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

Percentage Accurate: 99.4% → 99.7%

Time: 14.3s

Alternatives: 14

Speedup: 1.0×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

8. Simplified 99.8%

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

Alternative 2: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 4 \cdot 10^{-6}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 4e-6)
     (* t_1 (sqrt (* (* z 2.0) (fma t t 1.0))))
     (if (<= (* t t) 5e+307)
       (* (exp (/ (* t t) 2.0)) (* (sqrt (* z 2.0)) (- y)))
       (* t_1 (sqrt (* 2.0 (* z (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 * t) <= 4e-6) {
		tmp = t_1 * sqrt(((z * 2.0) * fma(t, t, 1.0)));
	} else if ((t * t) <= 5e+307) {
		tmp = exp(((t * t) / 2.0)) * (sqrt((z * 2.0)) * -y);
	} else {
		tmp = t_1 * sqrt((2.0 * (z * 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 (Float64(t * t) <= 4e-6)
		tmp = Float64(t_1 * sqrt(Float64(Float64(z * 2.0) * fma(t, t, 1.0))));
	elseif (Float64(t * t) <= 5e+307)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(sqrt(Float64(z * 2.0)) * Float64(-y)));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * (t ^ 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[N[(t * t), $MachinePrecision], 4e-6], N[(t$95$1 * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 5e+307], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 3.99999999999999982e-6

   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. Step-by-step derivation

      1. associate-*l* 99.7%

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

      2. remove-double-neg 99.7%

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

      3. remove-double-neg 99.7%

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

      4. exp-sqrt 99.6%

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

      5. exp-prod 99.6%

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

   3. Simplified 99.6%

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

      1. pow1 99.6%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 99.3%

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

       1. +-commutative 99.3%

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

       2. unpow2 99.3%

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

       3. fma-define 99.3%

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

   11. Simplified 99.3%

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

   if 3.99999999999999982e-6 < (*.f64 t t) < 5e307

   1. Initial program 98.4%

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

      1. add-sqr-sqrt 51.6%

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

      2. pow2 51.6%

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

   4. Applied egg-rr 51.6%

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

   5. Taylor expanded in x around 0 82.8%

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

      1. neg-mul-1 82.8%

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

   7. Simplified 82.8%

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

   if 5e307 < (*.f64 t 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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 100.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{-18}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(t\_2 \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= (* t t) 5e-18)
     (* t_1 t_2)
     (if (<= (* t t) 5e+307)
       (* (exp (/ (* t t) 2.0)) (* t_2 (- y)))
       (* t_1 (sqrt (* 2.0 (* z (pow t 2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 5e-18) {
		tmp = t_1 * t_2;
	} else if ((t * t) <= 5e+307) {
		tmp = exp(((t * t) / 2.0)) * (t_2 * -y);
	} else {
		tmp = t_1 * sqrt((2.0 * (z * 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) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((z * 2.0d0))
    if ((t * t) <= 5d-18) then
        tmp = t_1 * t_2
    else if ((t * t) <= 5d+307) then
        tmp = exp(((t * t) / 2.0d0)) * (t_2 * -y)
    else
        tmp = t_1 * sqrt((2.0d0 * (z * (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 t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 5e-18) {
		tmp = t_1 * t_2;
	} else if ((t * t) <= 5e+307) {
		tmp = Math.exp(((t * t) / 2.0)) * (t_2 * -y);
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z * Math.pow(t, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 5e-18:
		tmp = t_1 * t_2
	elif (t * t) <= 5e+307:
		tmp = math.exp(((t * t) / 2.0)) * (t_2 * -y)
	else:
		tmp = t_1 * math.sqrt((2.0 * (z * math.pow(t, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 5e-18)
		tmp = Float64(t_1 * t_2);
	elseif (Float64(t * t) <= 5e+307)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(t_2 * Float64(-y)));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * (t ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 5e-18)
		tmp = t_1 * t_2;
	elseif ((t * t) <= 5e+307)
		tmp = exp(((t * t) / 2.0)) * (t_2 * -y);
	else
		tmp = t_1 * sqrt((2.0 * (z * (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]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 5e-18], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 5e+307], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * (-y)), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 5.00000000000000036e-18

