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

Percentage Accurate: 99.4% → 99.4%

Time: 14.1s

Alternatives: 10

Speedup: 1.0×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 200000:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;x \cdot \left(t\_2 \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {\left({z}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 200000.0)
     (* t_1 t_2)
     (if (<= t 2.8e+92)
       (* x (* t_2 (- 0.5 (/ y x))))
       (* t_1 (pow (* (pow z 2.0) 4.0) 0.25))))))

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 <= 200000.0) {
		tmp = t_1 * t_2;
	} else if (t <= 2.8e+92) {
		tmp = x * (t_2 * (0.5 - (y / x)));
	} else {
		tmp = t_1 * pow((pow(z, 2.0) * 4.0), 0.25);
	}
	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 <= 200000.0d0) then
        tmp = t_1 * t_2
    else if (t <= 2.8d+92) then
        tmp = x * (t_2 * (0.5d0 - (y / x)))
    else
        tmp = t_1 * (((z ** 2.0d0) * 4.0d0) ** 0.25d0)
    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 <= 200000.0) {
		tmp = t_1 * t_2;
	} else if (t <= 2.8e+92) {
		tmp = x * (t_2 * (0.5 - (y / x)));
	} else {
		tmp = t_1 * Math.pow((Math.pow(z, 2.0) * 4.0), 0.25);
	}
	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 <= 200000.0:
		tmp = t_1 * t_2
	elif t <= 2.8e+92:
		tmp = x * (t_2 * (0.5 - (y / x)))
	else:
		tmp = t_1 * math.pow((math.pow(z, 2.0) * 4.0), 0.25)
	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 <= 200000.0)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 2.8e+92)
		tmp = Float64(x * Float64(t_2 * Float64(0.5 - Float64(y / x))));
	else
		tmp = Float64(t_1 * (Float64((z ^ 2.0) * 4.0) ^ 0.25));
	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 <= 200000.0)
		tmp = t_1 * t_2;
	elseif (t <= 2.8e+92)
		tmp = x * (t_2 * (0.5 - (y / x)));
	else
		tmp = t_1 * (((z ^ 2.0) * 4.0) ^ 0.25);
	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, 200000.0], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 2.8e+92], N[(x * N[(t$95$2 * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[(N[Power[z, 2.0], $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2e5

   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 74.0%

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

      1. *-rgt-identity 74.0%

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

      2. *-commutative 74.0%

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

      3. sub-neg 74.0%

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

      4. distribute-lft-in 73.0%

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

   5. Applied egg-rr 73.0%

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

      1. distribute-lft-out 74.0%

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

      2. *-commutative 74.0%

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

      3. sub-neg 74.0%

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

   7. Simplified 74.0%

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

   if 2e5 < t < 2.80000000000000001e92

   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 11.9%

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

      1. add-cube-cbrt 11.9%

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

      2. pow3 11.9%

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

   5. Applied egg-rr 11.9%

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

   6. Taylor expanded in x around inf 32.1%

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

      1. associate-*r* 32.1%

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

      2. rem-cube-cbrt 32.1%

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

      3. rem-cube-cbrt 32.1%

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

      4. distribute-rgt-out 32.1%

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

      5. +-commutative 32.1%

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

      6. mul-1-neg 32.1%

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

      7. unsub-neg 32.1%

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

   8. Simplified 32.1%

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

   if 2.80000000000000001e92 < 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 15.3%

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

      1. add-cube-cbrt 15.3%

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

      2. pow3 15.3%

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

   5. Applied egg-rr 15.3%

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

      1. pow1/2 15.3%

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

      2. rem-cube-cbrt 15.3%

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

      3. metadata-eval 15.3%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot {\left(z \cdot 2\right)}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \cdot 1 \]

      4. pow-sqr 15.3%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left({\left(z \cdot 2\right)}^{0.25} \cdot {\left(z \cdot 2\right)}^{0.25}\right)}\right) \cdot 1 \]

      5. pow-prod-down 39.2%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\left(z \cdot 2\right) \cdot \left(z \cdot 2\right)\right)}^{0.25}}\right) \cdot 1 \]

