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

Percentage Accurate: 99.5% → 99.8%

Time: 18.8s

Alternatives: 12

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[Sqrt[N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]], $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. Step-by-step derivation

   1. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

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. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

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. Add Preprocessing
5. Step-by-step derivation

   1. exp-sqrt 99.8%

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

   2. add0 99.8%

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

   3. exp-sqrt 99.8%

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

   4. sqrt-unprod 99.8%

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

   5. associate-*l* 99.8%

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

   6. pow2 99.8%

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

6. Applied egg-rr 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} + 0\right)} \]
7. Step-by-step derivation

   1. add0 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*r* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 80000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(x \cdot 0.5\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 80000000.0)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* (* x 0.5) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 80000000.0) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * ((x * 0.5) * 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 ((t * t) <= 80000000.0d0) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * ((x * 0.5d0) * 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 ((t * t) <= 80000000.0) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * ((x * 0.5) * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 80000000.0:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * ((x * 0.5) * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 80000000.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(Float64(x * 0.5) * 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 ((t * t) <= 80000000.0)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * ((x * 0.5) * 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[LessEqual[N[(t * t), $MachinePrecision], 80000000.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 8e7

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0 97.9%

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

      1. associate-*r* 97.9%

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

      2. sqrt-prod 98.3%

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

      3. *-commutative 98.3%

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

      4. pow1/2 98.3%

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

      5. metadata-eval 98.3%

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

      6. pow-div 98.1%

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

      7. pow1 98.1%

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

      8. pow1/2 98.1%

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

      9. pow1 98.1%

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

      10. metadata-eval 98.1%

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

      11. pow-div 58.6%

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

      12. pow1 58.6%

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

      13. *-commutative 58.6%

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

      14. add0 58.6%

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

   5. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} + 0} \]
   6. Step-by-step derivation

      1. add0 98.3%

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

      2. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   if 8e7 < (*.f64 t t)

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.3%

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

      2. flip-+ 83.7%

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

      3. *-commutative 83.7%

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

      4. *-commutative 83.7%

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

      5. swap-sqr 82.2%

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

      6. add-sqr-sqrt 82.2%

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

      7. pow2 82.2%

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

      8. fma-neg 82.2%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 0.5, -y\right)\right)}}^{2} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. metadata-eval 82.2%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot {\left(\mathsf{fma}\left(x, 0.5, -y\right)\right)}^{2} - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. fma-neg 82.2%

          \[\leadsto \frac{\left(z \cdot 2\right) \cdot {\left(\mathsf{fma}\left(x, 0.5, -y\right)\right)}^{2} - 0}{\color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)} \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 82.2%

      \[\leadsto \color{blue}{\frac{\left(z \cdot 2\right) \cdot {\left(\mathsf{fma}\left(x, 0.5, -y\right)\right)}^{2} - 0}{\mathsf{fma}\left(x, 0.5, -y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. --rgt-identity 82.2%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(\mathsf{fma}\left(x, 0.5, -y\right)\right)}^{2}}}{\mathsf{fma}\left(x, 0.5, -y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      2. --rgt-identity 82.2%

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

      3. *-commutative 82.2%

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

      4. times-frac 95.6%

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

      5. fma-neg 95.6%

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

      6. *-commutative 95.6%

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

      7. fma-neg 95.6%

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

      8. *-commutative 95.6%

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

      9. *-commutative 95.6%

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

      10. *-commutative 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in x around inf 68.1%

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

      1. pow1 68.1%

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

      2. pow1/2 68.1%

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

      3. pow-div 68.1%

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

      4. metadata-eval 68.1%

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

      5. pow1/2 68.1%

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

      6. add0 68.1%

         \[\leadsto \color{blue}{\left(\left(0.5 \cdot x\right) \cdot \sqrt{2 \cdot z} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot x\right) \cdot \sqrt{2 \cdot z} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   10. Step-by-step derivation

