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

Percentage Accurate: 99.6% → 99.8%

Time: 11.7s

Alternatives: 9

Speedup: 1.0×

• 45.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.2%

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

   2. expm1-udef 77.2%

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

   3. sqrt-unprod 77.2%

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

   4. associate-*l* 77.2%

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

   5. exp-prod 77.2%

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

5. Applied egg-rr 77.2%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 99.8%

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

   3. exp-prod 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 86.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;z \cdot 2 \leq 10^{+118}:\\ \;\;\;\;\left(0.5 \cdot \left(t \cdot t\right) + 1\right) \cdot \left(t_1 \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* z 2.0) 1e+118)
     (* (+ (* 0.5 (* t t)) 1.0) (* t_1 (sqrt (* z 2.0))))
     (* t_1 (sqrt (* 2.0 (+ z (* z (* t t)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((z * 2.0) <= 1e+118) {
		tmp = ((0.5 * (t * t)) + 1.0) * (t_1 * sqrt((z * 2.0)));
	} else {
		tmp = t_1 * sqrt((2.0 * (z + (z * (t * t)))));
	}
	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 ((z * 2.0d0) <= 1d+118) then
        tmp = ((0.5d0 * (t * t)) + 1.0d0) * (t_1 * sqrt((z * 2.0d0)))
    else
        tmp = t_1 * sqrt((2.0d0 * (z + (z * (t * t)))))
    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 ((z * 2.0) <= 1e+118) {
		tmp = ((0.5 * (t * t)) + 1.0) * (t_1 * Math.sqrt((z * 2.0)));
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z + (z * (t * t)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (z * 2.0) <= 1e+118:
		tmp = ((0.5 * (t * t)) + 1.0) * (t_1 * math.sqrt((z * 2.0)))
	else:
		tmp = t_1 * math.sqrt((2.0 * (z + (z * (t * t)))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(z * 2.0) <= 1e+118)
		tmp = Float64(Float64(Float64(0.5 * Float64(t * t)) + 1.0) * Float64(t_1 * sqrt(Float64(z * 2.0))));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z + Float64(z * Float64(t * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((z * 2.0) <= 1e+118)
		tmp = ((0.5 * (t * t)) + 1.0) * (t_1 * sqrt((z * 2.0)));
	else
		tmp = t_1 * sqrt((2.0 * (z + (z * (t * t)))));
	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[N[(z * 2.0), $MachinePrecision], 1e+118], N[(N[(N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z + N[(z * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 2) < 9.99999999999999967e117

   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. Taylor expanded in t around 0 78.9%

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

      1. unpow2 78.9%

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

   4. Simplified 78.9%

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

   if 9.99999999999999967e117 < (*.f64 z 2)

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

      2. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 96.1%

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

      2. expm1-udef 96.1%

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

      3. sqrt-unprod 96.1%

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

      4. associate-*l* 96.1%

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

      5. exp-prod 96.1%

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

   5. Applied egg-rr 96.1%

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

      1. expm1-def 96.1%

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

      2. expm1-log1p 99.9%

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

      3. exp-prod 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t around 0 95.3%

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

      1. distribute-lft-out 95.3%

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

      2. *-commutative 95.3%

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

      3. unpow2 95.3%

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

   10. Simplified 95.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 2 \leq 10^{+118}:\\ \;\;\;\;\left(0.5 \cdot \left(t \cdot t\right) + 1\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]

Alternative 3: 58.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 27.5:\\ \;\;\;\;t_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(t_1 \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 27.5) (* t_1 (sqrt (* z 2.0))) (sqrt (* (* z 2.0) (* t_1 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 27.5) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = sqrt(((z * 2.0) * (t_1 * 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 = (x * 0.5d0) - y
    if (t <= 27.5d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else
        tmp = sqrt(((z * 2.0d0) * (t_1 * t_1)))
    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 <= 27.5) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(((z * 2.0) * (t_1 * t_1)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 27.5:
		tmp = t_1 * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(((z * 2.0) * (t_1 * t_1)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 27.5)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 27.5)
		tmp = t_1 * sqrt((z * 2.0));
	else
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	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, 27.5], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 27.5

   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. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 69.6%

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

      1. sqrt-unprod 69.8%

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

      2. *-commutative 69.8%

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

   6. Applied egg-rr 69.8%

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

   if 27.5 < 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. 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. Simplified 100.0%