      6. swap-sqr 39.2%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left(\left(z \cdot z\right) \cdot \left(2 \cdot 2\right)\right)}}^{0.25}\right) \cdot 1 \]

      7. pow2 39.2%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot {\left(\color{blue}{{z}^{2}} \cdot \left(2 \cdot 2\right)\right)}^{0.25}\right) \cdot 1 \]

      8. metadata-eval 39.2%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot {\left({z}^{2} \cdot \color{blue}{4}\right)}^{0.25}\right) \cdot 1 \]

   7. Applied egg-rr 39.2%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left({z}^{2} \cdot 4\right)}^{0.25}}\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 200000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;x \cdot \left(\sqrt{z \cdot 2} \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot {\left({z}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 0.5:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+226}:\\ \;\;\;\;x \cdot \left(t\_2 \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 0.5)
     (* t_1 t_2)
     (if (<= t 3.2e+226)
       (* x (* t_2 (- 0.5 (/ y x))))
       (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 t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 0.5) {
		tmp = t_1 * t_2;
	} else if (t <= 3.2e+226) {
		tmp = x * (t_2 * (0.5 - (y / x)));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((z * 2.0d0))
    if (t <= 0.5d0) then
        tmp = t_1 * t_2
    else if (t <= 3.2d+226) then
        tmp = x * (t_2 * (0.5d0 - (y / x)))
    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 t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 0.5) {
		tmp = t_1 * t_2;
	} else if (t <= 3.2e+226) {
		tmp = x * (t_2 * (0.5 - (y / x)));
	} 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
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 0.5:
		tmp = t_1 * t_2
	elif t <= 3.2e+226:
		tmp = x * (t_2 * (0.5 - (y / x)))
	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)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 0.5)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 3.2e+226)
		tmp = Float64(x * Float64(t_2 * Float64(0.5 - Float64(y / x))));
	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;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 0.5)
		tmp = t_1 * t_2;
	elseif (t <= 3.2e+226)
		tmp = x * (t_2 * (0.5 - (y / x)));
	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]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.5], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 3.2e+226], N[(x * N[(t$95$2 * N[(0.5 - N[(y / x), $MachinePrecision]), $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 3 regimes

2. if t < 0.5

   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 74.8%

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

      1. *-rgt-identity 74.8%

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

      2. *-commutative 74.8%

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

      3. sub-neg 74.8%

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

      4. distribute-lft-in 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. distribute-lft-out 74.8%

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

      2. *-commutative 74.8%

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

      3. sub-neg 74.8%

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

   7. Simplified 74.8%

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

   if 0.5 < t < 3.19999999999999977e226

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 16.9%

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

      1. add-cube-cbrt 16.9%

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

      2. pow3 16.9%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in x around inf 29.5%

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

      1. associate-*r* 29.5%

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

      2. rem-cube-cbrt 29.5%

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

      3. rem-cube-cbrt 29.5%

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

      4. distribute-rgt-out 29.5%

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

      5. +-commutative 29.5%

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

      6. mul-1-neg 29.5%

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

      7. unsub-neg 29.5%

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

   8. Simplified 29.5%

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

   if 3.19999999999999977e226 < 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 8.6%

