Data.Number.Erf:$dmerfcx from erf-2.0.0.0

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 5

Speedup: 1.0×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (exp (* y y))))

C

double code(double x, double y) {
	return x * exp((y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * exp((y * y))
end function

Java

public static double code(double x, double y) {
	return x * Math.exp((y * y));
}

Python

def code(x, y):
	return x * math.exp((y * y))

Julia

function code(x, y)
	return Float64(x * exp(Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * exp((y * y));
end

Wolfram

code[x_, y_] := N[(x * N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (exp (* y y))))

C

double code(double x, double y) {
	return x * exp((y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * exp((y * y))
end function

Java

public static double code(double x, double y) {
	return x * Math.exp((y * y));
}

Python

def code(x, y):
	return x * math.exp((y * y))

Julia

function code(x, y)
	return Float64(x * exp(Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * exp((y * y));
end

Wolfram

code[x_, y_] := N[(x * N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{6}}{2}\right)}\\ x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \left(t_0 \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (pow (exp y) 2.0) (/ (/ y 6.0) 2.0))))
   (* x (* (pow (exp (* y 2.0)) (* y 0.3333333333333333)) (* t_0 t_0)))))

C

double code(double x, double y) {
	double t_0 = pow(pow(exp(y), 2.0), ((y / 6.0) / 2.0));
	return x * (pow(exp((y * 2.0)), (y * 0.3333333333333333)) * (t_0 * t_0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (exp(y) ** 2.0d0) ** ((y / 6.0d0) / 2.0d0)
    code = x * ((exp((y * 2.0d0)) ** (y * 0.3333333333333333d0)) * (t_0 * t_0))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.pow(Math.exp(y), 2.0), ((y / 6.0) / 2.0));
	return x * (Math.pow(Math.exp((y * 2.0)), (y * 0.3333333333333333)) * (t_0 * t_0));
}

Python

def code(x, y):
	t_0 = math.pow(math.pow(math.exp(y), 2.0), ((y / 6.0) / 2.0))
	return x * (math.pow(math.exp((y * 2.0)), (y * 0.3333333333333333)) * (t_0 * t_0))

Julia

function code(x, y)
	t_0 = (exp(y) ^ 2.0) ^ Float64(Float64(y / 6.0) / 2.0)
	return Float64(x * Float64((exp(Float64(y * 2.0)) ^ Float64(y * 0.3333333333333333)) * Float64(t_0 * t_0)))
end

MATLAB

function tmp = code(x, y)
	t_0 = (exp(y) ^ 2.0) ^ ((y / 6.0) / 2.0);
	tmp = x * ((exp((y * 2.0)) ^ (y * 0.3333333333333333)) * (t_0 * t_0));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Power[N[Exp[y], $MachinePrecision], 2.0], $MachinePrecision], N[(N[(y / 6.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(x * N[(N[Power[N[Exp[N[(y * 2.0), $MachinePrecision]], $MachinePrecision], N[(y * 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

   \[x \cdot e^{y \cdot y} \]
2. Step-by-step derivation

   1. *-un-lft-identity 99.9%

      \[\leadsto x \cdot e^{\color{blue}{1 \cdot \left(y \cdot y\right)}} \]

   2. exp-prod 100.0%

      \[\leadsto x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(y \cdot y\right)}} \]

   3. exp-1-e 100.0%

      \[\leadsto x \cdot {\color{blue}{e}}^{\left(y \cdot y\right)} \]

   4. pow2 100.0%

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

3. Applied egg-rr 100.0%

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

   1. unpow2 100.0%

      \[\leadsto x \cdot {e}^{\color{blue}{\left(y \cdot y\right)}} \]

   2. pow-unpow 99.9%

      \[\leadsto x \cdot \color{blue}{{\left({e}^{y}\right)}^{y}} \]

   3. e-exp-1 99.9%

      \[\leadsto x \cdot {\left({\color{blue}{\left(e^{1}\right)}}^{y}\right)}^{y} \]

   4. exp-prod 100.0%

      \[\leadsto x \cdot {\color{blue}{\left(e^{1 \cdot y}\right)}}^{y} \]

   5. *-un-lft-identity 100.0%

      \[\leadsto x \cdot {\left(e^{\color{blue}{y}}\right)}^{y} \]

   6. add-cbrt-cube 99.9%

      \[\leadsto x \cdot {\color{blue}{\left(\sqrt[3]{\left(e^{y} \cdot e^{y}\right) \cdot e^{y}}\right)}}^{y} \]

   7. unpow3 99.9%

      \[\leadsto x \cdot {\left(\sqrt[3]{\color{blue}{{\left(e^{y}\right)}^{3}}}\right)}^{y} \]