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

   4. Taylor expanded in t around 0 15.1%

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

      1. add-sqr-sqrt 3.2%

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

      2. sqrt-unprod 15.8%

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

      3. *-commutative 15.8%

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

      4. *-commutative 15.8%

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

      5. swap-sqr 23.6%

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

      6. sqrt-unprod 23.6%

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

      7. sqrt-unprod 23.6%

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

      8. add-sqr-sqrt 23.6%

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

      9. *-commutative 23.6%

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

      10. pow2 23.6%

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

   6. Applied egg-rr 23.6%

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

      1. unpow2 23.6%

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

   8. Applied egg-rr 23.6%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 27.5:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \end{array} \]

Alternative 4: 85.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. add-exp-log 99.8%

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

   2. pow1/2 99.8%

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

   3. log-pow 99.8%

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

   4. add-log-exp 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in t around 0 83.5%

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

   1. +-commutative 83.5%

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

   2. unpow2 83.5%

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

8. Simplified 83.5%

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

9. Final simplification 83.5%

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

Alternative 5: 83.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.2%

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

   2. expm1-udef 77.2%

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

   3. sqrt-unprod 77.2%

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

   4. associate-*l* 77.2%

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

   5. exp-prod 77.2%

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

5. Applied egg-rr 77.2%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 99.8%

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

   3. exp-prod 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 81.5%

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

   1. distribute-lft-out 81.5%

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

   2. *-commutative 81.5%

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

   3. unpow2 81.5%

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

10. Simplified 81.5%

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

11. Final simplification 81.5%

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

Alternative 6: 55.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.5e+229)
   (sqrt (* (* z 2.0) (* y y)))
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+229) {
		tmp = sqrt(((z * 2.0) * (y * y)));
	} else {
		tmp = ((x * 0.5) - y) * sqrt((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 (y <= (-9.5d+229)) then
        tmp = sqrt(((z * 2.0d0) * (y * y)))
    else
        tmp = ((x * 0.5d0) - y) * sqrt((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 (y <= -9.5e+229) {
		tmp = Math.sqrt(((z * 2.0) * (y * y)));
	} else {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9.5e+229:
		tmp = math.sqrt(((z * 2.0) * (y * y)))
	else:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.5e+229)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * y)));
	else
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9.5e+229)
		tmp = sqrt(((z * 2.0) * (y * y)));
	else
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.5e+229], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5e229

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

      2. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 48.0%

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

      1. add-sqr-sqrt 47.9%

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

      2. sqrt-unprod 79.3%

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

      3. *-commutative 79.3%

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

      4. *-commutative 79.3%

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

      5. swap-sqr 84.7%

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

      6. sqrt-unprod 84.7%

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

      7. sqrt-unprod 84.7%

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

      8. add-sqr-sqrt 84.7%

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

      9. *-commutative 84.7%

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

      10. pow2 84.7%

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

   6. Applied egg-rr 84.7%

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

   7. Taylor expanded in x around 0 84.7%

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

      1. unpow2 84.7%

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

   9. Simplified 84.7%

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

   if -9.5e229 < y

   1. Initial program 99.4%

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

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 57.3%

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

      1. sqrt-unprod 57.4%

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

      2. *-commutative 57.4%

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

   6. Applied egg-rr 57.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \]

Alternative 7: 42.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-48} \lor \neg \left(y \leq 7.4 \cdot 10^{+36}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.4e-48) (not (<= y 7.4e+36)))
   (* y (- (sqrt (* z 2.0))))
   (* x (sqrt (* 0.5 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.4e-48) || !(y <= 7.4e+36)) {
		tmp = y * -sqrt((z * 2.0));
	} else {
		tmp = x * sqrt((0.5 * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-2.4d-48)) .or. (.not. (y <= 7.4d+36))) then
        tmp = y * -sqrt((z * 2.0d0))
    else
        tmp = x * sqrt((0.5d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.4e-48) || !(y <= 7.4e+36)) {
		tmp = y * -Math.sqrt((z * 2.0));
	} else {
		tmp = x * Math.sqrt((0.5 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.4e-48) or not (y <= 7.4e+36):
		tmp = y * -math.sqrt((z * 2.0))
	else:
		tmp = x * math.sqrt((0.5 * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.4e-48) || !(y <= 7.4e+36))
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	else
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -2.4e-48) || ~((y <= 7.4e+36)))
		tmp = y * -sqrt((z * 2.0));
	else
		tmp = x * sqrt((0.5 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.4e-48], N[Not[LessEqual[y, 7.4e+36]], $MachinePrecision]], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e-48 or 7.40000000000000058e36 < 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. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 60.8%