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

      1. *-rgt-identity 8.6%

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

      2. *-commutative 8.6%

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

      3. pow1 8.6%

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

      4. metadata-eval 8.6%

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

      5. sqrt-pow1 23.7%

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

      6. sqrt-prod 27.9%

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

      7. pow1/2 27.9%

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

      8. associate-*l* 27.9%

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

   5. Applied egg-rr 27.9%

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

      1. unpow1/2 27.9%

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

      2. associate-*r* 27.9%

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

      3. *-commutative 27.9%

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

   7. Simplified 27.9%

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

      1. *-commutative 27.9%

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

      2. pow2 27.9%

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

   9. Applied egg-rr 27.9%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.5:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+226}:\\ \;\;\;\;x \cdot \left(\sqrt{z \cdot 2} \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t\_1 \cdot \left(x \cdot \frac{0.5}{y} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 2e-22)
     (* (- (* x 0.5) y) t_1)
     (* y (* t_1 (+ (* x (/ 0.5 y)) -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 2e-22) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = y * (t_1 * ((x * (0.5 / y)) + -1.0));
	}
	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 = sqrt((z * 2.0d0))
    if (t <= 2d-22) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = y * (t_1 * ((x * (0.5d0 / y)) + (-1.0d0)))
    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 <= 2e-22) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = y * (t_1 * ((x * (0.5 / y)) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 2e-22:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = y * (t_1 * ((x * (0.5 / y)) + -1.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 2e-22)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(y * Float64(t_1 * Float64(Float64(x * Float64(0.5 / y)) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 2e-22)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = y * (t_1 * ((x * (0.5 / y)) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2e-22], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(y * N[(t$95$1 * N[(N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.0000000000000001e-22

   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 74.5%

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

      1. *-rgt-identity 74.5%

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

      2. *-commutative 74.5%

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

      3. sub-neg 74.5%

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

      4. distribute-lft-in 73.4%

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

   5. Applied egg-rr 73.4%

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

      1. distribute-lft-out 74.5%

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

      2. *-commutative 74.5%

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

      3. sub-neg 74.5%

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

   7. Simplified 74.5%

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

   if 2.0000000000000001e-22 < 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 19.2%

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

      1. add-cube-cbrt 19.1%

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

      2. pow3 19.2%

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

   5. Applied egg-rr 19.2%

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

   6. Taylor expanded in y around inf 33.5%

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

      1. rem-cube-cbrt 33.5%

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

      2. associate-*r* 33.5%

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

      3. rem-cube-cbrt 33.6%

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

      4. distribute-rgt-out 33.6%

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

      5. +-commutative 33.6%

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

      6. fma-define 33.6%

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

   8. Simplified 33.6%

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

      1. fma-undefine 33.6%

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

      2. distribute-lft-in 33.6%

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

      3. clear-num 33.6%

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

      4. un-div-inv 33.6%

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

   10. Applied egg-rr 33.6%

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

       1. distribute-lft-out 33.6%

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

       2. *-commutative 33.6%

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

       3. associate-/r/ 33.6%

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

   12. Simplified 33.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{0.5}{y} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -1650 \lor \neg \left(y \leq 3.7 \cdot 10^{+168}\right):\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= y -1650.0) (not (<= y 3.7e+168)))
     (* y (- t_1))
     (* t_1 (* x 0.5)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((y <= -1650.0) || !(y <= 3.7e+168)) {
		tmp = y * -t_1;
	} else {
		tmp = t_1 * (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((y <= (-1650.0d0)) .or. (.not. (y <= 3.7d+168))) then
        tmp = y * -t_1
    else
        tmp = t_1 * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((y <= -1650.0) || !(y <= 3.7e+168)) {
		tmp = y * -t_1;
	} else {
		tmp = t_1 * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (y <= -1650.0) or not (y <= 3.7e+168):
		tmp = y * -t_1
	else:
		tmp = t_1 * (x * 0.5)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((y <= -1650.0) || !(y <= 3.7e+168))
		tmp = Float64(y * Float64(-t_1));
	else
		tmp = Float64(t_1 * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((y <= -1650.0) || ~((y <= 3.7e+168)))
		tmp = y * -t_1;
	else
		tmp = t_1 * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -1650.0], N[Not[LessEqual[y, 3.7e+168]], $MachinePrecision]], N[(y * (-t$95$1)), $MachinePrecision], N[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1650 or 3.70000000000000009e168 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 63.6%

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

      1. *-rgt-identity 63.6%

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

      2. *-commutative 63.6%

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

      3. sub-neg 63.6%

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

      4. distribute-lft-in 61.2%

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

   5. Applied egg-rr 61.2%

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

      1. distribute-lft-out 63.6%

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

      2. *-commutative 63.6%

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

      3. sub-neg 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in x around 0 56.5%