   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. Step-by-step derivation

      1. associate-*l* 99.7%

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

      2. remove-double-neg 99.7%

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

      3. remove-double-neg 99.7%

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

      4. exp-sqrt 99.7%

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

      5. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 99.3%

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

       1. unpow1/2 99.3%

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

       2. metadata-eval 99.3%

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

       3. pow-sqr 99.1%

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

       4. unpow1/2 99.1%

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

       5. metadata-eval 99.1%

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

       6. pow-sqr 99.0%

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

       7. unswap-sqr 99.0%

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

       8. *-commutative 99.0%

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

       9. exp-to-pow 99.0%

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

       10. exp-to-pow 95.0%

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

       11. exp-sum 94.6%

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

       12. distribute-rgt-in 94.6%

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

       13. *-commutative 94.6%

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

       14. exp-prod 94.6%

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

       15. exp-sum 95.0%

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

       16. rem-exp-log 95.0%

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

       17. rem-exp-log 99.3%

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

       18. *-commutative 99.3%

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

   11. Simplified 99.7%

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

   if 5.00000000000000036e-18 < (*.f64 t t) < 5e307

   1. Initial program 98.5%

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

      1. add-sqr-sqrt 50.7%

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

      2. pow2 50.7%

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

   4. Applied egg-rr 50.7%

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

   5. Taylor expanded in x around 0 81.8%

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

      1. neg-mul-1 81.8%

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

   7. Simplified 81.8%

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

   if 5e307 < (*.f64 t 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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 100.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 880:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {t\_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 880.0)
     (* t_1 t_2)
     (if (<= t 5.2e+220)
       (sqrt (* (* z 2.0) (pow t_1 2.0)))
       (* t_2 (* x (- 0.5 (/ y x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 880.0) {
		tmp = t_1 * t_2;
	} else if (t <= 5.2e+220) {
		tmp = sqrt(((z * 2.0) * pow(t_1, 2.0)));
	} else {
		tmp = t_2 * (x * (0.5 - (y / x)));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((z * 2.0d0))
    if (t <= 880.0d0) then
        tmp = t_1 * t_2
    else if (t <= 5.2d+220) then
        tmp = sqrt(((z * 2.0d0) * (t_1 ** 2.0d0)))
    else
        tmp = t_2 * (x * (0.5d0 - (y / x)))
    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 t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 880.0) {
		tmp = t_1 * t_2;
	} else if (t <= 5.2e+220) {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(t_1, 2.0)));
	} else {
		tmp = t_2 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 880.0:
		tmp = t_1 * t_2
	elif t <= 5.2e+220:
		tmp = math.sqrt(((z * 2.0) * math.pow(t_1, 2.0)))
	else:
		tmp = t_2 * (x * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 880.0)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 5.2e+220)
		tmp = sqrt(Float64(Float64(z * 2.0) * (t_1 ^ 2.0)));
	else
		tmp = Float64(t_2 * Float64(x * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 880.0)
		tmp = t_1 * t_2;
	elseif (t <= 5.2e+220)
		tmp = sqrt(((z * 2.0) * (t_1 ^ 2.0)));
	else
		tmp = t_2 * (x * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 880.0], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 5.2e+220], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$2 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 880

   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. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 67.7%

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

       1. unpow1/2 67.7%

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

       2. metadata-eval 67.7%

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

       3. pow-sqr 67.6%

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

       4. unpow1/2 67.6%

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

       5. metadata-eval 67.6%

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

       6. pow-sqr 67.6%

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

       7. unswap-sqr 67.5%

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

       8. *-commutative 67.5%

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

       9. exp-to-pow 67.5%

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

       10. exp-to-pow 65.1%

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

       11. exp-sum 64.8%

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

       12. distribute-rgt-in 64.8%

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

       13. *-commutative 64.8%

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

       14. exp-prod 64.8%

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

       15. exp-sum 65.1%

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

       16. rem-exp-log 65.1%

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

       17. rem-exp-log 67.7%

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

       18. *-commutative 67.7%

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

   11. Simplified 68.0%

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

   if 880 < t < 5.19999999999999988e220

   1. Initial program 97.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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 x around 0 63.6%

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

      1. +-commutative 63.6%

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

      2. mul-1-neg 63.6%

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

      3. associate-*l* 63.6%

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

      4. *-commutative 63.6%

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

      5. distribute-lft-neg-in 63.6%

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

      6. associate-*l* 63.6%

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

      7. *-commutative 63.6%

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

      8. associate-*r* 63.6%

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

      9. *-commutative 63.6%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 15.7%

