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

Percentage Accurate: 99.5% → 99.4%

Time: 14.0s

Alternatives: 7

Speedup: 1.0×

• 45.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := e^{\frac{t \cdot t}{2}}\\ \mathbf{if}\;t\_2 \leq 2.2 \cdot 10^{+301}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(y \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (exp (/ (* t t) 2.0))))
   (if (<= t_2 2.2e+301) (* (- (* x 0.5) y) t_1) (* t_2 (* y t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = exp(((t * t) / 2.0));
	double tmp;
	if (t_2 <= 2.2e+301) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_2 * (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) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = exp(((t * t) / 2.0d0))
    if (t_2 <= 2.2d+301) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = t_2 * (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 t_2 = Math.exp(((t * t) / 2.0));
	double tmp;
	if (t_2 <= 2.2e+301) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_2 * (y * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	t_2 = math.exp(((t * t) / 2.0))
	tmp = 0
	if t_2 <= 2.2e+301:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = t_2 * (y * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = exp(Float64(Float64(t * t) / 2.0))
	tmp = 0.0
	if (t_2 <= 2.2e+301)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(t_2 * Float64(y * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	t_2 = exp(((t * t) / 2.0));
	tmp = 0.0;
	if (t_2 <= 2.2e+301)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = t_2 * (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]}, Block[{t$95$2 = N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2.2e+301], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$2 * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))) < 2.2e301

   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. Step-by-step derivation

      1. exp-sqrt 99.6%

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

      2. pow1 99.6%

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

      3. exp-sqrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. exp-sqrt 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. pow2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 99.0%

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

   if 2.2e301 < (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64)))

   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 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. distribute-rgt-neg-in 67.2%

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

   5. Simplified 67.2%

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

      1. pow1 67.2%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 28.2%

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

      4. sqr-neg 28.2%

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

      5. add-sqr-sqrt 28.2%

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

      6. associate-*l* 28.2%

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

      7. sqrt-prod 28.2%

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

   7. Applied egg-rr 28.2%

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

      1. unpow1 28.2%

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

   9. Simplified 28.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{t \cdot t}{2}} \leq 2.2 \cdot 10^{+301}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 10^{-30}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(-t\_1\right)\right) \cdot e^{\frac{t \cdot t}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 1e-30)
     (* (- (* x 0.5) y) t_1)
     (* (* y (- t_1)) (exp (/ (* t t) 2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 1e-30) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = (y * -t_1) * exp(((t * t) / 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((t * t) <= 1d-30) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = (y * -t_1) * exp(((t * t) / 2.0d0))
    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 * t) <= 1e-30) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = (y * -t_1) * Math.exp(((t * t) / 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 1e-30:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = (y * -t_1) * math.exp(((t * t) / 2.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 1e-30)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(Float64(y * Float64(-t_1)) * exp(Float64(Float64(t * t) / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 1e-30)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = (y * -t_1) * exp(((t * t) / 2.0));
	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[N[(t * t), $MachinePrecision], 1e-30], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(y * (-t$95$1)), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1e-30

   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. Step-by-step derivation

      1. exp-sqrt 99.6%

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

      2. pow1 99.6%

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

      3. exp-sqrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. exp-sqrt 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. pow2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 99.6%

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

   if 1e-30 < (*.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 0 68.3%

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

      1. mul-1-neg 68.3%

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

      2. distribute-rgt-neg-in 68.3%

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

   5. Simplified 68.3%

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

      1. distribute-rgt-neg-out 68.3%

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

      2. neg-sub0 68.3%

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

      3. associate-*l* 68.3%

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

      4. sqrt-prod 68.3%

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

   7. Applied egg-rr 68.3%

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

      1. neg-sub0 68.3%

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

      2. distribute-lft-neg-in 68.3%

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

   9. Simplified 68.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-30}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(-\sqrt{z \cdot 2}\right)\right) \cdot e^{\frac{t \cdot t}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. exp-sqrt 99.8%

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

   2. pow1 99.8%

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

   3. exp-sqrt 99.8%

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

   4. associate-*l* 99.8%

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

   5. exp-sqrt 99.8%

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

   6. sqrt-unprod 99.8%

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

   7. pow2 99.8%

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

4. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

6. Simplified 99.8%

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

   1. pow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 55.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.4e+154)
   (sqrt (* (* z 2.0) (* y (- y x))))
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.4e+154) {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	} else {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-4.4d+154)) then
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    else
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.4e+154) {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	} else {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -4.4e+154:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	else:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.4e+154)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	else
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -4.4e+154)
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	else
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.4e+154], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000002e154

   1. Initial program 100.0%

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

      1. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 47.9%

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

      1. add-sqr-sqrt 36.3%

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

      2. sqrt-unprod 57.0%

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

      3. *-commutative 57.0%

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

      4. *-commutative 57.0%

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

      5. swap-sqr 64.1%

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

      6. add-sqr-sqrt 64.1%

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

      7. pow2 64.1%

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

   7. Applied egg-rr 64.1%

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

   8. Taylor expanded in x around 0 60.1%

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

      1. +-commutative 60.1%

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

      2. mul-1-neg 60.1%

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

      3. unsub-neg 60.1%

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

      4. unpow2 60.1%

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

      5. distribute-rgt-out-- 64.1%

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

   10. Simplified 64.1%

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

   if -4.4000000000000002e154 < 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. Step-by-step derivation

      1. exp-sqrt 99.8%

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

      2. pow1 99.8%

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

      3. exp-sqrt 99.8%

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

      4. associate-*l* 99.8%

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

      5. exp-sqrt 99.8%

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

      6. sqrt-unprod 99.8%

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

      7. pow2 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 57.5%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.4% 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. Step-by-step derivation

   1. exp-sqrt 99.8%

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

   2. pow1 99.8%

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

   3. exp-sqrt 99.8%

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

   4. associate-*l* 99.8%

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

   5. exp-sqrt 99.8%

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

   6. sqrt-unprod 99.8%

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

   7. pow2 99.8%

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

4. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 56.4%

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

8. Final simplification 56.4%

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

Alternative 7: 29.9% 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. Step-by-step derivation

   1. exp-sqrt 99.8%

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

   2. pow1 99.8%

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

   3. exp-sqrt 99.8%

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

   4. associate-*l* 99.8%

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

   5. exp-sqrt 99.8%

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

   6. sqrt-unprod 99.8%

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

   7. pow2 99.8%

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

4. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 56.4%

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

8. Taylor expanded in x around 0 28.4%

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

   1. mul-1-neg 28.4%

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

10. Simplified 28.4%

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

    1. pow1 28.4%

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

    2. associate-*l* 28.4%

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

    3. sqrt-prod 28.5%

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

12. Applied egg-rr 28.5%

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

    1. unpow1 28.5%

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

14. Simplified 28.5%

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

15. Final simplification 28.5%

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

Developer target: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024131 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

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