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

      1. sqrt-unprod 60.9%

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

      2. *-commutative 60.9%

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

   6. Applied egg-rr 60.9%

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

   7. Taylor expanded in x around 0 51.8%

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

      1. mul-1-neg 51.8%

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

      2. associate-*l* 51.8%

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

      3. distribute-rgt-neg-in 51.8%

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

      4. *-commutative 51.8%

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

      5. distribute-rgt-neg-in 51.8%

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

   9. Simplified 51.8%

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

       1. associate-*r* 51.7%

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

       2. sqrt-prod 51.9%

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

       3. *-commutative 51.9%

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

       4. distribute-rgt-neg-out 51.9%

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

   11. Applied egg-rr 51.9%

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

       1. distribute-rgt-neg-in 51.9%

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

       2. *-commutative 51.9%

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

   13. Simplified 51.9%

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

   if -2.4e-48 < y < 7.40000000000000058e36

   1. Initial program 98.9%

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

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 51.4%

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

      1. add-sqr-sqrt 25.1%

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

      2. sqrt-unprod 20.8%

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

      3. *-commutative 20.8%

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

      4. *-commutative 20.8%

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

      5. swap-sqr 23.3%

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

      6. sqrt-unprod 23.3%

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

      7. sqrt-unprod 23.4%

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

      8. add-sqr-sqrt 23.4%

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

      9. *-commutative 23.4%

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

      10. pow2 23.4%

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

   6. Applied egg-rr 23.4%

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

   7. Taylor expanded in x around inf 21.8%

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

      1. associate-*r* 21.8%

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

      2. *-commutative 21.8%

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

      3. unpow2 21.8%

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

   9. Simplified 21.8%

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

       1. *-commutative 21.8%

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

       2. sqrt-prod 24.2%

          \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot \sqrt{x \cdot x}} \]

       3. *-commutative 24.2%

          \[\leadsto \sqrt{\color{blue}{z \cdot 0.5}} \cdot \sqrt{x \cdot x} \]

       4. sqrt-prod 22.7%

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

       5. add-sqr-sqrt 43.2%

          \[\leadsto \sqrt{z \cdot 0.5} \cdot \color{blue}{x} \]

   11. Applied egg-rr 43.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-48} \lor \neg \left(y \leq 7.4 \cdot 10^{+36}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \end{array} \]

Alternative 8: 43.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.2e-47)
   (sqrt (* (* z 2.0) (* y y)))
   (if (<= y 3.3e+38) (* x (sqrt (* 0.5 z))) (* y (- (sqrt (* z 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.2e-47) {
		tmp = sqrt(((z * 2.0) * (y * y)));
	} else if (y <= 3.3e+38) {
		tmp = x * sqrt((0.5 * z));
	} else {
		tmp = y * -sqrt((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 (y <= (-8.2d-47)) then
        tmp = sqrt(((z * 2.0d0) * (y * y)))
    else if (y <= 3.3d+38) then
        tmp = x * sqrt((0.5d0 * z))
    else
        tmp = y * -sqrt((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 (y <= -8.2e-47) {
		tmp = Math.sqrt(((z * 2.0) * (y * y)));
	} else if (y <= 3.3e+38) {
		tmp = x * Math.sqrt((0.5 * z));
	} else {
		tmp = y * -Math.sqrt((z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -8.2e-47:
		tmp = math.sqrt(((z * 2.0) * (y * y)))
	elif y <= 3.3e+38:
		tmp = x * math.sqrt((0.5 * z))
	else:
		tmp = y * -math.sqrt((z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.2e-47)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * y)));
	elseif (y <= 3.3e+38)
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	else
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -8.2e-47)
		tmp = sqrt(((z * 2.0) * (y * y)));
	elseif (y <= 3.3e+38)
		tmp = x * sqrt((0.5 * z));
	else
		tmp = y * -sqrt((z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.2e-47], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 3.3e+38], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.20000000000000003e-47

