Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 4

Speedup: 1.0×

• 26.1% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot x\right) \cdot 2 - 1 \]
2. Step-by-step derivation

   1. associate-*l* 100.0%

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

   2. fma-neg 100.0%

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

   3. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 2, -1\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.5) -1.0 (* 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = 2.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.5d0) then
        tmp = -1.0d0
    else
        tmp = 2.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = 2.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.5:
		tmp = -1.0
	else:
		tmp = 2.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.5)
		tmp = -1.0;
	else
		tmp = Float64(2.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.5)
		tmp = -1.0;
	else
		tmp = 2.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.5], -1.0, N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.6%

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

   if 0.5 < (*.f64 x x)

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot 2 - 1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 29.1%

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

      2. metadata-eval 29.1%

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

      3. sub-neg 29.1%

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

      4. swap-sqr 29.1%

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

      5. associate-*l* 29.0%

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

      6. metadata-eval 29.0%

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{4} + \left(-1\right)}{\left(x \cdot x\right) \cdot 2 + 1} \]

      7. metadata-eval 29.0%

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 4 + \color{blue}{-1}}{\left(x \cdot x\right) \cdot 2 + 1} \]

      8. associate-*l* 29.0%

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 4 + -1}{\color{blue}{x \cdot \left(x \cdot 2\right)} + 1} \]

   4. Applied egg-rr 29.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 4 + -1}{x \cdot \left(x \cdot 2\right) + 1}} \]

   5. Taylor expanded in x around inf 98.1%

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

      1. unpow2 98.1%

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

   7. Simplified 98.1%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 4: 48.4% 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 52.1%

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

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2"
  :precision binary64
  (- (* (* x x) 2.0) 1.0))