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

      1. *-commutative 15.7%

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

      2. *-commutative 15.7%

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

      3. *-commutative 15.7%

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

      4. add-sqr-sqrt 10.6%

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

      5. unpow2 10.6%

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

      6. associate-*r* 10.6%

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

      7. sqrt-prod 10.6%

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

      8. add-sqr-sqrt 7.5%

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

      9. sqrt-unprod 18.9%

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

      10. *-commutative 18.9%

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

      11. *-commutative 18.9%

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

      12. swap-sqr 21.8%

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

   10. Applied egg-rr 25.4%

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

   if 5.19999999999999988e220 < 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 16.1%

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

      1. add-sqr-sqrt 53.1%

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

      2. pow2 53.1%

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

   5. Applied egg-rr 11.1%

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

   6. Taylor expanded in x around inf 28.0%

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

      1. unpow2 28.0%

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

      2. rem-square-sqrt 28.0%

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

      3. +-commutative 28.0%

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

      4. mul-1-neg 28.0%

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

      5. unsub-neg 28.0%

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

   8. Simplified 28.0%

      \[\leadsto \left(\color{blue}{\left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 880:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(t\_1 \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 5e-18)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* t_1 (- y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 5e-18) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * (t_1 * -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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((t * t) <= 5d-18) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * (t_1 * -y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 5e-18) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (t_1 * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 5e-18:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * (t_1 * -y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 5e-18)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(t_1 * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 5e-18)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * (t_1 * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 5e-18], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 5.00000000000000036e-18

   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. Step-by-step derivation

      1. associate-*l* 99.7%

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

      2. remove-double-neg 99.7%

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

      3. remove-double-neg 99.7%

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

      4. exp-sqrt 99.7%

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

      5. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 99.3%

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

       1. unpow1/2 99.3%

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

       2. metadata-eval 99.3%

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

       3. pow-sqr 99.1%

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

       4. unpow1/2 99.1%

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

       5. metadata-eval 99.1%

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

       6. pow-sqr 99.0%

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

       7. unswap-sqr 99.0%

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

       8. *-commutative 99.0%

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

       9. exp-to-pow 99.0%

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

       10. exp-to-pow 95.0%

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

       11. exp-sum 94.6%

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

       12. distribute-rgt-in 94.6%

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

       13. *-commutative 94.6%

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

       14. exp-prod 94.6%

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

       15. exp-sum 95.0%

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

       16. rem-exp-log 95.0%

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

       17. rem-exp-log 99.3%

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

       18. *-commutative 99.3%

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

   11. Simplified 99.7%

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

   if 5.00000000000000036e-18 < (*.f64 t t)

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 51.8%

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

      2. pow2 51.8%

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

   4. Applied egg-rr 51.8%

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

   5. Taylor expanded in x around 0 80.9%

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

      1. neg-mul-1 80.9%

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

   7. Simplified 80.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.0)
     (* t_1 (sqrt (* z 2.0)))
     (* (* t (* t_1 (sqrt 2.0))) (sqrt z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(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 <= 1.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else
        tmp = (t * (t_1 * sqrt(2.0d0))) * sqrt(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 <= 1.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = (t * (t_1 * Math.sqrt(2.0))) * Math.sqrt(z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 1.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	else:
		tmp = (t * (t_1 * math.sqrt(2.0))) * math.sqrt(z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(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 <= 1.0)
		tmp = t_1 * sqrt((z * 2.0));
	else
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(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, 1.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   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. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 67.7%