   8. exp-prod 99.9%

      \[\leadsto x \cdot {\left(\sqrt[3]{\color{blue}{e^{y \cdot 3}}}\right)}^{y} \]

   9. pow1/3 99.9%

      \[\leadsto x \cdot {\color{blue}{\left({\left(e^{y \cdot 3}\right)}^{0.3333333333333333}\right)}}^{y} \]

   10. pow-unpow 99.9%

       \[\leadsto x \cdot \color{blue}{{\left(e^{y \cdot 3}\right)}^{\left(0.3333333333333333 \cdot y\right)}} \]

   11. *-commutative 99.9%

       \[\leadsto x \cdot {\left(e^{y \cdot 3}\right)}^{\color{blue}{\left(y \cdot 0.3333333333333333\right)}} \]

   12. add-cube-cbrt 99.9%

       \[\leadsto x \cdot {\color{blue}{\left(\left(\sqrt[3]{e^{y \cdot 3}} \cdot \sqrt[3]{e^{y \cdot 3}}\right) \cdot \sqrt[3]{e^{y \cdot 3}}\right)}}^{\left(y \cdot 0.3333333333333333\right)} \]

   13. unpow-prod-down 99.9%

       \[\leadsto x \cdot \color{blue}{\left({\left(\sqrt[3]{e^{y \cdot 3}} \cdot \sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right)} \]

   14. pow2 99.9%

       \[\leadsto x \cdot \left({\color{blue}{\left({\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{2}\right)}}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

   15. exp-prod 99.9%

       \[\leadsto x \cdot \left({\left({\left(\sqrt[3]{\color{blue}{{\left(e^{y}\right)}^{3}}}\right)}^{2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

   16. unpow3 99.9%

       \[\leadsto x \cdot \left({\left({\left(\sqrt[3]{\color{blue}{\left(e^{y} \cdot e^{y}\right) \cdot e^{y}}}\right)}^{2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

   17. add-cbrt-cube 99.9%

       \[\leadsto x \cdot \left({\left({\color{blue}{\left(e^{y}\right)}}^{2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto x \cdot \color{blue}{\left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(e^{y}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right)} \]
6. Step-by-step derivation

   1. pow-exp 100.0%

      \[\leadsto x \cdot \left({\color{blue}{\left(e^{y \cdot 2}\right)}}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(e^{y}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left({\color{blue}{\left(e^{y \cdot 2}\right)}}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(e^{y}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]
8. Step-by-step derivation

   1. sqr-pow 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \color{blue}{\left({\left(e^{y}\right)}^{\left(\frac{y \cdot 0.3333333333333333}{2}\right)} \cdot {\left(e^{y}\right)}^{\left(\frac{y \cdot 0.3333333333333333}{2}\right)}\right)}\right) \]

   2. pow-prod-down 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(e^{y} \cdot e^{y}\right)}^{\left(\frac{y \cdot 0.3333333333333333}{2}\right)}}\right) \]

   3. exp-lft-sqr 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\color{blue}{\left(e^{y \cdot 2}\right)}}^{\left(\frac{y \cdot 0.3333333333333333}{2}\right)}\right) \]

   4. sqr-pow 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \color{blue}{\left({\left(e^{y \cdot 2}\right)}^{\left(\frac{\frac{y \cdot 0.3333333333333333}{2}}{2}\right)} \cdot {\left(e^{y \cdot 2}\right)}^{\left(\frac{\frac{y \cdot 0.3333333333333333}{2}}{2}\right)}\right)}\right) \]

   5. exp-prod 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \left({\color{blue}{\left({\left(e^{y}\right)}^{2}\right)}}^{\left(\frac{\frac{y \cdot 0.3333333333333333}{2}}{2}\right)} \cdot {\left(e^{y \cdot 2}\right)}^{\left(\frac{\frac{y \cdot 0.3333333333333333}{2}}{2}\right)}\right)\right) \]

   6. associate-/l* 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\color{blue}{\frac{y}{\frac{2}{0.3333333333333333}}}}{2}\right)} \cdot {\left(e^{y \cdot 2}\right)}^{\left(\frac{\frac{y \cdot 0.3333333333333333}{2}}{2}\right)}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{\color{blue}{6}}}{2}\right)} \cdot {\left(e^{y \cdot 2}\right)}^{\left(\frac{\frac{y \cdot 0.3333333333333333}{2}}{2}\right)}\right)\right) \]

   8. exp-prod 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{6}}{2}\right)} \cdot {\color{blue}{\left({\left(e^{y}\right)}^{2}\right)}}^{\left(\frac{\frac{y \cdot 0.3333333333333333}{2}}{2}\right)}\right)\right) \]

