Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 5

Speedup: 1.0×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 2 - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* (* x x) 2.0) 1.0))

C

double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) * 2.0d0) - 1.0d0
end function

Java

public static double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

Python

def code(x):
	return ((x * x) * 2.0) - 1.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * 2.0) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) * 2.0) - 1.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 2 - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* (* x x) 2.0) 1.0))

C

double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) * 2.0d0) - 1.0d0
end function

Java

public static double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

Python

def code(x):
	return ((x * x) * 2.0) - 1.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * 2.0) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) * 2.0) - 1.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 2 + -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* (* x x) 2.0) -1.0))

C

double code(double x) {
	return ((x * x) * 2.0) + -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) * 2.0d0) + (-1.0d0)
end function

Java

public static double code(double x) {
	return ((x * x) * 2.0) + -1.0;
}

Python

def code(x):
	return ((x * x) * 2.0) + -1.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * 2.0) + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) * 2.0) + -1.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot x\right) \cdot 2 - 1 \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(x \cdot x\right) \cdot 2 + -1 \]
4. Add Preprocessing

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 2e-13) -1.0 (* (* x x) 2.0)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = -1.0;
	} else {
		tmp = (x * x) * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = -1.0d0
    else
        tmp = (x * x) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = -1.0;
	} else {
		tmp = (x * x) * 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = -1.0
	else:
		tmp = (x * x) * 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = -1.0;
	else
		tmp = Float64(Float64(x * x) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = -1.0;
	else
		tmp = (x * x) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], -1.0, N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot 2 - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 99.8%

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

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot 2 - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 99.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.72) -1.0 (* x 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = -1.0;
	} else {
		tmp = x * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = -1.0d0
    else
        tmp = x * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = -1.0;
	} else {
		tmp = x * 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.72:
		tmp = -1.0
	else:
		tmp = x * 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.72)
		tmp = -1.0;
	else
		tmp = Float64(x * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = -1.0;
	else
		tmp = x * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.72], -1.0, N[(x * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot 2 - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 66.0%

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

   if 0.71999999999999997 < x

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot 2 - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot x\right)} \]
   6. Step-by-step derivation

      1. rem-exp-log N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\log \left(x \cdot x\right)}\right)\right) \]

      2. sum-log N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\log x + \log x}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\frac{\log x \cdot \log x - \log x \cdot \log x}{\log x - \log x}}\right)\right) \]

      4. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\frac{0}{\log x - \log x}}\right)\right) \]

      5. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\frac{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{\log x - \log x}}\right)\right) \]

      6. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\frac{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{0}}\right)\right) \]

      7. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\frac{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}}\right)\right) \]

      8. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) + \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}\right)\right) \]

      9. log-prod N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}\right)\right) \]

      10. sqr-pow N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\log \left({x}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\log \left({x}^{1}\right)}\right)\right) \]

      12. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(e^{\log x}\right)\right) \]

      13. rem-exp-log 6.9%

          \[\leadsto \mathsf{*.f64}\left(2, x\right) \]

   7. Applied egg-rr 6.9%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.72) -1.0 2.0))

C

double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = -1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.72:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.72)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.72], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot 2 - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 66.0%

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

   if 0.71999999999999997 < x

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot 2 - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 99.2%

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

   7. Applied egg-rr 4.3%

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

Alternative 5: 49.7% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

   \[\left(x \cdot x\right) \cdot 2 - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 49.9%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2"
  :precision binary64
  (- (* (* x x) 2.0) 1.0))