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

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 6

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * N[(x + -0.5), $MachinePrecision] + N[(0.918938533204673 - x), $MachinePrecision]), $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. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. associate-+r+ 100.0%

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

   6. associate-+l+ 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. distribute-lft-out 100.0%

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

   9. fma-def 100.0%

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

   10. +-commutative 100.0%

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

   11. metadata-eval 100.0%

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

   12. +-commutative 100.0%

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

   13. cancel-sign-sub-inv 100.0%

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

   14. *-lft-identity 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

2. Final simplification 100.0%

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

Alternative 3: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \lor \neg \left(y \leq 1.4\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3) (not (<= y 1.4)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3) || !(y <= 1.4)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.3d0)) .or. (.not. (y <= 1.4d0))) then
        tmp = y * (x - 0.5d0)
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.3) || !(y <= 1.4)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3) or not (y <= 1.4):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3) || !(y <= 1.4))
		tmp = Float64(y * Float64(x - 0.5));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.3) || ~((y <= 1.4)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3], N[Not[LessEqual[y, 1.4]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000004 or 1.3999999999999999 < 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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. Taylor expanded in y around inf 98.3%

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

   if -1.30000000000000004 < y < 1.3999999999999999

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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. Taylor expanded in y around 0 98.0%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 98.0%

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

      2. unsub-neg 98.0%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Simplified 98.0%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \lor \neg \left(y \leq 1.4\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 4: 49.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = y * x
	elif y <= 1.0:
		tmp = -x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y * x);
	elseif (y <= 1.0)
		tmp = Float64(-x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y * x;
	elseif (y <= 1.0)
		tmp = -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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. Taylor expanded in x around inf 47.2%

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

      1. sub-neg 47.2%

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

      2. metadata-eval 47.2%

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

      3. *-commutative 47.2%

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

      4. distribute-rgt-in 47.2%

         \[\leadsto \color{blue}{y \cdot x + -1 \cdot x} \]

      5. fma-def 47.2%

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

      6. neg-mul-1 47.2%

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

      7. fma-neg 47.2%

         \[\leadsto \color{blue}{y \cdot x - x} \]

   6. Simplified 47.2%

      \[\leadsto \color{blue}{y \cdot x - x} \]

   7. Taylor expanded in y around inf 46.3%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -1 < y < 1

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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. Taylor expanded in x around inf 45.1%

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

      1. sub-neg 45.1%

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

      2. metadata-eval 45.1%

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

      3. *-commutative 45.1%

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

      4. distribute-rgt-in 45.1%

         \[\leadsto \color{blue}{y \cdot x + -1 \cdot x} \]

      5. fma-def 45.1%

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

      6. neg-mul-1 45.1%

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

      7. fma-neg 45.1%

         \[\leadsto \color{blue}{y \cdot x - x} \]

   6. Simplified 45.1%

      \[\leadsto \color{blue}{y \cdot x - x} \]

   7. Taylor expanded in y around 0 43.4%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. neg-mul-1 43.4%

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

   9. Simplified 43.4%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -900:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.75:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -900.0) (* y x) (if (<= y 1.75) (- 0.918938533204673 x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -900.0) {
		tmp = y * x;
	} else if (y <= 1.75) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-900.0d0)) then
        tmp = y * x
    else if (y <= 1.75d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -900.0) {
		tmp = y * x;
	} else if (y <= 1.75) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -900.0:
		tmp = y * x
	elif y <= 1.75:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -900.0)
		tmp = Float64(y * x);
	elseif (y <= 1.75)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -900.0)
		tmp = y * x;
	elseif (y <= 1.75)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -900.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.75], N[(0.918938533204673 - x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -900 or 1.75 < 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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. Taylor expanded in x around inf 47.4%

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

      1. sub-neg 47.4%

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

      2. metadata-eval 47.4%

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

      3. *-commutative 47.4%

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

      4. distribute-rgt-in 47.4%

         \[\leadsto \color{blue}{y \cdot x + -1 \cdot x} \]

      5. fma-def 47.4%

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

      6. neg-mul-1 47.4%

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

      7. fma-neg 47.4%

         \[\leadsto \color{blue}{y \cdot x - x} \]

   6. Simplified 47.4%

      \[\leadsto \color{blue}{y \cdot x - x} \]

   7. Taylor expanded in y around inf 46.9%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -900 < y < 1.75

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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. Taylor expanded in y around 0 96.8%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 96.8%

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

      2. unsub-neg 96.8%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Simplified 96.8%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -900:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.75:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 26.9% accurate, 5.5× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

function code(x, y)
	return Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_] := (-x)

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. fma-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. remove-double-neg 100.0%

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

   6. sub-neg 100.0%

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

   7. fma-neg 100.0%

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

   8. sub-neg 100.0%

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

   9. remove-double-neg 100.0%

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

   10. +-commutative 100.0%

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

   11. 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. Taylor expanded in x around inf 46.0%

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

   1. sub-neg 46.0%

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

   2. metadata-eval 46.0%

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

   3. *-commutative 46.0%

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

   4. distribute-rgt-in 46.1%

      \[\leadsto \color{blue}{y \cdot x + -1 \cdot x} \]

   5. fma-def 46.1%

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

   6. neg-mul-1 46.1%

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

   7. fma-neg 46.1%

      \[\leadsto \color{blue}{y \cdot x - x} \]

6. Simplified 46.1%

   \[\leadsto \color{blue}{y \cdot x - x} \]

7. Taylor expanded in y around 0 24.4%

   \[\leadsto \color{blue}{-1 \cdot x} \]
8. Step-by-step derivation

   1. neg-mul-1 24.4%

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

9. Simplified 24.4%

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

10. Final simplification 24.4%

    \[\leadsto -x \]

Reproduce

herbie shell --seed 2023221 
(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))