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

      1. neg-mul-1 56.5%

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

   10. Simplified 56.5%

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

   if -1650 < y < 3.70000000000000009e168

   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 56.6%

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

      1. *-rgt-identity 56.6%

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

      2. *-commutative 56.6%

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

      3. sub-neg 56.6%

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

      4. distribute-lft-in 56.6%

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

   5. Applied egg-rr 56.6%

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

      1. distribute-lft-out 56.6%

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

      2. *-commutative 56.6%

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

      3. sub-neg 56.6%

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

   7. Simplified 56.6%

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

   8. Taylor expanded in x around inf 43.4%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1650 \lor \neg \left(y \leq 3.7 \cdot 10^{+168}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -720000:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+168}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= y -720000.0)
     (sqrt (* (* z 2.0) (* y (- y x))))
     (if (<= y 3.4e+168) (* t_1 (* x 0.5)) (* y (- t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (y <= -720000.0) {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	} else if (y <= 3.4e+168) {
		tmp = t_1 * (x * 0.5);
	} 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 (y <= (-720000.0d0)) then
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    else if (y <= 3.4d+168) then
        tmp = t_1 * (x * 0.5d0)
    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 (y <= -720000.0) {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	} else if (y <= 3.4e+168) {
		tmp = t_1 * (x * 0.5);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if y <= -720000.0:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	elif y <= 3.4e+168:
		tmp = t_1 * (x * 0.5)
	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 (y <= -720000.0)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	elseif (y <= 3.4e+168)
		tmp = Float64(t_1 * Float64(x * 0.5));
	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 (y <= -720000.0)
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	elseif (y <= 3.4e+168)
		tmp = t_1 * (x * 0.5);
	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[LessEqual[y, -720000.0], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 3.4e+168], N[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.2e5

   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. Add Preprocessing

   3. Taylor expanded in t around 0 57.1%

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

      1. *-rgt-identity 57.1%

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

      2. *-commutative 57.1%

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

      3. pow1 57.1%

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

      4. metadata-eval 57.1%

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

      5. sqrt-pow1 63.4%

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

      6. sqrt-prod 60.2%

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

      7. pow1/2 60.2%

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

      8. associate-*l* 60.2%

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

   5. Applied egg-rr 60.2%

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

      1. unpow1/2 60.2%

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

      2. associate-*r* 60.2%

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

      3. *-commutative 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in x around 0 48.2%

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

      1. +-commutative 48.2%

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

      2. unpow2 48.2%

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

      3. associate-*r* 48.2%

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

      4. distribute-rgt-in 57.0%

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

      5. mul-1-neg 57.0%

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

      6. unsub-neg 57.0%

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

   10. Simplified 57.0%

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

   if -7.2e5 < y < 3.40000000000000003e168

   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 56.6%

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

      1. *-rgt-identity 56.6%

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

      2. *-commutative 56.6%

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

      3. sub-neg 56.6%

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

      4. distribute-lft-in 56.6%

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

   5. Applied egg-rr 56.6%

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

      1. distribute-lft-out 56.6%

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

      2. *-commutative 56.6%

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

      3. sub-neg 56.6%

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

   7. Simplified 56.6%

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

   8. Taylor expanded in x around inf 43.4%

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

   if 3.40000000000000003e168 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 77.5%

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

      1. *-rgt-identity 77.5%

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

      2. *-commutative 77.5%

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

      3. sub-neg 77.5%

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

      4. distribute-lft-in 73.8%

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

   5. Applied egg-rr 73.8%

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

      1. distribute-lft-out 77.5%

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

      2. *-commutative 77.5%

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

      3. sub-neg 77.5%

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

   7. Simplified 77.5%

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

   8. Taylor expanded in x around 0 66.2%

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

      1. neg-mul-1 66.2%

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

   10. Simplified 66.2%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -720000:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+168}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 18.0) (* (- (* x 0.5) y) t_1) (* x (* t_1 (- 0.5 (/ y x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 18.0) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = x * (t_1 * (0.5 - (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t <= 18.0d0) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = x * (t_1 * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 18.0) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = x * (t_1 * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 18.0:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = x * (t_1 * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 18.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(x * Float64(t_1 * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 18.0)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = x * (t_1 * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 18.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(x * N[(t$95$1 * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 18

   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 74.8%

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

      1. *-rgt-identity 74.8%

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

      2. *-commutative 74.8%

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

      3. sub-neg 74.8%

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

      4. distribute-lft-in 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. distribute-lft-out 74.8%

