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

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 9

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(0.918938533204673 + N[(N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - 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. 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. Taylor expanded in y around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 49.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -130:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-77}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-206}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-289}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-258}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -130.0)
   (* y -0.5)
   (if (<= y -9.2e-77)
     (- x)
     (if (<= y -1.22e-206)
       0.918938533204673
       (if (<= y -7.5e-289)
         (- x)
         (if (<= y 2.5e-258)
           0.918938533204673
           (if (<= y 1.25) (- x) (* y -0.5))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -130.0) {
		tmp = y * -0.5;
	} else if (y <= -9.2e-77) {
		tmp = -x;
	} else if (y <= -1.22e-206) {
		tmp = 0.918938533204673;
	} else if (y <= -7.5e-289) {
		tmp = -x;
	} else if (y <= 2.5e-258) {
		tmp = 0.918938533204673;
	} else if (y <= 1.25) {
		tmp = -x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-130.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= (-9.2d-77)) then
        tmp = -x
    else if (y <= (-1.22d-206)) then
        tmp = 0.918938533204673d0
    else if (y <= (-7.5d-289)) then
        tmp = -x
    else if (y <= 2.5d-258) then
        tmp = 0.918938533204673d0
    else if (y <= 1.25d0) then
        tmp = -x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -130.0) {
		tmp = y * -0.5;
	} else if (y <= -9.2e-77) {
		tmp = -x;
	} else if (y <= -1.22e-206) {
		tmp = 0.918938533204673;
	} else if (y <= -7.5e-289) {
		tmp = -x;
	} else if (y <= 2.5e-258) {
		tmp = 0.918938533204673;
	} else if (y <= 1.25) {
		tmp = -x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -130.0:
		tmp = y * -0.5
	elif y <= -9.2e-77:
		tmp = -x
	elif y <= -1.22e-206:
		tmp = 0.918938533204673
	elif y <= -7.5e-289:
		tmp = -x
	elif y <= 2.5e-258:
		tmp = 0.918938533204673
	elif y <= 1.25:
		tmp = -x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -130.0)
		tmp = Float64(y * -0.5);
	elseif (y <= -9.2e-77)
		tmp = Float64(-x);
	elseif (y <= -1.22e-206)
		tmp = 0.918938533204673;
	elseif (y <= -7.5e-289)
		tmp = Float64(-x);
	elseif (y <= 2.5e-258)
		tmp = 0.918938533204673;
	elseif (y <= 1.25)
		tmp = Float64(-x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -130.0)
		tmp = y * -0.5;
	elseif (y <= -9.2e-77)
		tmp = -x;
	elseif (y <= -1.22e-206)
		tmp = 0.918938533204673;
	elseif (y <= -7.5e-289)
		tmp = -x;
	elseif (y <= 2.5e-258)
		tmp = 0.918938533204673;
	elseif (y <= 1.25)
		tmp = -x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -130.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -9.2e-77], (-x), If[LessEqual[y, -1.22e-206], 0.918938533204673, If[LessEqual[y, -7.5e-289], (-x), If[LessEqual[y, 2.5e-258], 0.918938533204673, If[LessEqual[y, 1.25], (-x), N[(y * -0.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -130 or 1.25 < 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. Taylor expanded in y around inf 97.0%

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

   5. Taylor expanded in x around 0 45.6%

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

   if -130 < y < -9.19999999999999994e-77 or -1.22000000000000002e-206 < y < -7.49999999999999998e-289 or 2.4999999999999999e-258 < y < 1.25

   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. Taylor expanded in y around 0 95.1%

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

      1. neg-mul-1 95.1%

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

      2. unsub-neg 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in x around inf 61.6%

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

      1. neg-mul-1 61.6%

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

   9. Simplified 61.6%

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

   if -9.19999999999999994e-77 < y < -1.22000000000000002e-206 or -7.49999999999999998e-289 < y < 2.4999999999999999e-258

