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

Percentage Accurate: 99.4% → 99.8%

Time: 13.7s

Alternatives: 11

Speedup: 1.0×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% 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: 99.8% accurate, 0.7× speedup

Math

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

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(z * Float64(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[(z * N[(2.0 * N[Exp[N[Power[t, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. 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. Step-by-step derivation

   1. expm1-log1p-u 98.6%

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

   2. expm1-udef 74.4%

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

   3. sqrt-unprod 74.4%

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

   4. associate-*l* 74.4%

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

   5. pow-exp 74.4%

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

   6. pow2 74.4%

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

5. Applied egg-rr 74.4%

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

   1. expm1-def 98.6%

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

   2. expm1-log1p 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 65:\\ \;\;\;\;t_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \sqrt{\frac{{z}^{2}}{z \cdot 2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 65.0)
     (* t_1 (sqrt (* z 2.0)))
     (if (<= t 1.42e+79)
       (sqrt (* (pow y 2.0) (* z 2.0)))
       (* t_1 (* 2.0 (sqrt (/ (pow z 2.0) (* z 2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 65.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else if (t <= 1.42e+79) {
		tmp = sqrt((pow(y, 2.0) * (z * 2.0)));
	} else {
		tmp = t_1 * (2.0 * sqrt((pow(z, 2.0) / (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) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 65.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else if (t <= 1.42d+79) then
        tmp = sqrt(((y ** 2.0d0) * (z * 2.0d0)))
    else
        tmp = t_1 * (2.0d0 * sqrt(((z ** 2.0d0) / (z * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 65.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else if (t <= 1.42e+79) {
		tmp = Math.sqrt((Math.pow(y, 2.0) * (z * 2.0)));
	} else {
		tmp = t_1 * (2.0 * Math.sqrt((Math.pow(z, 2.0) / (z * 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 65.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	elif t <= 1.42e+79:
		tmp = math.sqrt((math.pow(y, 2.0) * (z * 2.0)))
	else:
		tmp = t_1 * (2.0 * math.sqrt((math.pow(z, 2.0) / (z * 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 65.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	elseif (t <= 1.42e+79)
		tmp = sqrt(Float64((y ^ 2.0) * Float64(z * 2.0)));
	else
		tmp = Float64(t_1 * Float64(2.0 * sqrt(Float64((z ^ 2.0) / Float64(z * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 65.0)
		tmp = t_1 * sqrt((z * 2.0));
	elseif (t <= 1.42e+79)
		tmp = sqrt(((y ^ 2.0) * (z * 2.0)));
	else
		tmp = t_1 * (2.0 * sqrt(((z ^ 2.0) / (z * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 65.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.42e+79], N[Sqrt[N[(N[Power[y, 2.0], $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Sqrt[N[(N[Power[z, 2.0], $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 65

   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. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 69.5%

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

   if 65 < t < 1.41999999999999998e79

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 15.5%

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

   5. Taylor expanded in x around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. associate-*l* 8.7%

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

      3. distribute-rgt-neg-in 8.7%

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

   7. Simplified 8.7%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-udef 2.8%

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

   9. Applied egg-rr 1.2%

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

       1. expm1-def 1.2%

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

       2. expm1-log1p 1.7%

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

   11. Simplified 1.7%

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

       1. add-sqr-sqrt 1.0%

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

       2. sqrt-unprod 31.0%

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

       3. *-commutative 31.0%

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

       4. *-commutative 31.0%

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

       5. swap-sqr 53.7%

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

       6. add-sqr-sqrt 53.7%

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

       7. *-commutative 53.7%

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

       8. pow2 53.7%

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

   13. Applied egg-rr 53.7%

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

       1. *-commutative 53.7%

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

   15. Simplified 53.7%

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

   if 1.41999999999999998e79 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 8.4%

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

      1. add-cbrt-cube 18.3%

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

      2. sqrt-prod 18.3%

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

      3. sqrt-prod 18.3%

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

      4. add-sqr-sqrt 18.3%

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

      5. pow1 18.3%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{\color{blue}{{\left(z \cdot 2\right)}^{1}} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)} \]

