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

Percentage Accurate: 99.4% → 99.4%

Time: 12.3s

Alternatives: 7

Speedup: 1.0×

• 44.3% 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: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 5000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(x \cdot 0.5\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 5000000.0)
     (* (- (* 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) <= 5000000.0) {
		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) <= 5000000.0d0) 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) <= 5000000.0) {
		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) <= 5000000.0:
		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) <= 5000000.0)
		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) <= 5000000.0)
		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], 5000000.0], 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) < 5e6

   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 97.6%

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

   if 5e6 < (*.f64 t t)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.6%

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

      1. associate-*r* 83.6%

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

      2. *-commutative 83.6%

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

   5. Simplified 83.6%

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

      1. pow1 83.6%

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

      2. *-commutative 83.6%

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

      3. associate-*r* 83.6%

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

      4. associate-*l* 83.6%

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

      5. sqrt-prod 83.6%

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

      6. *-commutative 83.6%

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

      7. *-commutative 83.6%

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

   7. Applied egg-rr 83.6%

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

      1. unpow1 83.6%

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

   9. Simplified 83.6%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 0.0003:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t\_1 \cdot \mathsf{fma}\left(0.5, \frac{x}{y}, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 0.0003)
     (* (- (* x 0.5) y) t_1)
     (* y (* t_1 (fma 0.5 (/ x 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 <= 0.0003) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = y * (t_1 * fma(0.5, (x / y), -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 0.0003)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(y * Float64(t_1 * fma(0.5, Float64(x / y), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.0003], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(y * N[(t$95$1 * N[(0.5 * N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.99999999999999974e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 68.2%

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

   if 2.99999999999999974e-4 < 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 26.7%

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

   4. Taylor expanded in z around 0 26.7%

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

   5. Taylor expanded in y around inf 38.0%

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

      1. *-rgt-identity 38.0%

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

      2. pow1 38.0%

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

      3. associate-*l* 39.6%

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

      4. fmm-def 39.6%

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

      5. metadata-eval 39.6%

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

      6. sqrt-unprod 39.6%

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

   7. Applied egg-rr 39.6%

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

      1. unpow1 39.6%

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

      2. *-commutative 39.6%

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

   9. Simplified 39.6%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0003:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(0.5, \frac{x}{y}, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{+175} \lor \neg \left(x \leq 1.3 \cdot 10^{+72}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= x -2.75e+175) (not (<= x 1.3e+72)))
     (* (* x 0.5) t_1)
     (* t_1 (- y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((x <= -2.75e+175) || !(x <= 1.3e+72)) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((x <= (-2.75d+175)) .or. (.not. (x <= 1.3d+72))) then
        tmp = (x * 0.5d0) * t_1
    else
        tmp = t_1 * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((x <= -2.75e+175) || !(x <= 1.3e+72)) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (x <= -2.75e+175) or not (x <= 1.3e+72):
		tmp = (x * 0.5) * t_1
	else:
		tmp = t_1 * -y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((x <= -2.75e+175) || !(x <= 1.3e+72))
		tmp = Float64(Float64(x * 0.5) * t_1);
	else
		tmp = Float64(t_1 * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((x <= -2.75e+175) || ~((x <= 1.3e+72)))
		tmp = (x * 0.5) * t_1;
	else
		tmp = t_1 * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2.75e+175], N[Not[LessEqual[x, 1.3e+72]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.75000000000000009e175 or 1.29999999999999991e72 < 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 x around inf 91.3%

