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

Percentage Accurate: 99.4% → 99.4%

Time: 15.5s

Alternatives: 8

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: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 0.019:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(t\_1 \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 0.019)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* 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 ((t * t) <= 0.019) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * (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 ((t * t) <= 0.019d0) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * (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 ((t * t) <= 0.019) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (t_1 * (x * 0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 0.019:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * (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 (Float64(t * t) <= 0.019)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * 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 ((t * t) <= 0.019)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * (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[LessEqual[N[(t * t), $MachinePrecision], 0.019], 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[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 0.0189999999999999995

   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 98.8%

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

   if 0.0189999999999999995 < (*.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 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-rgt-in 100.0%

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

      10. *-commutative 100.0%

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

      11. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 80.6%

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

      1. associate-*l* 80.6%

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

      2. *-commutative 80.6%

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

      3. unpow1/2 80.6%

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

      4. metadata-eval 80.6%

         \[\leadsto \left(0.5 \cdot \left(x \cdot \left({z}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot \sqrt{2}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. pow-sqr 80.6%

         \[\leadsto \left(0.5 \cdot \left(x \cdot \left(\color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)} \cdot \sqrt{2}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. unpow1/2 80.6%

         \[\leadsto \left(0.5 \cdot \left(x \cdot \left(\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \color{blue}{{2}^{0.5}}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 80.6%

         \[\leadsto \left(0.5 \cdot \left(x \cdot \left(\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot {2}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. pow-sqr 80.6%

         \[\leadsto \left(0.5 \cdot \left(x \cdot \left(\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. unswap-sqr 80.6%

         \[\leadsto \left(0.5 \cdot \left(x \cdot \color{blue}{\left(\left({z}^{0.25} \cdot {2}^{0.25}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. exp-to-pow 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left(\left(\color{blue}{e^{\log z \cdot 0.25}} \cdot {2}^{0.25}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. exp-to-pow 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left(\left(e^{\log z \cdot 0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25}}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. exp-sum 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left(\color{blue}{e^{\log z \cdot 0.25 + \log 2 \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. distribute-rgt-in 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log z + \log 2\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      14. +-commutative 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left(e^{0.25 \cdot \color{blue}{\left(\log 2 + \log z\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      15. *-commutative 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left(e^{\color{blue}{\left(\log 2 + \log z\right) \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      16. exp-prod 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left(\color{blue}{{\left(e^{\log 2 + \log z}\right)}^{0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      17. exp-sum 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left({\color{blue}{\left(e^{\log 2} \cdot e^{\log z}\right)}}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      18. rem-exp-log 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left({\left(\color{blue}{2} \cdot e^{\log z}\right)}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      19. rem-exp-log 80.6%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left({\left(2 \cdot \color{blue}{z}\right)}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      20. *-commutative 80.6%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left({\color{blue}{\left(z \cdot 2\right)}}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      21. exp-to-pow 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left({\left(z \cdot 2\right)}^{0.25} \cdot \left(\color{blue}{e^{\log z \cdot 0.25}} \cdot {2}^{0.25}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      22. exp-to-pow 80.5%

          \[\leadsto \left(0.5 \cdot \left(x \cdot \left({\left(z \cdot 2\right)}^{0.25} \cdot \left(e^{\log z \cdot 0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25}}\right)\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   8. Simplified 80.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.019:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 2.1e-15)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* y (- 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) <= 2.1e-15) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * (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 ((t * t) <= 2.1d-15) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * (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 ((t * t) <= 2.1e-15) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (y * -t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 2.1e-15:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * (y * -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) <= 2.1e-15)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * 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 ((t * t) <= 2.1e-15)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * (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[N[(t * t), $MachinePrecision], 2.1e-15], 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[(y * (-t$95$1)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.09999999999999981e-15

   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 99.6%

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

   if 2.09999999999999981e-15 < (*.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 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-rgt-in 100.0%

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

      10. *-commutative 100.0%

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

      11. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 73.1%

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

      1. mul-1-neg 73.1%

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

      2. associate-*l* 73.1%

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

      3. *-commutative 73.1%

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

      4. distribute-lft-neg-in 73.1%

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

      5. unpow1/2 73.1%

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

      6. metadata-eval 73.1%

         \[\leadsto \left(\left(-y\right) \cdot \left({z}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot \sqrt{2}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. pow-sqr 73.1%

         \[\leadsto \left(\left(-y\right) \cdot \left(\color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)} \cdot \sqrt{2}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. unpow1/2 73.1%

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

      9. metadata-eval 73.1%

         \[\leadsto \left(\left(-y\right) \cdot \left(\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot {2}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. pow-sqr 73.1%

          \[\leadsto \left(\left(-y\right) \cdot \left(\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. unswap-sqr 73.1%

          \[\leadsto \left(\left(-y\right) \cdot \color{blue}{\left(\left({z}^{0.25} \cdot {2}^{0.25}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   8. Simplified 73.1%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 2.75 \cdot 10^{-36}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(y \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 2.75e-36)
     (* (- (* x 0.5) y) t_1)
     (* t_1 (* y (+ (* 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 <= 2.75e-36) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (y * ((0.5 * (x / 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 <= 2.75d-36) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = t_1 * (y * ((0.5d0 * (x / 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 <= 2.75e-36) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 2.75e-36:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = t_1 * (y * ((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 <= 2.75e-36)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(t_1 * Float64(y * Float64(Float64(0.5 * Float64(x / 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 <= 2.75e-36)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = t_1 * (y * ((0.5 * (x / 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, 2.75e-36], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(y * N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.74999999999999992e-36