      6. sqrt-prod 18.3%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1} \cdot \color{blue}{\sqrt{z \cdot 2}}} \]

      7. pow1/2 18.3%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}} \]

      8. pow-prod-up 18.3%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{\color{blue}{{\left(z \cdot 2\right)}^{\left(1 + 0.5\right)}}} \]

      9. metadata-eval 18.3%

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

      10. pow1/3 18.3%

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

   6. Applied egg-rr 18.3%

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

      1. pow-pow 8.4%

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

      2. *-commutative 8.4%

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

      3. metadata-eval 8.4%

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

      4. metadata-eval 8.4%

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

      5. pow-div 8.4%

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

      6. pow1 8.4%

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

      7. pow1/2 8.4%

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

      8. *-un-lft-identity 8.4%

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

      9. times-frac 8.4%

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

      10. metadata-eval 8.4%

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

      11. *-commutative 8.4%

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

   8. Applied egg-rr 8.4%

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

      1. add-sqr-sqrt 8.4%

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

      2. sqrt-unprod 8.4%

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

      3. frac-times 30.3%

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

      4. pow2 30.3%

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

      5. add-sqr-sqrt 30.3%

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

      6. *-commutative 30.3%

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

   10. Applied egg-rr 30.3%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 65:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(2 \cdot \sqrt{\frac{{z}^{2}}{z \cdot 2}}\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Final simplification 99.4%

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

Alternative 4: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 20:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+230}:\\ \;\;\;\;{\left({\left(z \cdot 2\right)}^{1.5}\right)}^{0.3333333333333333} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 20.0)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (if (<= t 3.8e+79)
     (sqrt (* (pow y 2.0) (* z 2.0)))
     (if (<= t 2.85e+230)
       (* (pow (pow (* z 2.0) 1.5) 0.3333333333333333) (- y))
       (sqrt (* z (* 0.5 (pow x 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 20.0) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else if (t <= 3.8e+79) {
		tmp = sqrt((pow(y, 2.0) * (z * 2.0)));
	} else if (t <= 2.85e+230) {
		tmp = pow(pow((z * 2.0), 1.5), 0.3333333333333333) * -y;
	} else {
		tmp = sqrt((z * (0.5 * pow(x, 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 <= 20.0d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else if (t <= 3.8d+79) then
        tmp = sqrt(((y ** 2.0d0) * (z * 2.0d0)))
    else if (t <= 2.85d+230) then
        tmp = (((z * 2.0d0) ** 1.5d0) ** 0.3333333333333333d0) * -y
    else
        tmp = sqrt((z * (0.5d0 * (x ** 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 <= 20.0) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else if (t <= 3.8e+79) {
		tmp = Math.sqrt((Math.pow(y, 2.0) * (z * 2.0)));
	} else if (t <= 2.85e+230) {
		tmp = Math.pow(Math.pow((z * 2.0), 1.5), 0.3333333333333333) * -y;
	} else {
		tmp = Math.sqrt((z * (0.5 * Math.pow(x, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 20.0:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	elif t <= 3.8e+79:
		tmp = math.sqrt((math.pow(y, 2.0) * (z * 2.0)))
	elif t <= 2.85e+230:
		tmp = math.pow(math.pow((z * 2.0), 1.5), 0.3333333333333333) * -y
	else:
		tmp = math.sqrt((z * (0.5 * math.pow(x, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 20.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	elseif (t <= 3.8e+79)
		tmp = sqrt(Float64((y ^ 2.0) * Float64(z * 2.0)));
	elseif (t <= 2.85e+230)
		tmp = Float64(((Float64(z * 2.0) ^ 1.5) ^ 0.3333333333333333) * Float64(-y));
	else
		tmp = sqrt(Float64(z * Float64(0.5 * (x ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 20.0)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	elseif (t <= 3.8e+79)
		tmp = sqrt(((y ^ 2.0) * (z * 2.0)));
	elseif (t <= 2.85e+230)
		tmp = (((z * 2.0) ^ 1.5) ^ 0.3333333333333333) * -y;
	else
		tmp = sqrt((z * (0.5 * (x ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 20.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e+79], N[Sqrt[N[(N[Power[y, 2.0], $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 2.85e+230], N[(N[Power[N[Power[N[(z * 2.0), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * (-y)), $MachinePrecision], N[Sqrt[N[(z * N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 20

