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

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 10

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. cancel-sign-sub-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. neg-mul-1 100.0%

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

   7. associate-+r+ 100.0%

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

   8. unsub-neg 100.0%

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

   9. associate-+l- 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. distribute-rgt-neg-in 100.0%

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

   12. distribute-lft-out 100.0%

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

   13. fma-neg 100.0%

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

   14. +-commutative 100.0%

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

   15. metadata-eval 100.0%

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

   16. neg-sub0 100.0%

       \[\leadsto \mathsf{fma}\left(y, x + -0.5, \color{blue}{0 - \left(x - 0.918938533204673\right)}\right) \]

   17. associate-+l- 100.0%

       \[\leadsto \mathsf{fma}\left(y, x + -0.5, \color{blue}{\left(0 - x\right) + 0.918938533204673}\right) \]

   18. neg-sub0 100.0%

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

   19. +-commutative 100.0%

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

   20. unsub-neg 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 49.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{+181}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -0.98:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-109}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-299}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-119}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-34}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.4e+181)
   (- x)
   (if (<= x -0.98)
     (* y x)
     (if (<= x -7e-109)
       0.918938533204673
       (if (<= x -1.75e-299)
         (* y -0.5)
         (if (<= x 1.6e-119)
           0.918938533204673
           (if (<= x 3.4e-34)
             (* y -0.5)
             (if (<= x 0.92) 0.918938533204673 (- x)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.4e+181) {
		tmp = -x;
	} else if (x <= -0.98) {
		tmp = y * x;
	} else if (x <= -7e-109) {
		tmp = 0.918938533204673;
	} else if (x <= -1.75e-299) {
		tmp = y * -0.5;
	} else if (x <= 1.6e-119) {
		tmp = 0.918938533204673;
	} else if (x <= 3.4e-34) {
		tmp = y * -0.5;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.4d+181)) then
        tmp = -x
    else if (x <= (-0.98d0)) then
        tmp = y * x
    else if (x <= (-7d-109)) then
        tmp = 0.918938533204673d0
    else if (x <= (-1.75d-299)) then
        tmp = y * (-0.5d0)
    else if (x <= 1.6d-119) then
        tmp = 0.918938533204673d0
    else if (x <= 3.4d-34) then
        tmp = y * (-0.5d0)
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.4e+181) {
		tmp = -x;
	} else if (x <= -0.98) {
		tmp = y * x;
	} else if (x <= -7e-109) {
		tmp = 0.918938533204673;
	} else if (x <= -1.75e-299) {
		tmp = y * -0.5;
	} else if (x <= 1.6e-119) {
		tmp = 0.918938533204673;
	} else if (x <= 3.4e-34) {
		tmp = y * -0.5;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.4e+181:
		tmp = -x
	elif x <= -0.98:
		tmp = y * x
	elif x <= -7e-109:
		tmp = 0.918938533204673
	elif x <= -1.75e-299:
		tmp = y * -0.5
	elif x <= 1.6e-119:
		tmp = 0.918938533204673
	elif x <= 3.4e-34:
		tmp = y * -0.5
	elif x <= 0.92:
		tmp = 0.918938533204673
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.4e+181)
		tmp = Float64(-x);
	elseif (x <= -0.98)
		tmp = Float64(y * x);
	elseif (x <= -7e-109)
		tmp = 0.918938533204673;
	elseif (x <= -1.75e-299)
		tmp = Float64(y * -0.5);
	elseif (x <= 1.6e-119)
		tmp = 0.918938533204673;
	elseif (x <= 3.4e-34)
		tmp = Float64(y * -0.5);
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.4e+181)
		tmp = -x;
	elseif (x <= -0.98)
		tmp = y * x;
	elseif (x <= -7e-109)
		tmp = 0.918938533204673;
	elseif (x <= -1.75e-299)
		tmp = y * -0.5;
	elseif (x <= 1.6e-119)
		tmp = 0.918938533204673;
	elseif (x <= 3.4e-34)
		tmp = y * -0.5;
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.4e+181], (-x), If[LessEqual[x, -0.98], N[(y * x), $MachinePrecision], If[LessEqual[x, -7e-109], 0.918938533204673, If[LessEqual[x, -1.75e-299], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 1.6e-119], 0.918938533204673, If[LessEqual[x, 3.4e-34], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 0.92], 0.918938533204673, (-x)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.40000000000000055e181 or 0.92000000000000004 < x