   9. associate-/l* 100.0%

      \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{6}}{2}\right)} \cdot {\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\color{blue}{\frac{y}{\frac{2}{0.3333333333333333}}}}{2}\right)}\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{6}}{2}\right)} \cdot {\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{\color{blue}{6}}}{2}\right)}\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \color{blue}{\left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{6}}{2}\right)} \cdot {\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{6}}{2}\right)}\right)}\right) \]

10. Final simplification 100.0%

    \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot \left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{6}}{2}\right)} \cdot {\left({\left(e^{y}\right)}^{2}\right)}^{\left(\frac{\frac{y}{6}}{2}\right)}\right)\right) \]

Alternative 2: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(e^{y}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (*
   (pow (exp (* y 2.0)) (* y 0.3333333333333333))
   (pow (exp y) (* y 0.3333333333333333)))))

C

double code(double x, double y) {
	return x * (pow(exp((y * 2.0)), (y * 0.3333333333333333)) * pow(exp(y), (y * 0.3333333333333333)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((exp((y * 2.0d0)) ** (y * 0.3333333333333333d0)) * (exp(y) ** (y * 0.3333333333333333d0)))
end function

Java

public static double code(double x, double y) {
	return x * (Math.pow(Math.exp((y * 2.0)), (y * 0.3333333333333333)) * Math.pow(Math.exp(y), (y * 0.3333333333333333)));
}

Python

def code(x, y):
	return x * (math.pow(math.exp((y * 2.0)), (y * 0.3333333333333333)) * math.pow(math.exp(y), (y * 0.3333333333333333)))

Julia

function code(x, y)
	return Float64(x * Float64((exp(Float64(y * 2.0)) ^ Float64(y * 0.3333333333333333)) * (exp(y) ^ Float64(y * 0.3333333333333333))))
end

MATLAB

function tmp = code(x, y)
	tmp = x * ((exp((y * 2.0)) ^ (y * 0.3333333333333333)) * (exp(y) ^ (y * 0.3333333333333333)));
end

Wolfram

code[x_, y_] := N[(x * N[(N[Power[N[Exp[N[(y * 2.0), $MachinePrecision]], $MachinePrecision], N[(y * 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[Power[N[Exp[y], $MachinePrecision], N[(y * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot e^{y \cdot y} \]
2. Step-by-step derivation

   1. *-un-lft-identity 99.9%

      \[\leadsto x \cdot e^{\color{blue}{1 \cdot \left(y \cdot y\right)}} \]

   2. exp-prod 100.0%

      \[\leadsto x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(y \cdot y\right)}} \]

   3. exp-1-e 100.0%

      \[\leadsto x \cdot {\color{blue}{e}}^{\left(y \cdot y\right)} \]

   4. pow2 100.0%

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

3. Applied egg-rr 100.0%

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

   1. unpow2 100.0%

      \[\leadsto x \cdot {e}^{\color{blue}{\left(y \cdot y\right)}} \]

   2. pow-unpow 99.9%

      \[\leadsto x \cdot \color{blue}{{\left({e}^{y}\right)}^{y}} \]

   3. e-exp-1 99.9%

      \[\leadsto x \cdot {\left({\color{blue}{\left(e^{1}\right)}}^{y}\right)}^{y} \]

   4. exp-prod 100.0%

      \[\leadsto x \cdot {\color{blue}{\left(e^{1 \cdot y}\right)}}^{y} \]

   5. *-un-lft-identity 100.0%

      \[\leadsto x \cdot {\left(e^{\color{blue}{y}}\right)}^{y} \]

   6. add-cbrt-cube 99.9%

      \[\leadsto x \cdot {\color{blue}{\left(\sqrt[3]{\left(e^{y} \cdot e^{y}\right) \cdot e^{y}}\right)}}^{y} \]

   7. unpow3 99.9%

      \[\leadsto x \cdot {\left(\sqrt[3]{\color{blue}{{\left(e^{y}\right)}^{3}}}\right)}^{y} \]

   8. exp-prod 99.9%

      \[\leadsto x \cdot {\left(\sqrt[3]{\color{blue}{e^{y \cdot 3}}}\right)}^{y} \]

   9. pow1/3 99.9%

      \[\leadsto x \cdot {\color{blue}{\left({\left(e^{y \cdot 3}\right)}^{0.3333333333333333}\right)}}^{y} \]

   10. pow-unpow 99.9%

       \[\leadsto x \cdot \color{blue}{{\left(e^{y \cdot 3}\right)}^{\left(0.3333333333333333 \cdot y\right)}} \]