   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. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 69.5%

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

   if 20 < t < 3.8000000000000002e79

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 15.5%

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

   5. Taylor expanded in x around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. associate-*l* 8.7%

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

      3. distribute-rgt-neg-in 8.7%

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

   7. Simplified 8.7%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-udef 2.8%

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

   9. Applied egg-rr 1.2%

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

       1. expm1-def 1.2%

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

       2. expm1-log1p 1.7%

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

   11. Simplified 1.7%

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

       1. add-sqr-sqrt 1.0%

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

       2. sqrt-unprod 31.0%

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

       3. *-commutative 31.0%

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

       4. *-commutative 31.0%

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

       5. swap-sqr 53.7%

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

       6. add-sqr-sqrt 53.7%

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

       7. *-commutative 53.7%

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

       8. pow2 53.7%

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

   13. Applied egg-rr 53.7%

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

       1. *-commutative 53.7%

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

   15. Simplified 53.7%

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

   if 3.8000000000000002e79 < t < 2.8500000000000001e230

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 10.7%

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

   5. Taylor expanded in x around 0 9.1%

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

      1. mul-1-neg 9.1%

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

      2. associate-*l* 9.1%

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

      3. distribute-rgt-neg-in 9.1%

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

   7. Simplified 9.1%

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

      1. *-commutative 9.1%

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

      2. sqrt-prod 9.1%

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

      3. pow1/2 9.1%

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

      4. metadata-eval 9.1%

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

      5. pow-pow 15.1%

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

   9. Applied egg-rr 15.1%

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

   if 2.8500000000000001e230 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 3.8%

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

   5. Taylor expanded in x around inf 3.2%

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

      1. add-sqr-sqrt 0.9%

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

      2. sqrt-unprod 7.7%

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

      3. *-commutative 7.7%

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

      4. *-commutative 7.7%

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

      5. swap-sqr 7.7%

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

      6. *-commutative 7.7%

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

      7. *-commutative 7.7%

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

      8. swap-sqr 19.7%

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

      9. add-sqr-sqrt 19.7%

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

      10. *-commutative 19.7%

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

      11. *-commutative 19.7%

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

      12. swap-sqr 19.7%

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

      13. rem-square-sqrt 19.7%

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

      14. pow2 19.7%

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

      15. metadata-eval 19.7%

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

   7. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot \left(2 \cdot {x}^{2}\right)\right) \cdot 0.25}} \]
   8. Step-by-step derivation

      1. associate-*l* 19.7%

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

      2. *-commutative 19.7%

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

      3. associate-*l* 19.7%

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

      4. metadata-eval 19.7%

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

   9. Simplified 19.7%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 20:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+230}:\\ \;\;\;\;{\left({\left(z \cdot 2\right)}^{1.5}\right)}^{0.3333333333333333} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \end{array} \]