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 58.3%

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

      1. add-sqr-sqrt 56.4%

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

      2. sqrt-unprod 61.2%

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

      3. *-commutative 61.2%

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

      4. *-commutative 61.2%

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

      5. swap-sqr 60.2%

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

      6. sqrt-unprod 60.2%

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

      7. sqrt-unprod 60.2%

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

      8. add-sqr-sqrt 60.2%

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

      9. *-commutative 60.2%

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

      10. pow2 60.2%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in x around 0 54.5%

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

      1. unpow2 54.5%

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

   9. Simplified 54.5%

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

   if -8.20000000000000003e-47 < y < 3.2999999999999999e38

   1. Initial program 98.9%

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

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 51.4%

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

      1. add-sqr-sqrt 25.1%

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

      2. sqrt-unprod 20.8%

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

      3. *-commutative 20.8%

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

      4. *-commutative 20.8%

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

      5. swap-sqr 23.3%

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

      6. sqrt-unprod 23.3%

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

      7. sqrt-unprod 23.4%

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

      8. add-sqr-sqrt 23.4%

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

      9. *-commutative 23.4%

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

      10. pow2 23.4%

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

   6. Applied egg-rr 23.4%

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

   7. Taylor expanded in x around inf 21.8%

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

      1. associate-*r* 21.8%

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

      2. *-commutative 21.8%

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

      3. unpow2 21.8%

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

   9. Simplified 21.8%

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

       1. *-commutative 21.8%

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

       2. sqrt-prod 24.2%

          \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot \sqrt{x \cdot x}} \]

       3. *-commutative 24.2%

          \[\leadsto \sqrt{\color{blue}{z \cdot 0.5}} \cdot \sqrt{x \cdot x} \]

       4. sqrt-prod 22.7%

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

       5. add-sqr-sqrt 43.2%

          \[\leadsto \sqrt{z \cdot 0.5} \cdot \color{blue}{x} \]

   11. Applied egg-rr 43.2%

       \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot x} \]

   if 3.2999999999999999e38 < y

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

      2. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 64.1%

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

      1. sqrt-unprod 64.3%

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

      2. *-commutative 64.3%

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

   6. Applied egg-rr 64.3%

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

   7. Taylor expanded in x around 0 55.9%

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

      1. mul-1-neg 55.9%

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

      2. associate-*l* 55.9%

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

      3. distribute-rgt-neg-in 55.9%

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

      4. *-commutative 55.9%

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

      5. distribute-rgt-neg-in 55.9%

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

   9. Simplified 55.9%

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

       1. associate-*r* 55.9%

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

       2. sqrt-prod 56.1%

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

       3. *-commutative 56.1%

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

       4. distribute-rgt-neg-out 56.1%

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

   11. Applied egg-rr 56.1%

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

       1. distribute-rgt-neg-in 56.1%

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

       2. *-commutative 56.1%

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

   13. Simplified 56.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]

Alternative 9: 29.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \sqrt{0.5 \cdot z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 56.6%

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

   1. add-sqr-sqrt 30.4%

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

   2. sqrt-unprod 30.2%

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

   3. *-commutative 30.2%

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

   4. *-commutative 30.2%

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

   5. swap-sqr 31.0%

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

   6. sqrt-unprod 31.0%

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

   7. sqrt-unprod 31.0%

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

   8. add-sqr-sqrt 31.1%

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

   9. *-commutative 31.1%

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

   10. pow2 31.1%

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

6. Applied egg-rr 31.1%

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

7. Taylor expanded in x around inf 17.0%

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

   1. associate-*r* 17.0%

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

   2. *-commutative 17.0%

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

   3. unpow2 17.0%

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

9. Simplified 17.0%

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

    1. *-commutative 17.0%

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

    2. sqrt-prod 17.1%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot \sqrt{x \cdot x}} \]

    3. *-commutative 17.1%

       \[\leadsto \sqrt{\color{blue}{z \cdot 0.5}} \cdot \sqrt{x \cdot x} \]

    4. sqrt-prod 14.7%

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

    5. add-sqr-sqrt 26.2%

       \[\leadsto \sqrt{z \cdot 0.5} \cdot \color{blue}{x} \]

11. Applied egg-rr 26.2%

    \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot x} \]

12. Final simplification 26.2%

    \[\leadsto x \cdot \sqrt{0.5 \cdot z} \]

Developer target: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023257 
(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))))