   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 75.2%

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

   if 2.74999999999999992e-36 < 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 26.9%

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

   4. Taylor expanded in y around inf 37.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-36}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(y \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-71} \lor \neg \left(y \leq 5 \cdot 10^{+38}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.7e-71) (not (<= y 5e+38)))
   (* y (- (sqrt (* z 2.0))))
   (* x (sqrt (* 0.5 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.7e-71) || !(y <= 5e+38)) {
		tmp = y * -sqrt((z * 2.0));
	} else {
		tmp = x * sqrt((0.5 * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.7d-71)) .or. (.not. (y <= 5d+38))) then
        tmp = y * -sqrt((z * 2.0d0))
    else
        tmp = x * sqrt((0.5d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.7e-71) || !(y <= 5e+38)) {
		tmp = y * -Math.sqrt((z * 2.0));
	} else {
		tmp = x * Math.sqrt((0.5 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.7e-71) or not (y <= 5e+38):
		tmp = y * -math.sqrt((z * 2.0))
	else:
		tmp = x * math.sqrt((0.5 * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.7e-71) || !(y <= 5e+38))
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	else
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.7e-71) || ~((y <= 5e+38)))
		tmp = y * -sqrt((z * 2.0));
	else
		tmp = x * sqrt((0.5 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.7e-71], N[Not[LessEqual[y, 5e+38]], $MachinePrecision]], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.70000000000000002e-71 or 4.9999999999999997e38 < 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 64.4%

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

   4. Taylor expanded in x around 0 54.4%

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

      1. neg-mul-1 54.4%

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

   6. Simplified 54.4%

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

      1. *-rgt-identity 54.4%

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

      2. distribute-lft-neg-out 54.4%

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

      3. neg-sub0 54.4%

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

   8. Applied egg-rr 54.4%

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

      1. neg-sub0 54.4%

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

      2. *-commutative 54.4%

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

      3. distribute-rgt-neg-in 54.4%

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

   10. Simplified 54.4%

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

   if -1.70000000000000002e-71 < y < 4.9999999999999997e38

   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 56.1%

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

   4. Taylor expanded in x around inf 46.4%

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

      1. *-rgt-identity 46.4%

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

      2. add-cube-cbrt 46.0%

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

      3. pow3 46.0%

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

      4. *-commutative 46.0%

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

      5. *-commutative 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-*r* 46.0%

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

      8. sqrt-prod 46.0%

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

      9. *-commutative 46.0%

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

      10. associate-*l* 46.0%

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

   6. Applied egg-rr 46.0%

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

      1. rem-cube-cbrt 46.5%

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

      2. add-sqr-sqrt 46.4%

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

      3. sqrt-unprod 46.5%

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

      4. swap-sqr 46.5%

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

      5. add-sqr-sqrt 46.5%

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

      6. metadata-eval 46.5%

         \[\leadsto x \cdot \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{0.25}} \]

   8. Applied egg-rr 46.5%

      \[\leadsto \color{blue}{x \cdot \sqrt{\left(z \cdot 2\right) \cdot 0.25}} \]
   9. Step-by-step derivation

      1. associate-*l* 46.5%

         \[\leadsto x \cdot \sqrt{\color{blue}{z \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 46.5%

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

   10. Simplified 46.5%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-71} \lor \neg \left(y \leq 5 \cdot 10^{+38}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 1.5e-5) (* (- (* x 0.5) y) t_1) (* t_1 (* x (- 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 <= 1.5e-5) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (x * (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 <= 1.5d-5) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = t_1 * (x * (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 <= 1.5e-5) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 1.5e-5:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = t_1 * (x * (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 <= 1.5e-5)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(t_1 * Float64(x * 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 <= 1.5e-5)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = t_1 * (x * (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, 1.5e-5], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.50000000000000004e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 76.1%

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

   if 1.50000000000000004e-5 < 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 20.1%

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

   4. Taylor expanded in x around inf 27.8%

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

      1. mul-1-neg 27.8%

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

      2. unsub-neg 27.8%

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

   6. Simplified 27.8%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.1% 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 59.9%

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

4. Final simplification 59.9%

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

Alternative 8: 29.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 59.9%

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

4. Taylor expanded in x around inf 31.9%

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

   1. *-rgt-identity 31.9%

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

   2. add-cube-cbrt 31.6%

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

   3. pow3 31.6%

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

   4. *-commutative 31.6%

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

   5. *-commutative 31.6%

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

   6. *-commutative 31.6%

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

   7. associate-*r* 31.6%

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

   8. sqrt-prod 31.6%

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

   9. *-commutative 31.6%

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

   10. associate-*l* 31.6%

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

6. Applied egg-rr 31.6%

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

   1. rem-cube-cbrt 32.0%

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

   2. add-sqr-sqrt 31.8%

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

   3. sqrt-unprod 32.0%

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

   4. swap-sqr 32.0%

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

   5. add-sqr-sqrt 32.0%

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

   6. metadata-eval 32.0%

      \[\leadsto x \cdot \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{0.25}} \]

8. Applied egg-rr 32.0%

   \[\leadsto \color{blue}{x \cdot \sqrt{\left(z \cdot 2\right) \cdot 0.25}} \]
9. Step-by-step derivation

   1. associate-*l* 32.0%

      \[\leadsto x \cdot \sqrt{\color{blue}{z \cdot \left(2 \cdot 0.25\right)}} \]

   2. metadata-eval 32.0%

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

10. Simplified 32.0%

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

11. Final simplification 32.0%

    \[\leadsto x \cdot \sqrt{0.5 \cdot z} \]
12. 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 2024137 
(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))))