   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. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 67.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-77}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-206}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-289}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-258}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 3: 48.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-76}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{-205}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-289}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-258}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* x y)
   (if (<= y -7.8e-76)
     (- x)
     (if (<= y -2.85e-205)
       0.918938533204673
       (if (<= y -2.5e-289)
         (- x)
         (if (<= y 2.5e-258)
           0.918938533204673
           (if (<= y 1.0) (- x) (* x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= -7.8e-76) {
		tmp = -x;
	} else if (y <= -2.85e-205) {
		tmp = 0.918938533204673;
	} else if (y <= -2.5e-289) {
		tmp = -x;
	} else if (y <= 2.5e-258) {
		tmp = 0.918938533204673;
	} else if (y <= 1.0) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	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 = x * y
    else if (y <= (-7.8d-76)) then
        tmp = -x
    else if (y <= (-2.85d-205)) then
        tmp = 0.918938533204673d0
    else if (y <= (-2.5d-289)) then
        tmp = -x
    else if (y <= 2.5d-258) then
        tmp = 0.918938533204673d0
    else if (y <= 1.0d0) then
        tmp = -x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= -7.8e-76) {
		tmp = -x;
	} else if (y <= -2.85e-205) {
		tmp = 0.918938533204673;
	} else if (y <= -2.5e-289) {
		tmp = -x;
	} else if (y <= 2.5e-258) {
		tmp = 0.918938533204673;
	} else if (y <= 1.0) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= -7.8e-76:
		tmp = -x
	elif y <= -2.85e-205:
		tmp = 0.918938533204673
	elif y <= -2.5e-289:
		tmp = -x
	elif y <= 2.5e-258:
		tmp = 0.918938533204673
	elif y <= 1.0:
		tmp = -x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= -7.8e-76)
		tmp = Float64(-x);
	elseif (y <= -2.85e-205)
		tmp = 0.918938533204673;
	elseif (y <= -2.5e-289)
		tmp = Float64(-x);
	elseif (y <= 2.5e-258)
		tmp = 0.918938533204673;
	elseif (y <= 1.0)
		tmp = Float64(-x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x * y;
	elseif (y <= -7.8e-76)
		tmp = -x;
	elseif (y <= -2.85e-205)
		tmp = 0.918938533204673;
	elseif (y <= -2.5e-289)
		tmp = -x;
	elseif (y <= 2.5e-258)
		tmp = 0.918938533204673;
	elseif (y <= 1.0)
		tmp = -x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, -7.8e-76], (-x), If[LessEqual[y, -2.85e-205], 0.918938533204673, If[LessEqual[y, -2.5e-289], (-x), If[LessEqual[y, 2.5e-258], 0.918938533204673, If[LessEqual[y, 1.0], (-x), N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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. 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. Taylor expanded in y around inf 96.4%

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

   5. Taylor expanded in x around inf 53.4%

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

      1. *-commutative 53.4%

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

   7. Simplified 53.4%

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

   if -1 < y < -7.8000000000000005e-76 or -2.85e-205 < y < -2.50000000000000014e-289 or 2.4999999999999999e-258 < 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. Taylor expanded in y around 0 96.0%

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

      1. neg-mul-1 96.0%

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

      2. unsub-neg 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in x around inf 62.2%

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

      1. neg-mul-1 62.2%

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

   9. Simplified 62.2%

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

   if -7.8000000000000005e-76 < y < -2.85e-205 or -2.50000000000000014e-289 < y < 2.4999999999999999e-258

   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. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 67.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-76}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{-205}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-289}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-258}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -34000000 \lor \neg \left(y \leq 13000000\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 - y \cdot 0.5\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -34000000.0) (not (<= y 13000000.0)))
   (* y (- x 0.5))
   (- (- 0.918938533204673 (* y 0.5)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -34000000.0) || !(y <= 13000000.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = (0.918938533204673 - (y * 0.5)) - 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 <= (-34000000.0d0)) .or. (.not. (y <= 13000000.0d0))) then
        tmp = y * (x - 0.5d0)
    else
        tmp = (0.918938533204673d0 - (y * 0.5d0)) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -34000000.0) || !(y <= 13000000.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = (0.918938533204673 - (y * 0.5)) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -34000000.0) or not (y <= 13000000.0):
		tmp = y * (x - 0.5)
	else:
		tmp = (0.918938533204673 - (y * 0.5)) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -34000000.0) || !(y <= 13000000.0))
		tmp = Float64(y * Float64(x - 0.5));
	else
		tmp = Float64(Float64(0.918938533204673 - Float64(y * 0.5)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -34000000.0) || ~((y <= 13000000.0)))
		tmp = y * (x - 0.5);
	else
		tmp = (0.918938533204673 - (y * 0.5)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -34000000.0], N[Not[LessEqual[y, 13000000.0]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e7 or 1.3e7 < 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. Taylor expanded in y around inf 99.2%

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

   if -3.4e7 < y < 1.3e7

   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. Taylor expanded in y around 0 97.4%