   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 x around inf 97.9%

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

   6. Taylor expanded in y around 0 56.5%

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

      1. neg-mul-1 56.5%

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

   8. Simplified 56.5%

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

   if -9.40000000000000055e181 < x < -0.97999999999999998

   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 x around inf 97.5%

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

   6. Taylor expanded in y around inf 60.2%

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

      1. *-commutative 60.2%

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

   8. Simplified 60.2%

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

   if -0.97999999999999998 < x < -7e-109 or -1.74999999999999995e-299 < x < 1.59999999999999997e-119 or 3.4000000000000001e-34 < x < 0.92000000000000004

   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 x around 0 95.3%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]

   6. Taylor expanded in y around 0 70.5%

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

   if -7e-109 < x < -1.74999999999999995e-299 or 1.59999999999999997e-119 < x < 3.4000000000000001e-34

   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 60.7%

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

   6. Taylor expanded in x around 0 60.7%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 60.7%

         \[\leadsto \color{blue}{y \cdot -0.5} \]

   8. Simplified 60.7%

      \[\leadsto \color{blue}{y \cdot -0.5} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 49.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1600:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-102}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-300}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-118}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-35}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1600.0)
   (- x)
   (if (<= x -5e-102)
     0.918938533204673
     (if (<= x -3.4e-300)
       (* y -0.5)
       (if (<= x 2.5e-118)
         0.918938533204673
         (if (<= x 5e-35)
           (* y -0.5)
           (if (<= x 0.92) 0.918938533204673 (- x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1600.0) {
		tmp = -x;
	} else if (x <= -5e-102) {
		tmp = 0.918938533204673;
	} else if (x <= -3.4e-300) {
		tmp = y * -0.5;
	} else if (x <= 2.5e-118) {
		tmp = 0.918938533204673;
	} else if (x <= 5e-35) {
		tmp = y * -0.5;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1600.0d0)) then
        tmp = -x
    else if (x <= (-5d-102)) then
        tmp = 0.918938533204673d0
    else if (x <= (-3.4d-300)) then
        tmp = y * (-0.5d0)
    else if (x <= 2.5d-118) then
        tmp = 0.918938533204673d0
    else if (x <= 5d-35) then
        tmp = y * (-0.5d0)
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1600.0) {
		tmp = -x;
	} else if (x <= -5e-102) {
		tmp = 0.918938533204673;
	} else if (x <= -3.4e-300) {
		tmp = y * -0.5;
	} else if (x <= 2.5e-118) {
		tmp = 0.918938533204673;
	} else if (x <= 5e-35) {
		tmp = y * -0.5;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1600.0:
		tmp = -x
	elif x <= -5e-102:
		tmp = 0.918938533204673
	elif x <= -3.4e-300:
		tmp = y * -0.5
	elif x <= 2.5e-118:
		tmp = 0.918938533204673
	elif x <= 5e-35:
		tmp = y * -0.5
	elif x <= 0.92:
		tmp = 0.918938533204673
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1600.0)
		tmp = Float64(-x);
	elseif (x <= -5e-102)
		tmp = 0.918938533204673;
	elseif (x <= -3.4e-300)
		tmp = Float64(y * -0.5);
	elseif (x <= 2.5e-118)
		tmp = 0.918938533204673;
	elseif (x <= 5e-35)
		tmp = Float64(y * -0.5);
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1600.0)
		tmp = -x;
	elseif (x <= -5e-102)
		tmp = 0.918938533204673;
	elseif (x <= -3.4e-300)
		tmp = y * -0.5;
	elseif (x <= 2.5e-118)
		tmp = 0.918938533204673;
	elseif (x <= 5e-35)
		tmp = y * -0.5;
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1600.0], (-x), If[LessEqual[x, -5e-102], 0.918938533204673, If[LessEqual[x, -3.4e-300], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 2.5e-118], 0.918938533204673, If[LessEqual[x, 5e-35], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 0.92], 0.918938533204673, (-x)]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1600 or 0.92000000000000004 < x

   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 x around inf 98.2%

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

   6. Taylor expanded in y around 0 52.0%

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

      1. neg-mul-1 52.0%

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

   8. Simplified 52.0%

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

   if -1600 < x < -5.00000000000000026e-102 or -3.40000000000000018e-300 < x < 2.50000000000000007e-118 or 4.99999999999999964e-35 < x < 0.92000000000000004

   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 x around 0 94.1%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]

   6. Taylor expanded in y around 0 69.4%

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

   if -5.00000000000000026e-102 < x < -3.40000000000000018e-300 or 2.50000000000000007e-118 < x < 4.99999999999999964e-35