       1. add0 68.1%

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

       2. *-commutative 68.1%

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

   11. Simplified 68.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 80000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 5.5 \cdot 10^{+37}:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {t\_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 5.5e+37)
     (* t_1 (sqrt (* z 2.0)))
     (sqrt (* (* z 2.0) (pow t_1 2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 5.5e+37) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = sqrt(((z * 2.0) * pow(t_1, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 5.5d+37) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else
        tmp = sqrt(((z * 2.0d0) * (t_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 5.5e+37) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(t_1, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 5.5e+37:
		tmp = t_1 * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(((z * 2.0) * math.pow(t_1, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 5.5e+37)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * (t_1 ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 5.5e+37)
		tmp = t_1 * sqrt((z * 2.0));
	else
		tmp = sqrt(((z * 2.0) * (t_1 ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 5.5e+37], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 5.50000000000000016e37

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0 68.0%

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

      1. associate-*r* 68.0%

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

      2. sqrt-prod 68.3%

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

      3. *-commutative 68.3%

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

      4. pow1/2 68.3%

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

      5. metadata-eval 68.3%

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

      6. pow-div 68.1%

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

      7. pow1 68.1%

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

      8. pow1/2 68.1%

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

      9. pow1 68.1%

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

      10. metadata-eval 68.1%

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

      11. pow-div 52.7%

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

      12. pow1 52.7%

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

      13. *-commutative 52.7%

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

      14. add0 52.7%

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

   5. Applied egg-rr 68.3%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} + 0} \]
   6. Step-by-step derivation

      1. add0 68.3%

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

      2. *-commutative 68.3%

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

   7. Simplified 68.3%

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

   if 5.50000000000000016e37 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 13.2%

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

   5. Applied egg-rr 22.7%

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

      1. associate-*r* 22.7%

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

      2. *-commutative 22.7%

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

   7. Simplified 22.7%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+37}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% 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. Add Preprocessing

3. Final simplification 99.4%

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

Alternative 6: 59.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 1.95 \cdot 10^{-10}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{t\_2} \cdot \left(t\_1 \cdot \left(z \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -2}{\frac{-t\_2}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 1.95e-10)
     (* t_1 t_2)
     (if (<= t 1.25e+181)
       (* (/ 1.0 t_2) (* t_1 (* z 2.0)))
       (/ (* z -2.0) (/ (- t_2) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 1.95e-10) {
		tmp = t_1 * t_2;
	} else if (t <= 1.25e+181) {
		tmp = (1.0 / t_2) * (t_1 * (z * 2.0));
	} else {
		tmp = (z * -2.0) / (-t_2 / 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) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((z * 2.0d0))
    if (t <= 1.95d-10) then
        tmp = t_1 * t_2
    else if (t <= 1.25d+181) then
        tmp = (1.0d0 / t_2) * (t_1 * (z * 2.0d0))
    else
        tmp = (z * (-2.0d0)) / (-t_2 / 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 t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 1.95e-10) {
		tmp = t_1 * t_2;
	} else if (t <= 1.25e+181) {
		tmp = (1.0 / t_2) * (t_1 * (z * 2.0));
	} else {
		tmp = (z * -2.0) / (-t_2 / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 1.95e-10:
		tmp = t_1 * t_2
	elif t <= 1.25e+181:
		tmp = (1.0 / t_2) * (t_1 * (z * 2.0))
	else:
		tmp = (z * -2.0) / (-t_2 / t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 1.95e-10)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 1.25e+181)
		tmp = Float64(Float64(1.0 / t_2) * Float64(t_1 * Float64(z * 2.0)));
	else
		tmp = Float64(Float64(z * -2.0) / Float64(Float64(-t_2) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 1.95e-10)
		tmp = t_1 * t_2;
	elseif (t <= 1.25e+181)
		tmp = (1.0 / t_2) * (t_1 * (z * 2.0));
	else
		tmp = (z * -2.0) / (-t_2 / 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]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.95e-10], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 1.25e+181], N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(t$95$1 * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * -2.0), $MachinePrecision] / N[((-t$95$2) / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.95e-10