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

       1. unpow1/2 67.7%

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

       2. metadata-eval 67.7%

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

       3. pow-sqr 67.6%

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

       4. unpow1/2 67.6%

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

       5. metadata-eval 67.6%

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

       6. pow-sqr 67.6%

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

       7. unswap-sqr 67.5%

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

       8. *-commutative 67.5%

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

       9. exp-to-pow 67.5%

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

       10. exp-to-pow 65.1%

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

       11. exp-sum 64.8%

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

       12. distribute-rgt-in 64.8%

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

       13. *-commutative 64.8%

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

       14. exp-prod 64.8%

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

       15. exp-sum 65.1%

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

       16. rem-exp-log 65.1%

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

       17. rem-exp-log 67.7%

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

       18. *-commutative 67.7%

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

   11. Simplified 68.0%

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

   if 1 < t

   1. Initial program 98.5%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 73.5%

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

       1. +-commutative 73.5%

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

       2. unpow2 73.5%

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

       3. fma-define 73.5%

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

   11. Simplified 73.5%

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

   12. Taylor expanded in t around inf 47.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0015:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(x \cdot \left(\sqrt{z} \cdot \left(0.5 - \frac{y}{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 0.0015)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (* (sqrt 2.0) (* x (* (sqrt z) (- 0.5 (/ y x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 0.0015) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt(2.0) * (x * (sqrt(z) * (0.5 - (y / x))));
	}
	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 <= 0.0015d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = sqrt(2.0d0) * (x * (sqrt(z) * (0.5d0 - (y / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 0.0015) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(2.0) * (x * (Math.sqrt(z) * (0.5 - (y / x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 0.0015:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(2.0) * (x * (math.sqrt(z) * (0.5 - (y / x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 0.0015)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(x * Float64(sqrt(z) * Float64(0.5 - Float64(y / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 0.0015)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = sqrt(2.0) * (x * (sqrt(z) * (0.5 - (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 0.0015], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(x * N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0015

   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. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 67.8%

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

       1. unpow1/2 67.8%

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

       2. metadata-eval 67.8%

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

       3. pow-sqr 67.7%

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

       4. unpow1/2 67.7%

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

       5. metadata-eval 67.7%

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

       6. pow-sqr 67.7%

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

       7. unswap-sqr 67.6%

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

       8. *-commutative 67.6%

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

       9. exp-to-pow 67.6%

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

       10. exp-to-pow 65.2%

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

       11. exp-sum 64.9%

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

       12. distribute-rgt-in 64.9%

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

       13. *-commutative 64.9%

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

       14. exp-prod 64.9%

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

       15. exp-sum 65.2%

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

       16. rem-exp-log 65.2%

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

       17. rem-exp-log 67.8%

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

       18. *-commutative 67.8%

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

   11. Simplified 68.1%

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

   if 0.0015 < t

   1. Initial program 98.5%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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 x around 0 60.6%

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

      1. +-commutative 60.6%

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

      2. mul-1-neg 60.6%

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

      3. associate-*l* 60.6%

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

      4. *-commutative 60.6%

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

      5. distribute-lft-neg-in 60.6%

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

      6. associate-*l* 60.6%

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

      7. *-commutative 60.6%

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

      8. associate-*r* 60.6%

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

      9. *-commutative 60.6%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 16.4%

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

   9. Taylor expanded in x around inf 30.7%

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

       1. associate-*r* 30.7%

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

       2. distribute-rgt-out 30.7%

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

       3. +-commutative 30.7%

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

       4. mul-1-neg 30.7%

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

       5. unsub-neg 30.7%

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

   11. Simplified 30.7%

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

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Add Preprocessing
3. Add Preprocessing

Alternative 9: 43.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq -1300 \lor \neg \left(x \leq 2.75 \cdot 10^{-81}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= x -1300.0) (not (<= x 2.75e-81)))
     (* 0.5 (* x t_1))
     (* t_1 (- y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((x <= -1300.0) || !(x <= 2.75e-81)) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = t_1 * -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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((x <= (-1300.0d0)) .or. (.not. (x <= 2.75d-81))) then
        tmp = 0.5d0 * (x * t_1)
    else
        tmp = t_1 * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((x <= -1300.0) || !(x <= 2.75e-81)) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (x <= -1300.0) or not (x <= 2.75e-81):
		tmp = 0.5 * (x * t_1)
	else:
		tmp = t_1 * -y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((x <= -1300.0) || !(x <= 2.75e-81))
		tmp = Float64(0.5 * Float64(x * t_1));
	else
		tmp = Float64(t_1 * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((x <= -1300.0) || ~((x <= 2.75e-81)))
		tmp = 0.5 * (x * t_1);
	else
		tmp = t_1 * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -1300.0], N[Not[LessEqual[x, 2.75e-81]], $MachinePrecision]], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1300 or 2.75000000000000013e-81 < x

