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

Percentage Accurate: 99.3% → 100.0%

Time: 1.7s

Alternatives: 5

Speedup: 1.0×

• 25.0% 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 99.1%

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

2. Taylor expanded in x around 0 99.1%

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

   1. unpow2 99.1%

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

   2. rem-exp-log 95.2%

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

   3. log-rec 95.2%

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

   4. unpow2 95.2%

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

   5. log-pow 49.5%

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

   6. *-commutative 49.5%

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

   7. distribute-rgt-neg-in 49.5%

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

   8. exp-to-pow 100.0%

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

   9. 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 99.1%

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

2. Final simplification 99.1%

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

Alternative 3: 16.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;0.125\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.6e+77) 0.125 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.6e+77) {
		tmp = 0.125;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.6d+77) then
        tmp = 0.125d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6e+77) {
		tmp = 0.125;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6e+77:
		tmp = 0.125
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6e+77)
		tmp = 0.125;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6e+77)
		tmp = 0.125;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6e+77], 0.125, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001e77

   1. Initial program 99.3%

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

   3. Applied egg-rr 4.6%

      \[\leadsto \color{blue}{0.125} \]

   if 1.6000000000000001e77 < x

   1. Initial program 98.6%

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

   3. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;0.125\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 16.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.2e+77) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.2e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1999999999999999e77

   1. Initial program 99.3%

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

   3. Applied egg-rr 4.7%

      \[\leadsto \color{blue}{0.5} \]

   if 1.1999999999999999e77 < x

   1. Initial program 98.6%

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

   3. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 26.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 99.1%

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

3. Applied egg-rr 28.9%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 28.9%

   \[\leadsto 0 \]

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 2023297 
(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)))