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. sqrt-prod 71.3%

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

      3. *-commutative 71.3%

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

      4. pow1/2 71.3%

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

      5. metadata-eval 71.3%

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

      6. pow-div 71.1%

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

      7. pow1 71.1%

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

      8. pow1/2 71.1%

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

      9. pow1 71.1%

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

      10. metadata-eval 71.1%

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

      11. pow-div 53.4%

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

      12. pow1 53.4%

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

      13. *-commutative 53.4%

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

      14. add0 53.4%

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

   5. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} + 0} \]
   6. Step-by-step derivation

      1. add0 71.3%

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

      2. *-commutative 71.3%

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

   7. Simplified 71.3%

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

   if 1.95e-10 < t < 1.2500000000000001e181

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 12.3%

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

      1. associate-*r* 12.3%

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

      2. sqrt-prod 12.3%

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

      3. *-commutative 12.3%

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

      4. pow1/2 12.3%

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

      5. metadata-eval 12.3%

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

      6. pow-div 12.3%

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

      7. pow1 12.3%

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

      8. pow1/2 12.3%

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

      9. pow1 12.3%

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

      10. metadata-eval 12.3%

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

      11. pow-div 50.6%

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

      12. pow1 50.6%

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

      13. *-commutative 50.6%

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

      14. associate-*r/ 52.9%

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

      15. clear-num 52.9%

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

   5. Applied egg-rr 25.6%

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

      1. associate-/r/ 25.6%

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

      2. *-commutative 25.6%

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

   7. Simplified 25.6%

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

   if 1.2500000000000001e181 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 14.7%

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

      1. associate-*r* 14.7%

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

      2. sqrt-prod 14.7%

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

      3. *-commutative 14.7%

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

      4. pow1/2 14.7%

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

      5. metadata-eval 14.7%

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

      6. pow-div 14.7%

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

      7. pow1 14.7%

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

      8. pow1/2 14.7%

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

      9. pow1 14.7%

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

      10. metadata-eval 14.7%

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

      11. pow-div 58.1%

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

      12. pow1 58.1%

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

      13. *-commutative 58.1%

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

      14. clear-num 58.1%

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

      15. frac-2neg 58.1%

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

      16. frac-times 58.1%

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

   5. Applied egg-rr 27.6%

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

      1. *-commutative 27.6%

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

      2. distribute-lft-neg-out 27.6%

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

      3. associate-*r/ 27.6%

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

      4. *-rgt-identity 27.6%

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

      5. *-commutative 27.6%

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

   7. Simplified 27.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-10}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{\sqrt{z \cdot 2}} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(z \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -2}{\frac{-\sqrt{z \cdot 2}}{x \cdot 0.5 - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+180}:\\ \;\;\;\;\frac{t\_1 \cdot \left(z \cdot 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -2}{\frac{-t\_2}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 5e-11)
     (* t_1 t_2)
     (if (<= t 6.4e+180)
       (/ (* t_1 (* z 2.0)) t_2)
       (/ (* z -2.0) (/ (- t_2) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 5e-11) {
		tmp = t_1 * t_2;
	} else if (t <= 6.4e+180) {
		tmp = (t_1 * (z * 2.0)) / t_2;
	} else {
		tmp = (z * -2.0) / (-t_2 / 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) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((z * 2.0d0))
    if (t <= 5d-11) then
        tmp = t_1 * t_2
    else if (t <= 6.4d+180) then
        tmp = (t_1 * (z * 2.0d0)) / t_2
    else
        tmp = (z * (-2.0d0)) / (-t_2 / 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 t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 5e-11) {
		tmp = t_1 * t_2;
	} else if (t <= 6.4e+180) {
		tmp = (t_1 * (z * 2.0)) / t_2;
	} else {
		tmp = (z * -2.0) / (-t_2 / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 5e-11:
		tmp = t_1 * t_2
	elif t <= 6.4e+180:
		tmp = (t_1 * (z * 2.0)) / t_2
	else:
		tmp = (z * -2.0) / (-t_2 / t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 5e-11)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 6.4e+180)
		tmp = Float64(Float64(t_1 * Float64(z * 2.0)) / t_2);
	else
		tmp = Float64(Float64(z * -2.0) / Float64(Float64(-t_2) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 5e-11)
		tmp = t_1 * t_2;
	elseif (t <= 6.4e+180)
		tmp = (t_1 * (z * 2.0)) / t_2;
	else
		tmp = (z * -2.0) / (-t_2 / 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]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 5e-11], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 6.4e+180], N[(N[(t$95$1 * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(z * -2.0), $MachinePrecision] / N[((-t$95$2) / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.00000000000000018e-11