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

      1. associate-*r* 91.3%

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

      2. *-commutative 91.3%

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

   5. Simplified 91.3%

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

      1. pow1 91.3%

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

      2. *-commutative 91.3%

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

      3. associate-*r* 91.3%

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

      4. associate-*l* 91.2%

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

      5. sqrt-prod 91.2%

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

      6. *-commutative 91.2%

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

      7. *-commutative 91.2%

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

   7. Applied egg-rr 91.2%

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

      1. unpow1 91.2%

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

   9. Simplified 91.2%

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

   10. Taylor expanded in t around 0 55.5%

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

   if -2.75000000000000009e175 < x < 1.29999999999999991e72

   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 56.8%

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

   4. Taylor expanded in x around 0 44.0%

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

      1. mul-1-neg 44.0%

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

      2. *-commutative 44.0%

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

      3. distribute-rgt-neg-in 44.0%

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

      4. *-commutative 44.0%

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

      5. distribute-rgt-neg-in 44.0%

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

   6. Simplified 44.0%

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

      1. *-rgt-identity 44.0%

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

      2. associate-*r* 44.0%

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

      3. distribute-rgt-neg-out 44.0%

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

      4. add-sqr-sqrt 20.6%

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

      5. sqrt-unprod 18.9%

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

      6. sqr-neg 18.9%

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

      7. sqrt-unprod 0.8%

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

      8. add-sqr-sqrt 1.5%

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

      9. *-commutative 1.5%

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

      10. add-sqr-sqrt 0.8%

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

      11. sqrt-unprod 18.9%

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

      12. sqr-neg 18.9%

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

      13. sqrt-unprod 20.6%

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

      14. add-sqr-sqrt 44.0%

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

      15. *-commutative 44.0%

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

      16. sqrt-prod 44.2%

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

   8. Applied egg-rr 44.2%

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

      1. distribute-rgt-neg-in 44.2%

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

   10. Simplified 44.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+175} \lor \neg \left(x \leq 1.3 \cdot 10^{+72}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 58.8%

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

4. Final simplification 58.8%

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

Alternative 6: 29.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 58.8%

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

4. Taylor expanded in x around 0 35.2%

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

   1. mul-1-neg 35.2%

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

   2. *-commutative 35.2%

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

   3. distribute-rgt-neg-in 35.2%

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

   4. *-commutative 35.2%

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

   5. distribute-rgt-neg-in 35.2%

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

6. Simplified 35.2%

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

   1. *-rgt-identity 35.2%

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

   2. associate-*r* 35.2%

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

   3. distribute-rgt-neg-out 35.2%

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

   4. add-sqr-sqrt 16.6%

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

   5. sqrt-unprod 15.8%

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

   6. sqr-neg 15.8%

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

   7. sqrt-unprod 1.1%

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

   8. add-sqr-sqrt 2.4%

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

   9. *-commutative 2.4%

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

   10. add-sqr-sqrt 1.1%

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

   11. sqrt-unprod 15.8%

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

   12. sqr-neg 15.8%

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

   13. sqrt-unprod 16.6%

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

   14. add-sqr-sqrt 35.2%

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

   15. *-commutative 35.2%

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

   16. sqrt-prod 35.3%

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

8. Applied egg-rr 35.3%

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

   1. distribute-rgt-neg-in 35.3%

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

10. Simplified 35.3%

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

11. Final simplification 35.3%

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

Alternative 7: 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 99.8%

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

3. Taylor expanded in t around 0 58.8%

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

4. Taylor expanded in x around 0 35.2%

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

   1. mul-1-neg 35.2%

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

   2. *-commutative 35.2%

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

   3. distribute-rgt-neg-in 35.2%

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

   4. *-commutative 35.2%

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

   5. distribute-rgt-neg-in 35.2%

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

6. Simplified 35.2%

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

   1. pow1 35.2%

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

   2. *-rgt-identity 35.2%

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

   3. associate-*r* 35.2%

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

   4. *-commutative 35.2%

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

   5. add-sqr-sqrt 18.5%

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

   6. sqrt-unprod 20.2%

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

   7. sqr-neg 20.2%

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

   8. sqrt-unprod 1.2%

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

   9. add-sqr-sqrt 2.4%

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

   10. *-commutative 2.4%

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

   11. sqrt-prod 2.4%

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

8. Applied egg-rr 2.4%

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

   1. unpow1 2.4%

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

10. Simplified 2.4%

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

11. Final simplification 2.4%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024163 
(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))))