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

Percentage Accurate: 99.3% → 100.0%

Time: 1.5s

Alternatives: 3

Speedup: 1.0×

• 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: 99.3% 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, 0.0× 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 99.6%

   \[\frac{1}{x \cdot x} \]

2. Taylor expanded in x around 0 99.6%

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

   1. unpow2 99.6%

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

   2. associate-/r* 99.7%

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

   3. *-rgt-identity 99.7%

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

   4. associate-*r/ 99.5%

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

   5. unpow-1 99.5%

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

   6. unpow-1 99.5%

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

   7. pow-sqr 100.0%

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

   8. metadata-eval 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.3% 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]

Derivation

1. Initial program 99.6%

   \[\frac{1}{x \cdot x} \]

2. Final simplification 99.6%

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

Alternative 3: 99.8% accurate, 1.0× 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]

Derivation

1. Initial program 99.6%

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

   1. inv-pow 99.6%

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

   2. unpow-prod-down 99.5%

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

   3. inv-pow 99.5%

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

   4. inv-pow 99.5%

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

3. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{x}} \]
4. Step-by-step derivation

   1. un-div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Developer target: 99.8% accurate, 1.0× 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 2023196 
(FPCore (x)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ 1.0 x) x)

  (/ 1.0 (* x x)))