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. sqrt-prod 71.3%

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

      3. *-commutative 71.3%

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

      4. pow1/2 71.3%

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

      5. metadata-eval 71.3%

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

      6. pow-div 71.1%

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

      7. pow1 71.1%

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

      8. pow1/2 71.1%

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

      9. pow1 71.1%

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

      10. metadata-eval 71.1%

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

      11. pow-div 53.4%

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

      12. pow1 53.4%

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

      13. *-commutative 53.4%

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

      14. add0 53.4%

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

   5. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} + 0} \]
   6. Step-by-step derivation

      1. add0 71.3%

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

      2. *-commutative 71.3%

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

   7. Simplified 71.3%

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

   if 5.00000000000000018e-11 < t < 6.39999999999999987e180

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 12.3%

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

      1. associate-*r* 12.3%

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

      2. sqrt-prod 12.3%

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

      3. *-commutative 12.3%

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

      4. pow1/2 12.3%

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

      5. metadata-eval 12.3%

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

      6. pow-div 12.3%

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

      7. pow1 12.3%

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

      8. pow1/2 12.3%

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

      9. pow1 12.3%

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

      10. metadata-eval 12.3%

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

      11. pow-div 50.6%

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

      12. pow1 50.6%

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

      13. *-commutative 50.6%

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

      14. associate-*r/ 52.9%

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

   5. Applied egg-rr 25.6%

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

   if 6.39999999999999987e180 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 14.7%

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

      1. associate-*r* 14.7%

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

      2. sqrt-prod 14.7%

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

      3. *-commutative 14.7%

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

      4. pow1/2 14.7%

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

      5. metadata-eval 14.7%

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

      6. pow-div 14.7%

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

      7. pow1 14.7%

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

      8. pow1/2 14.7%

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

      9. pow1 14.7%

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

      10. metadata-eval 14.7%

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

      11. pow-div 58.1%

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

      12. pow1 58.1%

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

      13. *-commutative 58.1%

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

      14. clear-num 58.1%

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

      15. frac-2neg 58.1%

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

      16. frac-times 58.1%

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

   5. Applied egg-rr 27.6%

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

      1. *-commutative 27.6%

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

      2. distribute-lft-neg-out 27.6%

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

      3. associate-*r/ 27.6%

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

      4. *-rgt-identity 27.6%

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

      5. *-commutative 27.6%

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

   7. Simplified 27.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+180}:\\ \;\;\;\;\frac{\left(x \cdot 0.5 - y\right) \cdot \left(z \cdot 2\right)}{\sqrt{z \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -2}{\frac{-\sqrt{z \cdot 2}}{x \cdot 0.5 - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 1.95 \cdot 10^{-10}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot \left(z \cdot 2\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 1.95e-10) (* t_1 t_2) (/ (* t_1 (* z 2.0)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 1.95e-10) {
		tmp = t_1 * t_2;
	} else {
		tmp = (t_1 * (z * 2.0)) / t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((z * 2.0d0))
    if (t <= 1.95d-10) then
        tmp = t_1 * t_2
    else
        tmp = (t_1 * (z * 2.0d0)) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 1.95e-10) {
		tmp = t_1 * t_2;
	} else {
		tmp = (t_1 * (z * 2.0)) / t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 1.95e-10:
		tmp = t_1 * t_2
	else:
		tmp = (t_1 * (z * 2.0)) / t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 1.95e-10)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(Float64(t_1 * Float64(z * 2.0)) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 1.95e-10)
		tmp = t_1 * t_2;
	else
		tmp = (t_1 * (z * 2.0)) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.95e-10], N[(t$95$1 * t$95$2), $MachinePrecision], N[(N[(t$95$1 * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.95e-10

