Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, A

Percentage Accurate: 99.4% → 100.0%

Time: 2.7s

Alternatives: 3

Speedup: 0.2×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* x x)))

C

double code(double x) {
	return 1.0 / (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (x * x);
}

Python

def code(x):
	return 1.0 / (x * x)

Julia

function code(x)
	return Float64(1.0 / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x * x);
end

Wolfram

code[x_] := N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* x x)))

C

double code(double x) {
	return 1.0 / (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (x * x);
}

Python

def code(x):
	return 1.0 / (x * x)

Julia

function code(x)
	return Float64(1.0 / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x * x);
end

Wolfram

code[x_] := N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {x}^{-2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x -2.0))

C

double code(double x) {
	return pow(x, -2.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, -2.0);
}

Python

def code(x):
	return math.pow(x, -2.0)

Julia

function code(x)
	return x ^ -2.0
end

MATLAB

function tmp = code(x)
	tmp = x ^ -2.0;
end

Wolfram

code[x_] := N[Power[x, -2.0], $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\frac{1}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot x}} \]

   2. inv-pow N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. pow-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval 100.0

      \[\leadsto {x}^{\color{blue}{-2}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{{x}^{-2}} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-1}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow x -1.0) x))

C

double code(double x) {
	return pow(x, -1.0) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-1.0d0)) / x
end function

Java

public static double code(double x) {
	return Math.pow(x, -1.0) / x;
}

Python

def code(x):
	return math.pow(x, -1.0) / x

Julia

function code(x)
	return Float64((x ^ -1.0) / x)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -1.0) / x;
end

Wolfram

code[x_] := N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\frac{1}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot x}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot -1}}{x \cdot x} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{-1 \cdot -1}{\color{blue}{x \cdot x}} \]

   4. frac-times N/A

      \[\leadsto \color{blue}{\frac{-1}{x} \cdot \frac{-1}{x}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{-1}{x} \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{x} \]

   6. distribute-neg-frac N/A

      \[\leadsto \frac{-1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

   7. inv-pow N/A

      \[\leadsto \frac{-1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{{x}^{-1}}\right)\right) \]

   8. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(\mathsf{neg}\left({x}^{-1}\right)\right)}{x}} \]

   9. unpow1 N/A

      \[\leadsto \frac{-1 \cdot \left(\mathsf{neg}\left({x}^{-1}\right)\right)}{\color{blue}{{x}^{1}}} \]

   10. sqr-pow N/A

       \[\leadsto \frac{-1 \cdot \left(\mathsf{neg}\left({x}^{-1}\right)\right)}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} \]

   11. times-frac N/A

       \[\leadsto \color{blue}{\frac{-1}{{x}^{\left(\frac{1}{2}\right)}} \cdot \frac{\mathsf{neg}\left({x}^{-1}\right)}{{x}^{\left(\frac{1}{2}\right)}}} \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{{x}^{\left(\frac{1}{2}\right)}} \cdot \frac{\mathsf{neg}\left({x}^{-1}\right)}{{x}^{\left(\frac{1}{2}\right)}}} \]

   13. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{{x}^{\left(\frac{1}{2}\right)}}} \cdot \frac{\mathsf{neg}\left({x}^{-1}\right)}{{x}^{\left(\frac{1}{2}\right)}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{-1}{{x}^{\color{blue}{\frac{1}{2}}}} \cdot \frac{\mathsf{neg}\left({x}^{-1}\right)}{{x}^{\left(\frac{1}{2}\right)}} \]

   15. unpow1/2 N/A

       \[\leadsto \frac{-1}{\color{blue}{\sqrt{x}}} \cdot \frac{\mathsf{neg}\left({x}^{-1}\right)}{{x}^{\left(\frac{1}{2}\right)}} \]

   16. lower-sqrt.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\sqrt{x}}} \cdot \frac{\mathsf{neg}\left({x}^{-1}\right)}{{x}^{\left(\frac{1}{2}\right)}} \]

   17. lower-/.f64 N/A

       \[\leadsto \frac{-1}{\sqrt{x}} \cdot \color{blue}{\frac{\mathsf{neg}\left({x}^{-1}\right)}{{x}^{\left(\frac{1}{2}\right)}}} \]

   18. inv-pow N/A

       \[\leadsto \frac{-1}{\sqrt{x}} \cdot \frac{\mathsf{neg}\left(\color{blue}{\frac{1}{x}}\right)}{{x}^{\left(\frac{1}{2}\right)}} \]

   19. distribute-neg-frac N/A

       \[\leadsto \frac{-1}{\sqrt{x}} \cdot \frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{x}}}{{x}^{\left(\frac{1}{2}\right)}} \]

   20. metadata-eval N/A

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

   21. lower-/.f64 N/A

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

   22. metadata-eval N/A

       \[\leadsto \frac{-1}{\sqrt{x}} \cdot \frac{\frac{-1}{x}}{{x}^{\color{blue}{\frac{1}{2}}}} \]

   23. unpow1/2 N/A

       \[\leadsto \frac{-1}{\sqrt{x}} \cdot \frac{\frac{-1}{x}}{\color{blue}{\sqrt{x}}} \]

   24. lower-sqrt.f64 46.6

       \[\leadsto \frac{-1}{\sqrt{x}} \cdot \frac{\frac{-1}{x}}{\color{blue}{\sqrt{x}}} \]

4. Applied rewrites 46.6%

   \[\leadsto \color{blue}{\frac{-1}{\sqrt{x}} \cdot \frac{\frac{-1}{x}}{\sqrt{x}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{x}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot x}} \]

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

   \[\leadsto \frac{{x}^{-1}}{x} \]
9. Add Preprocessing

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(x \cdot x\right)}^{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (* x x) -1.0))

C

double code(double x) {
	return pow((x * x), -1.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow((x * x), -1.0);
}

Python

def code(x):
	return math.pow((x * x), -1.0)

Julia

function code(x)
	return Float64(x * x) ^ -1.0
end

MATLAB

function tmp = code(x)
	tmp = (x * x) ^ -1.0;
end

Wolfram

code[x_] := N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\frac{1}{x \cdot x} \]
2. Add Preprocessing

3. Final simplification 99.0%

   \[\leadsto {\left(x \cdot x\right)}^{-1} \]
4. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 x) x))

C

double code(double x) {
	return (1.0 / x) / x;
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 / x) / x

Julia

function code(x)
	return Float64(Float64(1.0 / x) / x)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) / x;
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]

Reproduce

herbie shell --seed 2024313 
(FPCore (x)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ 1 x) x))

  (/ 1.0 (* x x)))