Alternative 5: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 27:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(z \cdot 2\right)}^{3}\right)}^{0.16666666666666666} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 27.0)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (if (<= t 6.3e+78)
     (sqrt (* (pow y 2.0) (* z 2.0)))
     (* (pow (pow (* z 2.0) 3.0) 0.16666666666666666) (- y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 27.0) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else if (t <= 6.3e+78) {
		tmp = sqrt((pow(y, 2.0) * (z * 2.0)));
	} else {
		tmp = pow(pow((z * 2.0), 3.0), 0.16666666666666666) * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 27.0d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else if (t <= 6.3d+78) then
        tmp = sqrt(((y ** 2.0d0) * (z * 2.0d0)))
    else
        tmp = (((z * 2.0d0) ** 3.0d0) ** 0.16666666666666666d0) * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 27.0) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else if (t <= 6.3e+78) {
		tmp = Math.sqrt((Math.pow(y, 2.0) * (z * 2.0)));
	} else {
		tmp = Math.pow(Math.pow((z * 2.0), 3.0), 0.16666666666666666) * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 27.0:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	elif t <= 6.3e+78:
		tmp = math.sqrt((math.pow(y, 2.0) * (z * 2.0)))
	else:
		tmp = math.pow(math.pow((z * 2.0), 3.0), 0.16666666666666666) * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 27.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	elseif (t <= 6.3e+78)
		tmp = sqrt(Float64((y ^ 2.0) * Float64(z * 2.0)));
	else
		tmp = Float64(((Float64(z * 2.0) ^ 3.0) ^ 0.16666666666666666) * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 27.0)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	elseif (t <= 6.3e+78)
		tmp = sqrt(((y ^ 2.0) * (z * 2.0)));
	else
		tmp = (((z * 2.0) ^ 3.0) ^ 0.16666666666666666) * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 27.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.3e+78], N[Sqrt[N[(N[Power[y, 2.0], $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[Power[N[(z * 2.0), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision] * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 27

   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. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 69.5%

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

   if 27 < t < 6.3000000000000002e78

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 15.5%

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

   5. Taylor expanded in x around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. associate-*l* 8.7%

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

      3. distribute-rgt-neg-in 8.7%

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

   7. Simplified 8.7%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-udef 2.8%

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

   9. Applied egg-rr 1.2%

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

       1. expm1-def 1.2%

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

       2. expm1-log1p 1.7%

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

   11. Simplified 1.7%

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

       1. add-sqr-sqrt 1.0%

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

       2. sqrt-unprod 31.0%

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

       3. *-commutative 31.0%

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

       4. *-commutative 31.0%

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

       5. swap-sqr 53.7%

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

       6. add-sqr-sqrt 53.7%

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

       7. *-commutative 53.7%

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

       8. pow2 53.7%

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

   13. Applied egg-rr 53.7%

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

       1. *-commutative 53.7%

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

   15. Simplified 53.7%

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

   if 6.3000000000000002e78 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 8.4%

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

   5. Taylor expanded in x around 0 6.7%

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

      1. mul-1-neg 6.7%

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

      2. associate-*l* 6.7%

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

      3. distribute-rgt-neg-in 6.7%

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

   7. Simplified 6.7%

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

      1. *-commutative 6.7%

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

      2. sqrt-prod 6.7%

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

      3. pow1/2 6.7%

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

      4. metadata-eval 6.7%

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

      5. pow-pow 12.7%

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

      6. sqr-pow 12.7%

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

      7. pow-prod-down 20.7%

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

      8. *-commutative 20.7%

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

      9. *-commutative 20.7%

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

      10. pow-prod-up 20.7%

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

      11. *-commutative 20.7%

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

      12. metadata-eval 20.7%

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

      13. metadata-eval 20.7%

          \[\leadsto y \cdot \left(-{\left({\left(z \cdot 2\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}}\right) \]

   9. Applied egg-rr 20.7%

      \[\leadsto y \cdot \left(-\color{blue}{{\left({\left(z \cdot 2\right)}^{3}\right)}^{0.16666666666666666}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 27:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(z \cdot 2\right)}^{3}\right)}^{0.16666666666666666} \cdot \left(-y\right)\\ \end{array} \]

Alternative 6: 57.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{+72}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.85e+72)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (sqrt (* z (* 0.5 (pow x 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.85e+72) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt((z * (0.5 * pow(x, 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 <= 1.85d+72) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = sqrt((z * (0.5d0 * (x ** 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 <= 1.85e+72) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt((z * (0.5 * Math.pow(x, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.85e+72:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt((z * (0.5 * math.pow(x, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.85e+72)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(z * Float64(0.5 * (x ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.85e+72)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = sqrt((z * (0.5 * (x ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.85e+72], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(z * N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000001e72