   11. *-commutative 99.9%

       \[\leadsto x \cdot {\left(e^{y \cdot 3}\right)}^{\color{blue}{\left(y \cdot 0.3333333333333333\right)}} \]

   12. add-cube-cbrt 99.9%

       \[\leadsto x \cdot {\color{blue}{\left(\left(\sqrt[3]{e^{y \cdot 3}} \cdot \sqrt[3]{e^{y \cdot 3}}\right) \cdot \sqrt[3]{e^{y \cdot 3}}\right)}}^{\left(y \cdot 0.3333333333333333\right)} \]

   13. unpow-prod-down 99.9%

       \[\leadsto x \cdot \color{blue}{\left({\left(\sqrt[3]{e^{y \cdot 3}} \cdot \sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right)} \]

   14. pow2 99.9%

       \[\leadsto x \cdot \left({\color{blue}{\left({\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{2}\right)}}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

   15. exp-prod 99.9%

       \[\leadsto x \cdot \left({\left({\left(\sqrt[3]{\color{blue}{{\left(e^{y}\right)}^{3}}}\right)}^{2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

   16. unpow3 99.9%

       \[\leadsto x \cdot \left({\left({\left(\sqrt[3]{\color{blue}{\left(e^{y} \cdot e^{y}\right) \cdot e^{y}}}\right)}^{2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

   17. add-cbrt-cube 99.9%

       \[\leadsto x \cdot \left({\left({\color{blue}{\left(e^{y}\right)}}^{2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(\sqrt[3]{e^{y \cdot 3}}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto x \cdot \color{blue}{\left({\left({\left(e^{y}\right)}^{2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(e^{y}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right)} \]
6. Step-by-step derivation

   1. pow-exp 100.0%

      \[\leadsto x \cdot \left({\color{blue}{\left(e^{y \cdot 2}\right)}}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(e^{y}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left({\color{blue}{\left(e^{y \cdot 2}\right)}}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(e^{y}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

8. Final simplification 100.0%

   \[\leadsto x \cdot \left({\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.3333333333333333\right)} \cdot {\left(e^{y}\right)}^{\left(y \cdot 0.3333333333333333\right)}\right) \]

Alternative 3: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x \cdot {\left(e^{y}\right)}^{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (pow (exp y) y)))

C

double code(double x, double y) {
	return x * pow(exp(y), y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (exp(y) ** y)
end function

Java

public static double code(double x, double y) {
	return x * Math.pow(Math.exp(y), y);
}

Python

def code(x, y):
	return x * math.pow(math.exp(y), y)

Julia

function code(x, y)
	return Float64(x * (exp(y) ^ y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (exp(y) ^ y);
end

Wolfram

code[x_, y_] := N[(x * N[Power[N[Exp[y], $MachinePrecision], y], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot e^{y \cdot y} \]
2. Step-by-step derivation

   1. exp-prod 100.0%

      \[\leadsto x \cdot \color{blue}{{\left(e^{y}\right)}^{y}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot {\left(e^{y}\right)}^{y}} \]

4. Final simplification 100.0%

   \[\leadsto x \cdot {\left(e^{y}\right)}^{y} \]

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (exp (* y y))))

C

double code(double x, double y) {
	return x * exp((y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * exp((y * y))
end function

Java

public static double code(double x, double y) {
	return x * Math.exp((y * y));
}

Python

def code(x, y):
	return x * math.exp((y * y))

Julia

function code(x, y)
	return Float64(x * exp(Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * exp((y * y));
end

Wolfram

code[x_, y_] := N[(x * N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot e^{y \cdot y} \]

2. Final simplification 99.9%

   \[\leadsto x \cdot e^{y \cdot y} \]

Alternative 5: 50.8% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x \cdot e^{y \cdot y} \]

2. Taylor expanded in y around 0 47.2%

   \[\leadsto \color{blue}{x} \]

3. Final simplification 47.2%

   \[\leadsto x \]

Developer target: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x \cdot {\left(e^{y}\right)}^{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (pow (exp y) y)))

C

double code(double x, double y) {
	return x * pow(exp(y), y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (exp(y) ** y)
end function

Java

public static double code(double x, double y) {
	return x * Math.pow(Math.exp(y), y);
}

Python

def code(x, y):
	return x * math.pow(math.exp(y), y)

Julia

function code(x, y)
	return Float64(x * (exp(y) ^ y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (exp(y) ^ y);
end

Wolfram

code[x_, y_] := N[(x * N[Power[N[Exp[y], $MachinePrecision], y], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :herbie-target
  (* x (pow (exp y) y))

  (* x (exp (* y y))))