   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 60.7%

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

   6. Taylor expanded in x around 0 60.7%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 60.7%

         \[\leadsto \color{blue}{y \cdot -0.5} \]

   8. Simplified 60.7%

      \[\leadsto \color{blue}{y \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+289}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{+194}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -8000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-18}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.85e+289)
   (* y -0.5)
   (if (<= y -4.4e+194)
     (* y x)
     (if (<= y -8000.0)
       (* y -0.5)
       (if (<= y 4e-18) (- 0.918938533204673 x) (* x (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.85e+289) {
		tmp = y * -0.5;
	} else if (y <= -4.4e+194) {
		tmp = y * x;
	} else if (y <= -8000.0) {
		tmp = y * -0.5;
	} else if (y <= 4e-18) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.85d+289)) then
        tmp = y * (-0.5d0)
    else if (y <= (-4.4d+194)) then
        tmp = y * x
    else if (y <= (-8000.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 4d-18) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.85e+289) {
		tmp = y * -0.5;
	} else if (y <= -4.4e+194) {
		tmp = y * x;
	} else if (y <= -8000.0) {
		tmp = y * -0.5;
	} else if (y <= 4e-18) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.85e+289:
		tmp = y * -0.5
	elif y <= -4.4e+194:
		tmp = y * x
	elif y <= -8000.0:
		tmp = y * -0.5
	elif y <= 4e-18:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.85e+289)
		tmp = Float64(y * -0.5);
	elseif (y <= -4.4e+194)
		tmp = Float64(y * x);
	elseif (y <= -8000.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 4e-18)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.85e+289)
		tmp = y * -0.5;
	elseif (y <= -4.4e+194)
		tmp = y * x;
	elseif (y <= -8000.0)
		tmp = y * -0.5;
	elseif (y <= 4e-18)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.85e+289], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -4.4e+194], N[(y * x), $MachinePrecision], If[LessEqual[y, -8000.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 4e-18], N[(0.918938533204673 - x), $MachinePrecision], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.8499999999999999e289 or -4.4000000000000002e194 < y < -8e3

   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 98.0%

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

   6. Taylor expanded in x around 0 64.2%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 64.2%

         \[\leadsto \color{blue}{y \cdot -0.5} \]

   8. Simplified 64.2%

      \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -2.8499999999999999e289 < y < -4.4000000000000002e194

   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 x around inf 79.9%

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

   6. Taylor expanded in y around inf 79.9%

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

      1. *-commutative 79.9%

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

   8. Simplified 79.9%

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

   if -8e3 < y < 4.0000000000000003e-18

   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 98.9%

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

      1. neg-mul-1 98.9%

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

      2. sub-neg 98.9%

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

   7. Simplified 98.9%

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

   if 4.0000000000000003e-18 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 56.5%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+289}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{+194}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -8000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-18}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+290}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{+195}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -8000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.02e+290)
   (* y -0.5)
   (if (<= y -1.22e+195)
     (* y x)
     (if (<= y -8000.0)
       (* y -0.5)
       (if (<= y 1.0) (- 0.918938533204673 x) (* y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.02e+290) {
		tmp = y * -0.5;
	} else if (y <= -1.22e+195) {
		tmp = y * x;
	} else if (y <= -8000.0) {
		tmp = y * -0.5;
	} else if (y <= 1.0) {
		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 <= (-1.02d+290)) then
        tmp = y * (-0.5d0)
    else if (y <= (-1.22d+195)) then
        tmp = y * x
    else if (y <= (-8000.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.0d0) 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 <= -1.02e+290) {
		tmp = y * -0.5;
	} else if (y <= -1.22e+195) {
		tmp = y * x;
	} else if (y <= -8000.0) {
		tmp = y * -0.5;
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.02e+290:
		tmp = y * -0.5
	elif y <= -1.22e+195:
		tmp = y * x
	elif y <= -8000.0:
		tmp = y * -0.5
	elif y <= 1.0:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.02e+290)
		tmp = Float64(y * -0.5);
	elseif (y <= -1.22e+195)
		tmp = Float64(y * x);
	elseif (y <= -8000.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.0)
		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 <= -1.02e+290)
		tmp = y * -0.5;
	elseif (y <= -1.22e+195)
		tmp = y * x;
	elseif (y <= -8000.0)
		tmp = y * -0.5;
	elseif (y <= 1.0)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.02e+290], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -1.22e+195], N[(y * x), $MachinePrecision], If[LessEqual[y, -8000.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.0], N[(0.918938533204673 - x), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02e290 or -1.2200000000000001e195 < y < -8e3