   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. Add Preprocessing

   3. Taylor expanded in t around 0 59.6%

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

   4. Taylor expanded in x around inf 47.7%

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

      1. associate-*l* 47.7%

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

      2. *-commutative 47.7%

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

      3. unpow1/2 47.7%

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

      4. metadata-eval 47.7%

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

      5. pow-sqr 47.7%

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

      6. unpow1/2 47.7%

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

      7. metadata-eval 47.7%

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

      8. pow-sqr 47.7%

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

      9. unswap-sqr 47.6%

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

      10. *-commutative 47.6%

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

      11. exp-to-pow 47.6%

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

      12. exp-to-pow 46.2%

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

      13. exp-sum 46.1%

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

      14. distribute-rgt-in 46.1%

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

      15. *-commutative 46.1%

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

      16. exp-prod 46.1%

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

      17. exp-sum 46.2%

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

      18. rem-exp-log 46.2%

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

      19. rem-exp-log 47.7%

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

      20. *-commutative 47.7%

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

   6. Simplified 47.9%

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

   if -1300 < x < 2.75000000000000013e-81

   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 49.3%

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

   4. Taylor expanded in x around 0 41.5%

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

      1. mul-1-neg 41.5%

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

      2. associate-*l* 41.5%

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

      3. *-commutative 41.5%

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

      4. unpow1/2 41.5%

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

      5. metadata-eval 41.5%

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

      6. pow-sqr 41.4%

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

      7. unpow1/2 41.4%

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

      8. metadata-eval 41.4%

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

      9. pow-sqr 41.4%

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

      10. unswap-sqr 41.4%

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

      11. *-commutative 41.4%

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

      12. exp-to-pow 41.4%

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

      13. exp-to-pow 40.1%

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

      14. exp-sum 39.7%

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

      15. distribute-rgt-in 39.7%

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

      16. *-commutative 39.7%

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

      17. exp-prod 39.7%

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

      18. exp-sum 40.1%

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

      19. rem-exp-log 40.1%

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

      20. rem-exp-log 41.5%

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

      21. *-commutative 41.5%

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

   6. Simplified 41.6%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1300 \lor \neg \left(x \leq 2.75 \cdot 10^{-81}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 0.0015:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 0.0015) (* (- (* x 0.5) y) t_1) (* t_1 (* x (- 0.5 (/ y x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 0.0015) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	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((z * 2.0d0))
    if (t <= 0.0015d0) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = t_1 * (x * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 0.0015) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 0.0015:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = t_1 * (x * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 0.0015)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(t_1 * Float64(x * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 0.0015)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = t_1 * (x * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.0015], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0015

   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. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 67.8%

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

       1. unpow1/2 67.8%

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

       2. metadata-eval 67.8%

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

       3. pow-sqr 67.7%

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

       4. unpow1/2 67.7%

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

       5. metadata-eval 67.7%

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

       6. pow-sqr 67.7%

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

       7. unswap-sqr 67.6%

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

       8. *-commutative 67.6%

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

       9. exp-to-pow 67.6%

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

       10. exp-to-pow 65.2%

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

       11. exp-sum 64.9%

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

       12. distribute-rgt-in 64.9%

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

       13. *-commutative 64.9%

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

       14. exp-prod 64.9%

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

       15. exp-sum 65.2%

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

       16. rem-exp-log 65.2%

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

       17. rem-exp-log 67.8%

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

       18. *-commutative 67.8%

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

   11. Simplified 68.1%

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

   if 0.0015 < t

   1. Initial program 98.5%

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

   3. Taylor expanded in t around 0 16.4%

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

      1. add-sqr-sqrt 50.0%

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

      2. pow2 50.0%

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

   5. Applied egg-rr 10.7%

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

   6. Taylor expanded in x around inf 26.4%

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

      1. unpow2 26.4%

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

      2. rem-square-sqrt 26.4%

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

      3. +-commutative 26.4%

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

      4. mul-1-neg 26.4%

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

      5. unsub-neg 26.4%

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

   8. Simplified 26.4%

      \[\leadsto \left(\color{blue}{\left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0015:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+40}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\frac{{\left(z \cdot 2\right)}^{1.5}}{z \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 6e+40)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (* y (- (/ (pow (* z 2.0) 1.5) (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 6e+40) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = y * -(pow((z * 2.0), 1.5) / (z * 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) :: tmp
    if (t <= 6d+40) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = y * -(((z * 2.0d0) ** 1.5d0) / (z * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 6e+40) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = y * -(Math.pow((z * 2.0), 1.5) / (z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 6e+40:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = y * -(math.pow((z * 2.0), 1.5) / (z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 6e+40)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(y * Float64(-Float64((Float64(z * 2.0) ^ 1.5) / Float64(z * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 6e+40)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = y * -(((z * 2.0) ^ 1.5) / (z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 6e+40], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * (-N[(N[Power[N[(z * 2.0), $MachinePrecision], 1.5], $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000004e40

