto y \cdot \sqrt{\color{blue}{z \cdot 2}} \]

12. Simplified 2.7%

    \[\leadsto \color{blue}{y \cdot \sqrt{z \cdot 2}} \]

13. Final simplification 2.7%

    \[\leadsto y \cdot \sqrt{2 \cdot z} \]
14. Add Preprocessing

Developer target: 99.6% 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 2024086 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (* (* (- (* 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))))