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

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 3

Speedup: 1.0×

• 24.9% 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

\[\begin{array}{l} \\ \frac{{\left({\left(e^{0.1111111111111111}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)} \cdot \sqrt[3]{{\left(e^{0.6666666666666666}\right)}^{\left(1 + x \cdot {y}^{2}\right)}}\right)}^{3}}{e} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (pow
   (*
    (pow (exp 0.1111111111111111) (fma x (pow y 2.0) 1.0))
    (cbrt (pow (exp 0.6666666666666666) (+ 1.0 (* x (pow y 2.0))))))
   3.0)
  E))

C

double code(double x, double y) {
	return pow((pow(exp(0.1111111111111111), fma(x, pow(y, 2.0), 1.0)) * cbrt(pow(exp(0.6666666666666666), (1.0 + (x * pow(y, 2.0)))))), 3.0) / ((double) M_E);
}

Julia

function code(x, y)
	return Float64((Float64((exp(0.1111111111111111) ^ fma(x, (y ^ 2.0), 1.0)) * cbrt((exp(0.6666666666666666) ^ Float64(1.0 + Float64(x * (y ^ 2.0)))))) ^ 3.0) / exp(1))
end

Wolfram

code[x_, y_] := N[(N[Power[N[(N[Power[N[Exp[0.1111111111111111], $MachinePrecision], N[(x * N[Power[y, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[Exp[0.6666666666666666], $MachinePrecision], N[(1.0 + N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 70.3%

      \[\leadsto e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot y\right) \cdot y\right)\right)}} \]

   2. expm1-udef 70.3%

      \[\leadsto e^{\color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot y\right) \cdot y\right)} - 1}} \]

   3. exp-diff 70.3%

      \[\leadsto \color{blue}{\frac{e^{e^{\mathsf{log1p}\left(\left(x \cdot y\right) \cdot y\right)}}}{e^{1}}} \]

   4. log1p-udef 70.3%

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

   5. rem-exp-log 100.0%

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

   6. associate-*l* 100.0%

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

   7. pow2 100.0%

      \[\leadsto \frac{e^{1 + x \cdot \color{blue}{{y}^{2}}}}{e^{1}} \]

   8. exp-1-e 100.0%

      \[\leadsto \frac{e^{1 + x \cdot {y}^{2}}}{\color{blue}{e}} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{e^{1 + x \cdot {y}^{2}}}{e}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 98.5%

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

   2. pow3 98.5%

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

   3. +-commutative 98.5%

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

   4. fma-def 98.5%

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

5. Applied egg-rr 98.5%

   \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{e^{\mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}}{e} \]
6. Step-by-step derivation

   1. add-cube-cbrt 98.9%

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

   2. associate-*l* 98.9%

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

   3. cbrt-unprod 98.9%

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

   4. add-exp-log 98.9%

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

   5. pow1/3 98.9%

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

   6. pow1/3 99.9%

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

   7. pow-prod-up 99.9%

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

   8. log-pow 99.9%

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

   9. metadata-eval 99.9%

      \[\leadsto \frac{{\left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(x, {y}^{2}, 1\right)}}} \cdot \sqrt[3]{e^{\color{blue}{0.6666666666666666} \cdot \log \left(e^{\mathsf{fma}\left(x, {y}^{2}, 1\right)}\right)}}\right)}^{3}}{e} \]

   10. add-log-exp 99.9%

       \[\leadsto \frac{{\left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(x, {y}^{2}, 1\right)}}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \color{blue}{\mathsf{fma}\left(x, {y}^{2}, 1\right)}}}\right)}^{3}}{e} \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{{\color{blue}{\left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(x, {y}^{2}, 1\right)}}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}}^{3}}{e} \]
8. Step-by-step derivation

   1. unpow1/3 99.9%

      \[\leadsto \frac{{\left(\color{blue}{{\left(\sqrt[3]{e^{\mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   2. unpow1/3 99.9%

      \[\leadsto \frac{{\left({\color{blue}{\left({\left(e^{\mathsf{fma}\left(x, {y}^{2}, 1\right)}\right)}^{0.3333333333333333}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   3. exp-prod 99.9%

      \[\leadsto \frac{{\left({\color{blue}{\left(e^{\mathsf{fma}\left(x, {y}^{2}, 1\right) \cdot 0.3333333333333333}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   4. exp-prod 99.9%

      \[\leadsto \frac{{\left(\color{blue}{e^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right) \cdot 0.3333333333333333\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   5. *-commutative 99.9%

      \[\leadsto \frac{{\left(e^{\color{blue}{0.3333333333333333 \cdot \left(\mathsf{fma}\left(x, {y}^{2}, 1\right) \cdot 0.3333333333333333\right)}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   6. *-commutative 99.9%

      \[\leadsto \frac{{\left(e^{0.3333333333333333 \cdot \color{blue}{\left(0.3333333333333333 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)\right)}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   7. associate-*r* 99.9%

      \[\leadsto \frac{{\left(e^{\color{blue}{\left(0.3333333333333333 \cdot 0.3333333333333333\right) \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   8. metadata-eval 99.9%

      \[\leadsto \frac{{\left(e^{\color{blue}{0.1111111111111111} \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   9. exp-prod 99.9%

      \[\leadsto \frac{{\left(\color{blue}{{\left(e^{0.1111111111111111}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \mathsf{fma}\left(x, {y}^{2}, 1\right)}}\right)}^{3}}{e} \]

   10. exp-prod 100.0%

       \[\leadsto \frac{{\left({\left(e^{0.1111111111111111}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)} \cdot \sqrt[3]{\color{blue}{{\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)}}}\right)}^{3}}{e} \]

9. Simplified 100.0%

   \[\leadsto \frac{{\color{blue}{\left({\left(e^{0.1111111111111111}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)} \cdot \sqrt[3]{{\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)}}\right)}}^{3}}{e} \]
10. Step-by-step derivation

    1. fma-udef 100.0%

       \[\leadsto \frac{{\left({\left(e^{0.1111111111111111}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)} \cdot \sqrt[3]{{\left(e^{0.6666666666666666}\right)}^{\color{blue}{\left(x \cdot {y}^{2} + 1\right)}}}\right)}^{3}}{e} \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left({\left(e^{0.1111111111111111}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)} \cdot \sqrt[3]{{\left(e^{0.6666666666666666}\right)}^{\color{blue}{\left(x \cdot {y}^{2} + 1\right)}}}\right)}^{3}}{e} \]

12. Final simplification 100.0%

    \[\leadsto \frac{{\left({\left(e^{0.1111111111111111}\right)}^{\left(\mathsf{fma}\left(x, {y}^{2}, 1\right)\right)} \cdot \sqrt[3]{{\left(e^{0.6666666666666666}\right)}^{\left(1 + x \cdot {y}^{2}\right)}}\right)}^{3}}{e} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 50.8% 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. Taylor expanded in x around 0 43.6%

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

3. Final simplification 43.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))