   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. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 65.7%

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

       1. unpow1/2 65.7%

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

       2. metadata-eval 65.7%

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

       3. pow-sqr 65.5%

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

       4. unpow1/2 65.5%

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

       5. metadata-eval 65.5%

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

       6. pow-sqr 65.5%

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

       7. unswap-sqr 65.5%

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

       8. *-commutative 65.5%

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

       9. exp-to-pow 65.5%

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

       10. exp-to-pow 63.1%

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

       11. exp-sum 62.9%

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

       12. distribute-rgt-in 62.9%

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

       13. *-commutative 62.9%

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

       14. exp-prod 62.9%

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

       15. exp-sum 63.1%

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

       16. rem-exp-log 63.1%

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

       17. rem-exp-log 65.6%

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

       18. *-commutative 65.6%

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

   11. Simplified 65.9%

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

   if 6.0000000000000004e40 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0 15.9%

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

   4. Taylor expanded in x around 0 10.1%

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

      1. mul-1-neg 10.1%

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

      2. associate-*l* 10.1%

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

      3. *-commutative 10.1%

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

      4. unpow1/2 10.1%

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

      5. metadata-eval 10.1%

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

      6. pow-sqr 10.1%

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

      7. unpow1/2 10.1%

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

      8. metadata-eval 10.1%

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

      9. pow-sqr 10.1%

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

      10. unswap-sqr 10.1%

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

      11. *-commutative 10.1%

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

      12. exp-to-pow 10.1%

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

      13. exp-to-pow 10.1%

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

      14. exp-sum 10.1%

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

      15. distribute-rgt-in 10.1%

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

      16. *-commutative 10.1%

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

      17. exp-prod 10.1%

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

      18. exp-sum 10.1%

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

      19. rem-exp-log 10.1%

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

      20. rem-exp-log 10.1%

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

      21. *-commutative 10.1%

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

   6. Simplified 10.1%

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

      1. neg-sub0 10.1%

         \[\leadsto y \cdot \color{blue}{\left(0 - \sqrt{z \cdot 2}\right)} \]

      2. flip3-- 20.0%

         \[\leadsto y \cdot \color{blue}{\frac{{0}^{3} - {\left(\sqrt{z \cdot 2}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)}} \]

      3. metadata-eval 20.0%

         \[\leadsto y \cdot \frac{\color{blue}{0} - {\left(\sqrt{z \cdot 2}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      4. pow3 20.0%

         \[\leadsto y \cdot \frac{0 - \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{z \cdot 2}}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      5. add-sqr-sqrt 20.0%

         \[\leadsto y \cdot \frac{0 - \color{blue}{\left(z \cdot 2\right)} \cdot \sqrt{z \cdot 2}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      6. pow1 20.0%

         \[\leadsto y \cdot \frac{0 - \color{blue}{{\left(z \cdot 2\right)}^{1}} \cdot \sqrt{z \cdot 2}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      7. pow1/2 20.0%

         \[\leadsto y \cdot \frac{0 - {\left(z \cdot 2\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      8. pow-prod-up 20.0%

         \[\leadsto y \cdot \frac{0 - \color{blue}{{\left(z \cdot 2\right)}^{\left(1 + 0.5\right)}}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      9. metadata-eval 20.0%

         \[\leadsto y \cdot \frac{0 - {\left(z \cdot 2\right)}^{\color{blue}{1.5}}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      10. metadata-eval 20.0%

          \[\leadsto y \cdot \frac{0 - {\left(z \cdot 2\right)}^{1.5}}{\color{blue}{0} + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      11. add-sqr-sqrt 20.0%

          \[\leadsto y \cdot \frac{0 - {\left(z \cdot 2\right)}^{1.5}}{0 + \left(\color{blue}{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

   8. Applied egg-rr 20.0%

      \[\leadsto y \cdot \color{blue}{\frac{0 - {\left(z \cdot 2\right)}^{1.5}}{0 + \left(z \cdot 2 + 0 \cdot \sqrt{z \cdot 2}\right)}} \]
   9. Step-by-step derivation

