Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2

Percentage Accurate: 100.0% → 100.0%

Time: 12.1s

Alternatives: 3

Speedup: 1.0×

• 25.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot {y}^{2}\\ \sqrt[3]{{\left({e}^{t_0} \cdot \sqrt[3]{e^{t_0}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\left({y}^{2} \cdot \left(x \cdot 0.05555555555555555\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (pow y 2.0))))
   (*
    (cbrt (pow (* (pow E t_0) (cbrt (exp t_0))) 2.0))
    (pow (exp 2.0) (* (pow y 2.0) (* x 0.05555555555555555))))))

C

double code(double x, double y) {
	double t_0 = x * pow(y, 2.0);
	return cbrt(pow((pow(((double) M_E), t_0) * cbrt(exp(t_0))), 2.0)) * pow(exp(2.0), (pow(y, 2.0) * (x * 0.05555555555555555)));
}

Java

public static double code(double x, double y) {
	double t_0 = x * Math.pow(y, 2.0);
	return Math.cbrt(Math.pow((Math.pow(Math.E, t_0) * Math.cbrt(Math.exp(t_0))), 2.0)) * Math.pow(Math.exp(2.0), (Math.pow(y, 2.0) * (x * 0.05555555555555555)));
}

Julia

function code(x, y)
	t_0 = Float64(x * (y ^ 2.0))
	return Float64(cbrt((Float64((exp(1) ^ t_0) * cbrt(exp(t_0))) ^ 2.0)) * (exp(2.0) ^ Float64((y ^ 2.0) * Float64(x * 0.05555555555555555))))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[Power[N[(N[Power[E, t$95$0], $MachinePrecision] * N[Power[N[Exp[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[Power[y, 2.0], $MachinePrecision] * N[(x * 0.05555555555555555), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

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

   3. add-cube-cbrt 99.9%

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

   4. associate-*r* 99.9%

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

   5. unpow-prod-down 100.0%

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

5. Applied egg-rr 77.0%

   \[\leadsto \color{blue}{{\left({\left({\left(e^{x}\right)}^{\left({y}^{2}\right)} \cdot \sqrt[3]{{\left(e^{x}\right)}^{\left({y}^{2}\right)}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{{\left(e^{x}\right)}^{\left({y}^{2}\right)}}}} \]
6. Step-by-step derivation

   1. unpow1/3 77.0%

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

   2. exp-prod 92.5%

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

   3. exp-prod 92.5%

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

   4. exp-prod 99.9%

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

7. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. pow-exp 100.0%

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

   3. e-exp-1 100.0%

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

9. Applied egg-rr 100.0%

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

    1. pow1/3 100.0%

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

    2. pow1/3 100.0%

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

    3. pow-pow 100.0%

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

    4. *-un-lft-identity 100.0%

       \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{\color{blue}{1 \cdot \left(x \cdot {y}^{2}\right)}}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \]

    5. pow-exp 100.0%

       \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\color{blue}{\left({\left(e^{1}\right)}^{\left(x \cdot {y}^{2}\right)}\right)}}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \]

    6. e-exp-1 100.0%

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

    7. sqr-pow 100.0%

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

    8. pow-prod-down 100.0%

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

    9. pow-pow 100.0%

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

    10. e-exp-1 100.0%

        \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(\color{blue}{e^{1}} \cdot e\right)}^{\left(\frac{x \cdot {y}^{2}}{2} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right)} \]

    11. e-exp-1 100.0%

        \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{1} \cdot \color{blue}{e^{1}}\right)}^{\left(\frac{x \cdot {y}^{2}}{2} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right)} \]

    12. prod-exp 100.0%

        \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\color{blue}{\left(e^{1 + 1}\right)}}^{\left(\frac{x \cdot {y}^{2}}{2} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right)} \]

    13. metadata-eval 100.0%

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

    14. div-inv 100.0%

        \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\left(\color{blue}{\left(\left(x \cdot {y}^{2}\right) \cdot \frac{1}{2}\right)} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right)} \]

    15. metadata-eval 100.0%

        \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\left(\left(\left(x \cdot {y}^{2}\right) \cdot \color{blue}{0.5}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right)} \]

    16. metadata-eval 100.0%

        \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\left(\left(\left(x \cdot {y}^{2}\right) \cdot 0.5\right) \cdot \color{blue}{0.1111111111111111}\right)} \]

11. Applied egg-rr 100.0%

    \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot \color{blue}{{\left(e^{2}\right)}^{\left(\left(\left(x \cdot {y}^{2}\right) \cdot 0.5\right) \cdot 0.1111111111111111\right)}} \]

12. Taylor expanded in x around 0 100.0%

    \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\color{blue}{\left(0.05555555555555555 \cdot \left(x \cdot {y}^{2}\right)\right)}} \]
13. Step-by-step derivation

    1. *-commutative 100.0%

       \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\color{blue}{\left(\left(x \cdot {y}^{2}\right) \cdot 0.05555555555555555\right)}} \]

    2. *-commutative 100.0%

       \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\left(\color{blue}{\left({y}^{2} \cdot x\right)} \cdot 0.05555555555555555\right)} \]

    3. associate-*r* 100.0%

       \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\color{blue}{\left({y}^{2} \cdot \left(x \cdot 0.05555555555555555\right)\right)}} \]

14. Simplified 100.0%

    \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\color{blue}{\left({y}^{2} \cdot \left(x \cdot 0.05555555555555555\right)\right)}} \]

15. Final simplification 100.0%

    \[\leadsto \sqrt[3]{{\left({e}^{\left(x \cdot {y}^{2}\right)} \cdot \sqrt[3]{e^{x \cdot {y}^{2}}}\right)}^{2}} \cdot {\left(e^{2}\right)}^{\left({y}^{2} \cdot \left(x \cdot 0.05555555555555555\right)\right)} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 51.0% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 54.3%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 54.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023321 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))