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. sqrt-prod 71.3%

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

      3. *-commutative 71.3%

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

      4. pow1/2 71.3%

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

      5. metadata-eval 71.3%

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

      6. pow-div 71.1%

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

      7. pow1 71.1%

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

      8. pow1/2 71.1%

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

      9. pow1 71.1%

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

      10. metadata-eval 71.1%

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

      11. pow-div 53.4%

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

      12. pow1 53.4%

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

      13. *-commutative 53.4%

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

      14. add0 53.4%

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

   5. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} + 0} \]
   6. Step-by-step derivation

      1. add0 71.3%

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

      2. *-commutative 71.3%

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

   7. Simplified 71.3%

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

   if 1.95e-10 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 13.4%

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

      1. associate-*r* 13.4%

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

      2. sqrt-prod 13.4%

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

      3. *-commutative 13.4%

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

      4. pow1/2 13.4%

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

      5. metadata-eval 13.4%

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

      6. pow-div 13.4%

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

      7. pow1 13.4%

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

      8. pow1/2 13.4%

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

      9. pow1 13.4%

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

      10. metadata-eval 13.4%

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

      11. pow-div 54.2%

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

      12. pow1 54.2%

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

      13. *-commutative 54.2%

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

      14. associate-*r/ 56.6%

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

   5. Applied egg-rr 22.8%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-10}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 0.5 - y\right) \cdot \left(z \cdot 2\right)}{\sqrt{z \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{+72} \lor \neg \left(x \leq 2.9 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{x \cdot z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= x -6.4e+72) (not (<= x 2.9e+61)))
     (/ (* x z) t_1)
     (* y (- t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((x <= -6.4e+72) || !(x <= 2.9e+61)) {
		tmp = (x * z) / t_1;
	} else {
		tmp = y * -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 ((x <= (-6.4d+72)) .or. (.not. (x <= 2.9d+61))) then
        tmp = (x * z) / t_1
    else
        tmp = y * -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 ((x <= -6.4e+72) || !(x <= 2.9e+61)) {
		tmp = (x * z) / t_1;
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (x <= -6.4e+72) or not (x <= 2.9e+61):
		tmp = (x * z) / t_1
	else:
		tmp = y * -t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((x <= -6.4e+72) || !(x <= 2.9e+61))
		tmp = Float64(Float64(x * z) / t_1);
	else
		tmp = Float64(y * Float64(-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 ((x <= -6.4e+72) || ~((x <= 2.9e+61)))
		tmp = (x * z) / t_1;
	else
		tmp = y * -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[x, -6.4e+72], N[Not[LessEqual[x, 2.9e+61]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] / t$95$1), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.4000000000000003e72 or 2.9000000000000001e61 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 58.0%