      1. sub0-neg 20.0%

         \[\leadsto y \cdot \frac{\color{blue}{-{\left(z \cdot 2\right)}^{1.5}}}{0 + \left(z \cdot 2 + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      2. *-commutative 20.0%

         \[\leadsto y \cdot \frac{-{\color{blue}{\left(2 \cdot z\right)}}^{1.5}}{0 + \left(z \cdot 2 + 0 \cdot \sqrt{z \cdot 2}\right)} \]

      3. +-lft-identity 20.0%

         \[\leadsto y \cdot \frac{-{\left(2 \cdot z\right)}^{1.5}}{\color{blue}{z \cdot 2 + 0 \cdot \sqrt{z \cdot 2}}} \]

      4. mul0-lft 20.0%

         \[\leadsto y \cdot \frac{-{\left(2 \cdot z\right)}^{1.5}}{z \cdot 2 + \color{blue}{0}} \]

      5. +-rgt-identity 20.0%

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

      6. *-commutative 20.0%

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

   10. Simplified 20.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+40}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\frac{{\left(z \cdot 2\right)}^{1.5}}{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in t around 0 54.6%

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

    1. unpow1/2 54.6%

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

    2. metadata-eval 54.6%

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

    3. pow-sqr 54.5%

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

    4. unpow1/2 54.5%

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

    5. metadata-eval 54.5%

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

    6. pow-sqr 54.5%

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

    7. unswap-sqr 54.4%

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

    8. *-commutative 54.4%

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

    9. exp-to-pow 54.4%

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

    10. exp-to-pow 52.6%

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

    11. exp-sum 52.4%

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

    12. distribute-rgt-in 52.4%

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

    13. *-commutative 52.4%

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

    14. exp-prod 52.4%

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

    15. exp-sum 52.6%

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

    16. rem-exp-log 52.6%

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

    17. rem-exp-log 54.6%

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

    18. *-commutative 54.6%

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

11. Simplified 54.8%

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

Alternative 13: 29.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in t around 0 54.8%

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

4. Taylor expanded in x around 0 26.2%

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

   1. mul-1-neg 26.2%

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

   2. associate-*l* 26.2%

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

   3. *-commutative 26.2%

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

   4. unpow1/2 26.2%

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

   5. metadata-eval 26.2%

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

   6. pow-sqr 26.1%

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

   7. unpow1/2 26.1%

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

   8. metadata-eval 26.1%

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

   9. pow-sqr 26.1%

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

   10. unswap-sqr 26.1%

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

   11. *-commutative 26.1%

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

   12. exp-to-pow 26.1%

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

   13. exp-to-pow 25.4%

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

   14. exp-sum 25.2%

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

   15. distribute-rgt-in 25.2%

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

   16. *-commutative 25.2%

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

   17. exp-prod 25.2%

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

   18. exp-sum 25.4%

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

   19. rem-exp-log 25.4%

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

   20. rem-exp-log 26.2%

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

   21. *-commutative 26.2%

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

6. Simplified 26.2%

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

7. Final simplification 26.2%

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

Alternative 14: 2.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0 54.8%

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

4. Taylor expanded in x around 0 26.2%

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

   1. mul-1-neg 26.2%

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

   2. associate-*l* 26.2%

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

   3. *-commutative 26.2%

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

   4. unpow1/2 26.2%

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

   5. metadata-eval 26.2%

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

   6. pow-sqr 26.1%

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

   7. unpow1/2 26.1%

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

   8. metadata-eval 26.1%

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

   9. pow-sqr 26.1%

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

   10. unswap-sqr 26.1%

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

   11. *-commutative 26.1%

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

   12. exp-to-pow 26.1%

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

   13. exp-to-pow 25.4%

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

   14. exp-sum 25.2%

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

   15. distribute-rgt-in 25.2%

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

   16. *-commutative 25.2%

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

   17. exp-prod 25.2%

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

   18. exp-sum 25.4%

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

   19. rem-exp-log 25.4%

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

   20. rem-exp-log 26.2%

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

   21. *-commutative 26.2%

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

6. Simplified 26.2%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 2.6%

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

   3. sqr-neg 2.6%

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

   4. add-sqr-sqrt 2.6%

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

8. Applied egg-rr 2.6%

   \[\leadsto y \cdot \color{blue}{\sqrt{z \cdot 2}} \]
9. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

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