Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 2.8s

Alternatives: 3

Speedup: 1.2×

• 24.5% 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, 1.2× speedup

Math

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

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(Float64(x * x), 2.0, -1.0)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 4e-7) -1.0 (* (* 2.0 x) x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 4e-7) {
		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) <= 4d-7) 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) <= 4e-7) {
		tmp = -1.0;
	} else {
		tmp = (2.0 * x) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 4e-7:
		tmp = -1.0
	else:
		tmp = (2.0 * x) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 4e-7)
		tmp = -1.0;
	else
		tmp = Float64(Float64(2.0 * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 4e-7)
		tmp = -1.0;
	else
		tmp = (2.0 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e-7], -1.0, N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.9999999999999998e-7

   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. Applied rewrites 99.6%

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

   if 3.9999999999999998e-7 < (*.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. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x)
  :name "Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2"
  :precision binary64
  (- (* (* x x) 2.0) 1.0))