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

      1. neg-mul-1 97.4%

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

   6. Simplified 97.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -34000000 \lor \neg \left(y \leq 13000000\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 - y \cdot 0.5\right) - x\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.5) || !(x <= 0.5)) {
		tmp = 0.918938533204673 + ((x * y) - x);
	} else {
		tmp = (0.918938533204673 - (y * 0.5)) - 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 <= (-0.5d0)) .or. (.not. (x <= 0.5d0))) then
        tmp = 0.918938533204673d0 + ((x * y) - x)
    else
        tmp = (0.918938533204673d0 - (y * 0.5d0)) - x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.5 or 0.5 < 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. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 48.8%

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

   7. Simplified 99.0%

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

   if -0.5 < x < 0.5

   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. Taylor expanded in y around 0 99.8%

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

      1. neg-mul-1 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.4%

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

Alternative 6: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \lor \neg \left(y \leq 1.55\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.22) (not (<= y 1.55)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.22) || !(y <= 1.55)) {
		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 <= (-1.22d0)) .or. (.not. (y <= 1.55d0))) 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 <= -1.22) || !(y <= 1.55)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.22) or not (y <= 1.55):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.22) || !(y <= 1.55))
		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 <= -1.22) || ~((y <= 1.55)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.22], N[Not[LessEqual[y, 1.55]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.21999999999999997 or 1.55000000000000004 < 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. Taylor expanded in y around inf 96.4%

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

   if -1.21999999999999997 < y < 1.55000000000000004

   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. Taylor expanded in y around 0 97.4%

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

      1. neg-mul-1 97.4%

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

      2. unsub-neg 97.4%

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

   6. Simplified 97.4%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \lor \neg \left(y \leq 1.55\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 7: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2800000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.6:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2800000.0)
   (* x y)
   (if (<= y 1.6) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2800000.0) {
		tmp = x * y;
	} else if (y <= 1.6) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2800000.0d0)) then
        tmp = x * y
    else if (y <= 1.6d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2800000.0) {
		tmp = x * y;
	} else if (y <= 1.6) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2800000.0:
		tmp = x * y
	elif y <= 1.6:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2800000.0)
		tmp = Float64(x * y);
	elseif (y <= 1.6)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2800000.0)
		tmp = x * y;
	elseif (y <= 1.6)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2800000.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.6], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e6 or 1.6000000000000001 < 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. Taylor expanded in y around inf 97.9%

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

   5. Taylor expanded in x around inf 54.5%

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

      1. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   if -2.8e6 < y < 1.6000000000000001

   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. Taylor expanded in y around 0 95.6%

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

      1. neg-mul-1 95.6%

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

      2. unsub-neg 95.6%

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

   6. Simplified 95.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2800000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.6:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 50.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 6000:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.92) (- x) (if (<= x 6000.0) 0.918938533204673 (- x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = -x;
	} else if (x <= 6000.0) {
		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 <= (-0.92d0)) then
        tmp = -x
    else if (x <= 6000.0d0) 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 <= -0.92) {
		tmp = -x;
	} else if (x <= 6000.0) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.92:
		tmp = -x
	elif x <= 6000.0:
		tmp = 0.918938533204673
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.92)
		tmp = Float64(-x);
	elseif (x <= 6000.0)
		tmp = 0.918938533204673;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.92)
		tmp = -x;
	elseif (x <= 6000.0)
		tmp = 0.918938533204673;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.92], (-x), If[LessEqual[x, 6000.0], 0.918938533204673, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 6e3 < 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. Taylor expanded in y around 0 51.4%

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

      1. neg-mul-1 51.4%

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

      2. unsub-neg 51.4%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in x around inf 50.9%

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

      1. neg-mul-1 50.9%

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

   9. Simplified 50.9%

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

   if -0.92000000000000004 < x < 6e3

   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. Taylor expanded in y around 0 52.9%

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

      1. neg-mul-1 52.9%

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

      2. unsub-neg 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in x around 0 52.4%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 6000:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9: 26.5% 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. Taylor expanded in y around 0 52.1%

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

   1. neg-mul-1 52.1%

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

   2. unsub-neg 52.1%

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

6. Simplified 52.1%

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

7. Taylor expanded in x around 0 26.0%

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

8. Final simplification 26.0%

   \[\leadsto 0.918938533204673 \]

Reproduce

herbie shell --seed 2023279 
(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))