Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 + \left(\left(y - 1\right) \cdot x - 0.5 \cdot y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 0.918938533204673 (- (* (- y 1.0) x) (* 0.5 y))))

C

double code(double x, double y) {
	return 0.918938533204673 + (((y - 1.0) * x) - (0.5 * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0 + (((y - 1.0d0) * x) - (0.5d0 * y))
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673 + (((y - 1.0) * x) - (0.5 * y));
}

Python

def code(x, y):
	return 0.918938533204673 + (((y - 1.0) * x) - (0.5 * y))

Julia

function code(x, y)
	return Float64(0.918938533204673 + Float64(Float64(Float64(y - 1.0) * x) - Float64(0.5 * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673 + (((y - 1.0) * x) - (0.5 * y));
end

Wolfram

code[x_, y_] := N[(0.918938533204673 + N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] - N[(0.5 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto 0.918938533204673 + \left(\left(y - 1\right) \cdot x - 0.5 \cdot y\right) \]
4. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - 1\right) \cdot x\\ \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y 1.0) x)))
   (if (<= x -0.68) t_0 (if (<= x 0.55) (fma y -0.5 0.918938533204673) t_0))))

C

double code(double x, double y) {
	double t_0 = (y - 1.0) * x;
	double tmp;
	if (x <= -0.68) {
		tmp = t_0;
	} else if (x <= 0.55) {
		tmp = fma(y, -0.5, 0.918938533204673);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y - 1.0) * x)
	tmp = 0.0
	if (x <= -0.68)
		tmp = t_0;
	elseif (x <= 0.55)
		tmp = fma(y, -0.5, 0.918938533204673);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.68], t$95$0, If[LessEqual[x, 0.55], N[(y * -0.5 + 0.918938533204673), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049 or 0.55000000000000004 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -0.680000000000000049 < x < 0.55000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) + \frac{918938533204673}{1000000000000000}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} + \frac{918938533204673}{1000000000000000} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y + \frac{918938533204673}{1000000000000000} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{-1}{2}} + \frac{918938533204673}{1000000000000000} \]

      6. lower-fma.f64 98.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 0.918938533204673\right)} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 0.918938533204673\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))