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

      2. *-commutative 74.8%

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

      3. sub-neg 74.8%

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

   7. Simplified 74.8%

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

   if 18 < 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 14.2%

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

      1. add-cube-cbrt 14.2%

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

      2. pow3 14.2%

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

   5. Applied egg-rr 14.2%

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

   6. Taylor expanded in x around inf 24.1%

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

      1. associate-*r* 24.1%

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

      2. rem-cube-cbrt 24.1%

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

      3. rem-cube-cbrt 24.1%

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

      4. distribute-rgt-out 24.1%

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

      5. +-commutative 24.1%

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

      6. mul-1-neg 24.1%

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

      7. unsub-neg 24.1%

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

   8. Simplified 24.1%

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

4. Final simplification 61.5%

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

Alternative 8: 56.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{+249}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 4.2e+249)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (sqrt (* (* z 2.0) (* y (- y x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.2e+249) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 4.2d+249) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.2e+249) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 4.2e+249:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 4.2e+249)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 4.2e+249)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 4.2e+249], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.1999999999999997e249

   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 62.2%

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

      1. *-rgt-identity 62.2%

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

      2. *-commutative 62.2%

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

      3. sub-neg 62.2%

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

      4. distribute-lft-in 61.3%

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

   5. Applied egg-rr 61.3%

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

      1. distribute-lft-out 62.2%

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

      2. *-commutative 62.2%

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

      3. sub-neg 62.2%

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

   7. Simplified 62.2%

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

   if 4.1999999999999997e249 < 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 10.3%

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

      1. *-rgt-identity 10.3%

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

      2. *-commutative 10.3%

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

      3. pow1 10.3%

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

      4. metadata-eval 10.3%

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

      5. sqrt-pow1 32.0%

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

      6. sqrt-prod 32.0%

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

      7. pow1/2 32.0%

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

      8. associate-*l* 32.0%

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

   5. Applied egg-rr 32.0%

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

      1. unpow1/2 32.0%

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

      2. associate-*r* 32.0%

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

      3. *-commutative 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in x around 0 19.4%

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

      1. +-commutative 19.4%

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

      2. unpow2 19.4%

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

      3. associate-*r* 19.4%

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

      4. distribute-rgt-in 25.7%

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

      5. mul-1-neg 25.7%

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

      6. unsub-neg 25.7%

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

   10. Simplified 25.7%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{+249}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.3% 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 99.8%

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

3. Taylor expanded in t around 0 58.9%

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

   1. *-rgt-identity 58.9%

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

   2. *-commutative 58.9%

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

   3. sub-neg 58.9%

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

   4. distribute-lft-in 58.2%

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

5. Applied egg-rr 58.2%

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

   1. distribute-lft-out 58.9%

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

   2. *-commutative 58.9%

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

   3. sub-neg 58.9%

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

7. Simplified 58.9%

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

8. Taylor expanded in x around 0 28.5%

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

   1. neg-mul-1 28.5%

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

10. Simplified 28.5%

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

11. Final simplification 28.5%

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

Alternative 10: 2.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 58.9%

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

   1. *-rgt-identity 58.9%

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

   2. *-commutative 58.9%

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

   3. sub-neg 58.9%

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

   4. distribute-lft-in 58.2%

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

5. Applied egg-rr 58.2%

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

   1. distribute-lft-out 58.9%

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

   2. *-commutative 58.9%

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

   3. sub-neg 58.9%

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

7. Simplified 58.9%

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

8. Taylor expanded in x around 0 28.5%

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

   1. neg-mul-1 28.5%

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

10. Simplified 28.5%

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

    1. neg-sub0 28.5%

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

    2. sub-neg 28.5%

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

    3. add-sqr-sqrt 15.2%

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

    4. sqrt-unprod 17.6%

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

    5. sqr-neg 17.6%

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

    6. sqrt-unprod 1.3%

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

    7. add-sqr-sqrt 2.2%

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

12. Applied egg-rr 2.2%

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

    1. +-lft-identity 2.2%

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

14. Simplified 2.2%

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

15. Final simplification 2.2%

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

Developer Target 1: 99.3% 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 2024170 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

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