   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 98.0%

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

   6. Taylor expanded in x around 0 64.2%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 64.2%

         \[\leadsto \color{blue}{y \cdot -0.5} \]

   8. Simplified 64.2%

      \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -1.02e290 < y < -1.2200000000000001e195 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. 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 x around inf 59.6%

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

   6. Taylor expanded in y around inf 59.6%

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

      1. *-commutative 59.6%

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

   8. Simplified 59.6%

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

   if -8e3 < 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. 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 98.4%

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

      1. neg-mul-1 98.4%

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

      2. sub-neg 98.4%

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

   7. Simplified 98.4%

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

Alternative 6: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.35) (not (<= y 1.0)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.35) || !(y <= 1.0)) {
		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 <= (-1.35d0)) .or. (.not. (y <= 1.0d0))) 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 <= -1.35) || !(y <= 1.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.35) or not (y <= 1.0):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.35) || !(y <= 1.0))
		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 <= -1.35) || ~((y <= 1.0)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.35], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3500000000000001 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. 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 97.0%

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

   if -1.3500000000000001 < 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. 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 99.0%

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

      1. neg-mul-1 99.0%

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

      2. sub-neg 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1600 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1600.0) (not (<= x 0.92))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1600.0) || !(x <= 0.92)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1600.0d0)) .or. (.not. (x <= 0.92d0))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1600.0) || !(x <= 0.92)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1600.0) or not (x <= 0.92):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1600.0) || !(x <= 0.92))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1600.0) || ~((x <= 0.92)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1600.0], N[Not[LessEqual[x, 0.92]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -1600 or 0.92000000000000004 < x

   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 x around inf 98.2%

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

   6. Taylor expanded in y around 0 52.0%

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

      1. neg-mul-1 52.0%

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

   8. Simplified 52.0%

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

   if -1600 < x < 0.92000000000000004

   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 x around 0 97.1%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]

   6. Taylor expanded in y around 0 54.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1600 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 9: 25.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

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

Wolfram

code[x_, y_] := 0.918938533204673

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 x around 0 51.4%

   \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]

6. Taylor expanded in y around 0 29.0%

   \[\leadsto \color{blue}{0.918938533204673} \]
7. Add Preprocessing

Alternative 10: 2.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -0.918938533204673 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -0.918938533204673

Julia

function code(x, y)
	return -0.918938533204673
end

MATLAB

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

Wolfram

code[x_, y_] := -0.918938533204673

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 x around inf 90.4%

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

   1. associate-*r/ 90.4%

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

   2. metadata-eval 90.4%

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

   3. associate-*r/ 90.4%

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

7. Simplified 90.4%

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

8. Taylor expanded in y around 0 54.2%

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

   1. sub-neg 54.2%

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

   2. associate-*r/ 54.2%

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

   3. metadata-eval 54.2%

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

   4. metadata-eval 54.2%

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

10. Simplified 54.2%

    \[\leadsto \color{blue}{x \cdot \left(\frac{0.918938533204673}{x} + -1\right)} \]
11. Step-by-step derivation

    1. distribute-lft-in 54.2%

       \[\leadsto \color{blue}{x \cdot \frac{0.918938533204673}{x} + x \cdot -1} \]

    2. fma-define 54.2%

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

    3. *-commutative 54.2%

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

    4. neg-mul-1 54.2%

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

    5. fma-neg 54.2%

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

    6. frac-2neg 54.2%

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

    7. associate-*r/ 54.2%

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

    8. metadata-eval 54.2%

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

    9. add-sqr-sqrt 23.7%

       \[\leadsto \frac{x \cdot -0.918938533204673}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} - x \]

    10. sqrt-unprod 33.5%

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

    11. sqr-neg 33.5%

        \[\leadsto \frac{x \cdot -0.918938533204673}{\sqrt{\color{blue}{x \cdot x}}} - x \]

    12. sqrt-unprod 15.3%

        \[\leadsto \frac{x \cdot -0.918938533204673}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} - x \]

    13. add-sqr-sqrt 26.4%

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

12. Applied egg-rr 26.4%

    \[\leadsto \color{blue}{\frac{x \cdot -0.918938533204673}{x} - x} \]
13. Step-by-step derivation

    1. *-commutative 26.4%

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

    2. associate-/l* 26.4%

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

    3. *-inverses 26.4%

       \[\leadsto -0.918938533204673 \cdot \color{blue}{1} - x \]

    4. metadata-eval 26.4%

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

14. Simplified 26.4%

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

15. Taylor expanded in x around 0 2.7%

    \[\leadsto \color{blue}{-0.918938533204673} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))