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

      1. associate-*r* 58.0%

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

      2. sqrt-prod 58.2%

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

      3. *-commutative 58.2%

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

      4. pow1/2 58.2%

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

      5. metadata-eval 58.2%

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

      6. pow-div 58.1%

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

      7. pow1 58.1%

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

      8. pow1/2 58.1%

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

      9. pow1 58.1%

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

      10. metadata-eval 58.1%

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

      11. pow-div 57.0%

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

      12. pow1 57.0%

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

      13. *-commutative 57.0%

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

      14. associate-*r/ 56.1%

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

   5. Applied egg-rr 59.9%

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

   6. Taylor expanded in x around inf 48.0%

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

   if -6.4000000000000003e72 < x < 2.9000000000000001e61

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0 50.6%

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

   4. Taylor expanded in x around 0 39.7%

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

      1. mul-1-neg 39.7%

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

      2. associate-*l* 39.8%

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

      3. *-commutative 39.8%

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

      4. distribute-lft-neg-in 39.8%

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

      5. *-commutative 39.8%

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

   6. Simplified 39.8%

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

      1. sqrt-prod 39.8%

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

      2. distribute-lft-neg-out 39.8%

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

      3. neg-sub0 39.8%

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

      4. add-sqr-sqrt 15.4%

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

      5. sqrt-unprod 21.9%

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

      6. sqr-neg 21.9%

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

      7. sqrt-unprod 1.9%

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

      8. add-sqr-sqrt 2.5%

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

      9. *-commutative 2.5%

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

      10. add-sqr-sqrt 1.9%

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

      11. sqrt-unprod 21.9%

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

      12. sqr-neg 21.9%

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

      13. sqrt-unprod 15.4%

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

      14. add-sqr-sqrt 39.8%

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

   8. Applied egg-rr 39.8%

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

      1. neg-sub0 39.8%

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

      2. distribute-rgt-neg-in 39.8%

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

   10. Simplified 39.8%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+72} \lor \neg \left(x \leq 2.9 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{x \cdot z}{\sqrt{z \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.0% 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. Add Preprocessing

3. Taylor expanded in t around 0 53.5%

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

   1. associate-*r* 53.5%

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

   2. sqrt-prod 53.6%

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

   3. *-commutative 53.6%

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

   4. pow1/2 53.6%

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

   5. metadata-eval 53.6%

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

   6. pow-div 53.5%

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

   7. pow1 53.5%

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

   8. pow1/2 53.5%

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

   9. pow1 53.5%

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

   10. metadata-eval 53.5%

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

   11. pow-div 53.6%

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

   12. pow1 53.6%

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

   13. *-commutative 53.6%

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

   14. add0 53.6%

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

5. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} + 0} \]
6. Step-by-step derivation

   1. add0 53.6%

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

   2. *-commutative 53.6%

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

7. Simplified 53.6%

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

8. Final simplification 53.6%

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

Alternative 11: 30.4% 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. Add Preprocessing

3. Taylor expanded in t around 0 53.5%

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

4. Taylor expanded in x around 0 28.8%

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

   1. mul-1-neg 28.8%

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

   2. associate-*l* 28.8%

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

   3. *-commutative 28.8%

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

   4. distribute-lft-neg-in 28.8%

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

   5. *-commutative 28.8%

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

6. Simplified 28.8%

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

   1. sqrt-prod 28.9%

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

   2. distribute-lft-neg-out 28.9%

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

   3. neg-sub0 28.9%

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

   4. add-sqr-sqrt 11.6%

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

   5. sqrt-unprod 16.4%

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

   6. sqr-neg 16.4%

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

   7. sqrt-unprod 1.6%

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

   8. add-sqr-sqrt 2.7%

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

   9. *-commutative 2.7%

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

   10. add-sqr-sqrt 1.6%

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

   11. sqrt-unprod 16.4%

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

   12. sqr-neg 16.4%

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

   13. sqrt-unprod 11.6%

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

   14. add-sqr-sqrt 28.9%

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

8. Applied egg-rr 28.9%

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

   1. neg-sub0 28.9%

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

   2. distribute-rgt-neg-in 28.9%

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

10. Simplified 28.9%

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

11. Final simplification 28.9%

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

Alternative 12: 2.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0 53.5%

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

4. Taylor expanded in x around 0 28.8%

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

   1. mul-1-neg 28.8%

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

   2. associate-*l* 28.8%

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

   3. *-commutative 28.8%

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

   4. distribute-lft-neg-in 28.8%

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

   5. *-commutative 28.8%

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

6. Simplified 28.8%

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

   1. sqrt-prod 28.9%

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

   2. add0 28.9%

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

   3. *-commutative 28.9%

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

   4. add-sqr-sqrt 17.2%

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

   5. sqrt-unprod 16.6%

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

   6. sqr-neg 16.6%

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

   7. sqrt-unprod 1.2%

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

   8. add-sqr-sqrt 2.7%

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

8. Applied egg-rr 2.7%

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

   1. *-commutative 2.7%

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

   2. add0 2.7%

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

10. Simplified 2.7%

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

11. Final simplification 2.7%

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

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 2024034 
(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))))