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

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 12

Speedup: 1.5×

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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 100.0%

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

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 0.5\right) \cdot y + 0.918938533204673\\ \mathbf{if}\;y \leq -800:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-13}:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(-x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* (- x 0.5) y) 0.918938533204673)))
   (if (<= y -800.0)
     t_0
     (if (<= y 3.4e-13) (- 0.918938533204673 (fma (- x) y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 0.5) * y) + 0.918938533204673;
	double tmp;
	if (y <= -800.0) {
		tmp = t_0;
	} else if (y <= 3.4e-13) {
		tmp = 0.918938533204673 - fma(-x, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 0.5) * y) + 0.918938533204673)
	tmp = 0.0
	if (y <= -800.0)
		tmp = t_0;
	elseif (y <= 3.4e-13)
		tmp = Float64(0.918938533204673 - fma(Float64(-x), y, x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision] + 0.918938533204673), $MachinePrecision]}, If[LessEqual[y, -800.0], t$95$0, If[LessEqual[y, 3.4e-13], N[(0.918938533204673 - N[((-x) * y + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -800 or 3.40000000000000015e-13 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} + \frac{918938533204673}{1000000000000000} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -800 < y < 3.40000000000000015e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(-1 \cdot x, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.3%

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

Alternative 3: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -13200000:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(0.5, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(-x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -13200000.0)
   (fma y x (- x))
   (if (<= x 1.3e-7)
     (- 0.918938533204673 (fma 0.5 y x))
     (- 0.918938533204673 (fma (- x) y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -13200000.0) {
		tmp = fma(y, x, -x);
	} else if (x <= 1.3e-7) {
		tmp = 0.918938533204673 - fma(0.5, y, x);
	} else {
		tmp = 0.918938533204673 - fma(-x, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -13200000.0)
		tmp = fma(y, x, Float64(-x));
	elseif (x <= 1.3e-7)
		tmp = Float64(0.918938533204673 - fma(0.5, y, x));
	else
		tmp = Float64(0.918938533204673 - fma(Float64(-x), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -13200000.0], N[(y * x + (-x)), $MachinePrecision], If[LessEqual[x, 1.3e-7], N[(0.918938533204673 - N[(0.5 * y + x), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - N[((-x) * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 99.4

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

   8. Applied rewrites 99.4%

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

   10. Applied rewrites 99.5%

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

   if -1.32e7 < x < 1.29999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2}, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

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

   if 1.29999999999999999e-7 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(-1 \cdot x, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

Alternative 4: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, x, -x\right)\\ \mathbf{if}\;x \leq -13200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 560000000:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(0.5, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y x (- x))))
   (if (<= x -13200000.0)
     t_0
     (if (<= x 560000000.0) (- 0.918938533204673 (fma 0.5 y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, x, -x);
	double tmp;
	if (x <= -13200000.0) {
		tmp = t_0;
	} else if (x <= 560000000.0) {
		tmp = 0.918938533204673 - fma(0.5, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, x, Float64(-x))
	tmp = 0.0
	if (x <= -13200000.0)
		tmp = t_0;
	elseif (x <= 560000000.0)
		tmp = Float64(0.918938533204673 - fma(0.5, y, x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * x + (-x)), $MachinePrecision]}, If[LessEqual[x, -13200000.0], t$95$0, If[LessEqual[x, 560000000.0], N[(0.918938533204673 - N[(0.5 * y + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32e7 or 5.6e8 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 99.6

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.7%

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

   if -1.32e7 < x < 5.6e8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2}, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.5%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, x, -x\right)\\ \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y x (- x))))
   (if (<= x -0.7) t_0 (if (<= x 0.62) (fma -0.5 y 0.918938533204673) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, x, -x);
	double tmp;
	if (x <= -0.7) {
		tmp = t_0;
	} else if (x <= 0.62) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, x, Float64(-x))
	tmp = 0.0
	if (x <= -0.7)
		tmp = t_0;
	elseif (x <= 0.62)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * x + (-x)), $MachinePrecision]}, If[LessEqual[x, -0.7], t$95$0, If[LessEqual[x, 0.62], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.619999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 98.9

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

   8. Applied rewrites 98.9%

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

   10. Applied rewrites 98.9%

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

   if -0.69999999999999996 < x < 0.619999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 6: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - 1\right) \cdot x\\ \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y 1.0) x)))
   (if (<= x -0.7) t_0 (if (<= x 0.62) (fma -0.5 y 0.918938533204673) t_0))))

C

double code(double x, double y) {
	double t_0 = (y - 1.0) * x;
	double tmp;
	if (x <= -0.7) {
		tmp = t_0;
	} else if (x <= 0.62) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y - 1.0) * x)
	tmp = 0.0
	if (x <= -0.7)
		tmp = t_0;
	elseif (x <= 0.62)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.7], t$95$0, If[LessEqual[x, 0.62], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.619999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if -0.69999999999999996 < x < 0.619999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 7: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 0.5\right) \cdot y\\ \mathbf{if}\;y \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.35:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x 0.5) y)))
   (if (<= y -1.55) t_0 (if (<= y 1.35) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (x - 0.5) * y;
	double tmp;
	if (y <= -1.55) {
		tmp = t_0;
	} else if (y <= 1.35) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - 0.5d0) * y
    if (y <= (-1.55d0)) then
        tmp = t_0
    else if (y <= 1.35d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - 0.5) * y;
	double tmp;
	if (y <= -1.55) {
		tmp = t_0;
	} else if (y <= 1.35) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - 0.5) * y
	tmp = 0
	if y <= -1.55:
		tmp = t_0
	elif y <= 1.35:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - 0.5) * y)
	tmp = 0.0
	if (y <= -1.55)
		tmp = t_0;
	elseif (y <= 1.35)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - 0.5) * y;
	tmp = 0.0;
	if (y <= -1.55)
		tmp = t_0;
	elseif (y <= 1.35)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.55], t$95$0, If[LessEqual[y, 1.35], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000004 or 1.3500000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(\frac{1}{2} + -1 \cdot x\right) \cdot y\right)} \]