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 65.7%

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

   if 1.8500000000000001e72 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 8.2%

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

   5. Taylor expanded in x around inf 4.9%

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

      1. add-sqr-sqrt 1.0%

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

      2. sqrt-unprod 11.4%

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

      3. *-commutative 11.4%

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

      4. *-commutative 11.4%

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

      5. swap-sqr 11.4%

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

      6. *-commutative 11.4%

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

      7. *-commutative 11.4%

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

      8. swap-sqr 17.3%

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

      9. add-sqr-sqrt 17.3%

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

      10. *-commutative 17.3%

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

      11. *-commutative 17.3%

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

      12. swap-sqr 17.3%

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

      13. rem-square-sqrt 17.3%

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

      14. pow2 17.3%

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

      15. metadata-eval 17.3%

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

   7. Applied egg-rr 17.3%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot \left(2 \cdot {x}^{2}\right)\right) \cdot 0.25}} \]
   8. Step-by-step derivation

      1. associate-*l* 17.3%

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

      2. *-commutative 17.3%

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

      3. associate-*l* 17.3%

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

      4. metadata-eval 17.3%

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

   9. Simplified 17.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{+72}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \end{array} \]

Alternative 7: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 14.6:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 14.6)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (sqrt (* (pow y 2.0) (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 14.6) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt((pow(y, 2.0) * (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 <= 14.6d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = sqrt(((y ** 2.0d0) * (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 <= 14.6) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt((Math.pow(y, 2.0) * (z * 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 14.6:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt((math.pow(y, 2.0) * (z * 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 14.6)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64((y ^ 2.0) * Float64(z * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 14.6)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = sqrt(((y ^ 2.0) * (z * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 14.6], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[y, 2.0], $MachinePrecision] * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 14.5999999999999996

   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. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 69.5%

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

   if 14.5999999999999996 < t

   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. Step-by-step derivation

      1. *-commutative 98.4%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 10.3%

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

   5. Taylor expanded in x around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. associate-*l* 7.2%

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

      3. distribute-rgt-neg-in 7.2%

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

   7. Simplified 7.2%

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

      1. expm1-log1p-u 3.6%

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

      2. expm1-udef 3.5%

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

   9. Applied egg-rr 1.0%

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

       1. expm1-def 1.0%

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

       2. expm1-log1p 1.3%

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

   11. Simplified 1.3%

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

       1. add-sqr-sqrt 0.7%

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

       2. sqrt-unprod 13.9%

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

       3. *-commutative 13.9%

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

       4. *-commutative 13.9%

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

       5. swap-sqr 19.9%

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

       6. add-sqr-sqrt 19.9%

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

       7. *-commutative 19.9%

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

       8. pow2 19.9%

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

   13. Applied egg-rr 19.9%

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

       1. *-commutative 19.9%

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

   15. Simplified 19.9%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 14.6:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}\\ \end{array} \]

Alternative 8: 42.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+16} \lor \neg \left(y \leq 1.85 \cdot 10^{-38}\right):\\ \;\;\;\;y \cdot \left(-t_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= y -3.8e+16) (not (<= y 1.85e-38)))
     (* y (- t_1))
     (* 0.5 (* x t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((y <= -3.8e+16) || !(y <= 1.85e-38)) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((y <= (-3.8d+16)) .or. (.not. (y <= 1.85d-38))) then
        tmp = y * -t_1
    else
        tmp = 0.5d0 * (x * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((y <= -3.8e+16) || !(y <= 1.85e-38)) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (y <= -3.8e+16) or not (y <= 1.85e-38):
		tmp = y * -t_1
	else:
		tmp = 0.5 * (x * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((y <= -3.8e+16) || !(y <= 1.85e-38))
		tmp = Float64(y * Float64(-t_1));
	else
		tmp = Float64(0.5 * Float64(x * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((y <= -3.8e+16) || ~((y <= 1.85e-38)))
		tmp = y * -t_1;
	else
		tmp = 0.5 * (x * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -3.8e+16], N[Not[LessEqual[y, 1.85e-38]], $MachinePrecision]], N[(y * (-t$95$1)), $MachinePrecision], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8e16 or 1.85e-38 < y

