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

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 3

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

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

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]

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%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.918938533204673 - y \cdot 0.5\\ \mathbf{if}\;y \leq -400000 \lor \neg \left(y \leq 3.6 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot y + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.918938533204673 (* y 0.5))))
   (if (or (<= y -400000.0) (not (<= y 3.6e-13))) (+ (* x y) t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = 0.918938533204673 - (y * 0.5);
	double tmp;
	if ((y <= -400000.0) || !(y <= 3.6e-13)) {
		tmp = (x * y) + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.918938533204673d0 - (y * 0.5d0)
    if ((y <= (-400000.0d0)) .or. (.not. (y <= 3.6d-13))) then
        tmp = (x * y) + t_0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.918938533204673 - (y * 0.5);
	double tmp;
	if ((y <= -400000.0) || !(y <= 3.6e-13)) {
		tmp = (x * y) + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.918938533204673 - (y * 0.5)
	tmp = 0
	if (y <= -400000.0) or not (y <= 3.6e-13):
		tmp = (x * y) + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.918938533204673 - Float64(y * 0.5))
	tmp = 0.0
	if ((y <= -400000.0) || !(y <= 3.6e-13))
		tmp = Float64(Float64(x * y) + t_0);
	else
		tmp = Float64(t_0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.918938533204673 - (y * 0.5);
	tmp = 0.0;
	if ((y <= -400000.0) || ~((y <= 3.6e-13)))
		tmp = (x * y) + t_0;
	else
		tmp = t_0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -400000.0], N[Not[LessEqual[y, 3.6e-13]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4e5 or 3.5999999999999998e-13 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 98.7%

      \[\leadsto \color{blue}{x \cdot y} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   if -4e5 < y < 3.5999999999999998e-13

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 99.4%

         \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Simplified 99.4%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -400000 \lor \neg \left(y \leq 3.6 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 - y \cdot 0.5\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. sub-neg 100.0%

      \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. metadata-eval 100.0%

      \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in y around 0 77.0%

   \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
6. Step-by-step derivation

   1. neg-mul-1 77.0%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

7. Simplified 77.0%

   \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

8. Final simplification 77.0%

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

Reproduce

herbie shell --seed 2024130 
(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))