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto -1 \cdot \left(\color{blue}{\left(\frac{1}{2} - x\right)} \cdot y\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{1}{2} - x\right)\right) \cdot y} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{1}{2} - x\right)\right) \cdot y} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} - x\right)\right)\right)} \cdot y \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(\frac{1}{2} - x\right)\right)} \cdot y \]

      8. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(0 - \frac{1}{2}\right) + x\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} + x\right) \cdot y \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot y \]

      11. metadata-eval N/A

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

      12. sub-neg N/A

          \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right)} \cdot y \]

      13. lower--.f64 96.5

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

   8. Applied rewrites 96.5%

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

   if -1.55000000000000004 < y < 1.3500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 8: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-8}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e-8)
   (- 0.918938533204673 x)
   (if (<= x 0.62) (fma -0.5 y 0.918938533204673) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e-8) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 0.62) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e-8)
		tmp = Float64(0.918938533204673 - x);
	elseif (x <= 0.62)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e-8], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[x, 0.62], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.99999999999999946e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -5.99999999999999946e-8 < x < 0.619999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.619999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(\frac{1}{2} + -1 \cdot x\right) \cdot y\right)} \]

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto -1 \cdot \left(\color{blue}{\left(\frac{1}{2} - x\right)} \cdot y\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{1}{2} - x\right)\right) \cdot y} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{1}{2} - x\right)\right) \cdot y} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} - x\right)\right)\right)} \cdot y \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(\frac{1}{2} - x\right)\right)} \cdot y \]

      8. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(0 - \frac{1}{2}\right) + x\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} + x\right) \cdot y \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot y \]

      11. metadata-eval N/A

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

      12. sub-neg N/A

          \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right)} \cdot y \]

      13. lower--.f64 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 54.2%

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

Alternative 9: 73.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -800:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -800.0)
   (* -0.5 y)
   (if (<= y 1.85) (- 0.918938533204673 x) (* -0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -800.0) {
		tmp = -0.5 * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = -0.5 * 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 <= (-800.0d0)) then
        tmp = (-0.5d0) * y
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -800.0) {
		tmp = -0.5 * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -800.0:
		tmp = -0.5 * y
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -800.0)
		tmp = Float64(-0.5 * y);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -800.0)
		tmp = -0.5 * y;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -800.0], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -800 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(\frac{1}{2} + -1 \cdot x\right) \cdot y\right)} \]

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto -1 \cdot \left(\color{blue}{\left(\frac{1}{2} - x\right)} \cdot y\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{1}{2} - x\right)\right) \cdot y} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{1}{2} - x\right)\right) \cdot y} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} - x\right)\right)\right)} \cdot y \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(\frac{1}{2} - x\right)\right)} \cdot y \]

      8. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(0 - \frac{1}{2}\right) + x\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} + x\right) \cdot y \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot y \]

      11. metadata-eval N/A

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

      12. sub-neg N/A

          \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right)} \cdot y \]

      13. lower--.f64 98.2

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 56.5%

       \[\leadsto -0.5 \cdot y \]

   if -800 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 10: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -120:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.15:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -120.0) (* y x) (if (<= y 1.15) (- 0.918938533204673 x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -120.0) {
		tmp = y * x;
	} else if (y <= 1.15) {
		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 <= (-120.0d0)) then
        tmp = y * x
    else if (y <= 1.15d0) 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 <= -120.0) {
		tmp = y * x;
	} else if (y <= 1.15) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -120.0:
		tmp = y * x
	elif y <= 1.15:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -120.0)
		tmp = Float64(y * x);
	elseif (y <= 1.15)
		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 <= -120.0)
		tmp = y * x;
	elseif (y <= 1.15)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -120.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.15], N[(0.918938533204673 - x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -120 or 1.1499999999999999 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(\frac{1}{2} + -1 \cdot x\right) \cdot y\right)} \]

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto -1 \cdot \left(\color{blue}{\left(\frac{1}{2} - x\right)} \cdot y\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{1}{2} - x\right)\right) \cdot y} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{1}{2} - x\right)\right) \cdot y} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} - x\right)\right)\right)} \cdot y \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(\frac{1}{2} - x\right)\right)} \cdot y \]

      8. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(0 - \frac{1}{2}\right) + x\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} + x\right) \cdot y \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot y \]

      11. metadata-eval N/A

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

      12. sub-neg N/A

          \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right)} \cdot y \]

      13. lower--.f64 97.7

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 43.3%

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

   if -120 < y < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 11: 50.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(0.918938533204673 - x), $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. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

   3. lower--.f64 50.0

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

5. Applied rewrites 50.0%

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

Alternative 12: 26.1% accurate, 20.0× 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. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

   3. lower--.f64 50.0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{918938533204673}{1000000000000000} \]
7. Step-by-step derivation

8. Applied rewrites 24.0%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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))