   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. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 56.0%

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

   5. Taylor expanded in x around 0 43.4%

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

      1. mul-1-neg 43.4%

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

      2. associate-*l* 43.5%

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

      3. distribute-rgt-neg-in 43.5%

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

   7. Simplified 43.5%

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

      1. sqrt-prod 43.6%

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

      2. distribute-rgt-neg-out 43.6%

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

      3. neg-sub0 43.6%

         \[\leadsto \color{blue}{0 - y \cdot \sqrt{2 \cdot z}} \]

      4. *-commutative 43.6%

         \[\leadsto 0 - y \cdot \sqrt{\color{blue}{z \cdot 2}} \]

   9. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{0 - y \cdot \sqrt{z \cdot 2}} \]
   10. Step-by-step derivation

       1. neg-sub0 43.6%

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

       2. *-commutative 43.6%

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

       3. distribute-rgt-neg-in 43.6%

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

   11. Simplified 43.6%

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

   if -3.8e16 < y < 1.85e-38

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 53.2%

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

   5. Taylor expanded in x around inf 44.5%

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

      1. expm1-log1p-u 29.6%

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

      2. expm1-udef 18.7%

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

      3. *-commutative 18.7%

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

      4. *-commutative 18.7%

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

      5. associate-*r* 18.7%

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

      6. sqrt-prod 18.7%

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

      7. *-commutative 18.7%

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

      8. *-commutative 18.7%

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

   7. Applied egg-rr 18.7%

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

      1. expm1-def 29.7%

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

      2. expm1-log1p 44.6%

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

   9. Simplified 44.6%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+16} \lor \neg \left(y \leq 1.85 \cdot 10^{-38}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]

Alternative 9: 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. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 54.7%

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

5. Final simplification 54.7%

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

Alternative 10: 29.2% accurate, 2.0× speedup

Math

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

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 * Float64(-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. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 54.7%

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

5. Taylor expanded in x around 0 28.1%

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

   1. mul-1-neg 28.1%

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

   2. associate-*l* 28.1%

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

   3. distribute-rgt-neg-in 28.1%

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

7. Simplified 28.1%

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

   1. sqrt-prod 28.2%

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

   2. distribute-rgt-neg-out 28.2%

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

   3. neg-sub0 28.2%

      \[\leadsto \color{blue}{0 - y \cdot \sqrt{2 \cdot z}} \]

   4. *-commutative 28.2%

      \[\leadsto 0 - y \cdot \sqrt{\color{blue}{z \cdot 2}} \]

9. Applied egg-rr 28.2%

   \[\leadsto \color{blue}{0 - y \cdot \sqrt{z \cdot 2}} \]
10. Step-by-step derivation

    1. neg-sub0 28.2%

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

    2. *-commutative 28.2%

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

    3. distribute-rgt-neg-in 28.2%

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

11. Simplified 28.2%

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

12. Final simplification 28.2%

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

Alternative 11: 2.6% 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. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 54.7%

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

5. Taylor expanded in x around 0 28.1%

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

   1. mul-1-neg 28.1%

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

   2. associate-*l* 28.1%

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

   3. distribute-rgt-neg-in 28.1%

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

7. Simplified 28.1%

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

   1. expm1-log1p-u 16.5%

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

   2. expm1-udef 10.6%

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

9. Applied egg-rr 2.4%

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

    1. expm1-def 2.5%

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

    2. expm1-log1p 2.8%

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

11. Simplified 2.8%

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

12. Final simplification 2.8%

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

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023322 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

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