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

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right) - y \cdot 0.5\\ t_1 := y \cdot \left(x + -0.5\right) - x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10000:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, 0.918938533204673\right) - x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* x (+ y -1.0)) (* y 0.5))) (t_1 (- (* y (+ x -0.5)) x)))
   (if (<= t_0 -5e+14)
     t_1
     (if (<= t_0 10000.0) (- (fma y -0.5 0.918938533204673) x) t_1))))

C

double code(double x, double y) {
	double t_0 = (x * (y + -1.0)) - (y * 0.5);
	double t_1 = (y * (x + -0.5)) - x;
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = t_1;
	} else if (t_0 <= 10000.0) {
		tmp = fma(y, -0.5, 0.918938533204673) - x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(y + -1.0)) - Float64(y * 0.5))
	t_1 = Float64(Float64(y * Float64(x + -0.5)) - x)
	tmp = 0.0
	if (t_0 <= -5e+14)
		tmp = t_1;
	elseif (t_0 <= 10000.0)
		tmp = Float64(fma(y, -0.5, 0.918938533204673) - x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+14], t$95$1, If[LessEqual[t$95$0, 10000.0], N[(N[(y * -0.5 + 0.918938533204673), $MachinePrecision] - x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) < -5e14 or 1e4 < (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64)))

   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%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 99.7%

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

   if -5e14 < (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) < 1e4

   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%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(y + -1\right) - y \cdot 0.5 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(x + -0.5\right) - x\\ \mathbf{elif}\;x \cdot \left(y + -1\right) - y \cdot 0.5 \leq 10000:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, 0.918938533204673\right) - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6800:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+80}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+241}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6800.0)
   (* x y)
   (if (<= x 4.2e-10)
     (fma -0.5 y 0.918938533204673)
     (if (<= x 8.2e+80)
       (- 0.918938533204673 x)
       (if (<= x 2e+241) (* x y) (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6800.0) {
		tmp = x * y;
	} else if (x <= 4.2e-10) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (x <= 8.2e+80) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 2e+241) {
		tmp = x * y;
	} else {
		tmp = -x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6800.0)
		tmp = Float64(x * y);
	elseif (x <= 4.2e-10)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (x <= 8.2e+80)
		tmp = Float64(0.918938533204673 - x);
	elseif (x <= 2e+241)
		tmp = Float64(x * y);
	else
		tmp = Float64(-x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6800.0], N[(x * y), $MachinePrecision], If[LessEqual[x, 4.2e-10], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[x, 8.2e+80], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[x, 2e+241], N[(x * y), $MachinePrecision], (-x)]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6800 or 8.20000000000000003e80 < x < 2.0000000000000001e241

   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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.5%

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

   if -6800 < x < 4.2e-10

   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 97.8

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

   5. Applied rewrites 97.8%

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

   if 4.2e-10 < x < 8.20000000000000003e80

   1. Initial program 99.9%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   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 66.2

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

   5. Applied rewrites 66.2%

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

   if 2.0000000000000001e241 < 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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.7%

      \[\leadsto -x \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6800:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+80}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+241}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+74}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -75:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+161}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+74)
   (* x y)
   (if (<= y -75.0)
     (* y -0.5)
     (if (<= y 1.4)
       (- 0.918938533204673 x)
       (if (<= y 3.2e+161) (* x y) (* y -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+74) {
		tmp = x * y;
	} else if (y <= -75.0) {
		tmp = y * -0.5;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 3.2e+161) {
		tmp = x * y;
	} 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 <= (-4.8d+74)) then
        tmp = x * y
    else if (y <= (-75.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.4d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 3.2d+161) then
        tmp = x * y
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+74) {
		tmp = x * y;
	} else if (y <= -75.0) {
		tmp = y * -0.5;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 3.2e+161) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+74:
		tmp = x * y
	elif y <= -75.0:
		tmp = y * -0.5
	elif y <= 1.4:
		tmp = 0.918938533204673 - x
	elif y <= 3.2e+161:
		tmp = x * y
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+74)
		tmp = Float64(x * y);
	elseif (y <= -75.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.4)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 3.2e+161)
		tmp = Float64(x * y);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e+74)
		tmp = x * y;
	elseif (y <= -75.0)
		tmp = y * -0.5;
	elseif (y <= 1.4)
		tmp = 0.918938533204673 - x;
	elseif (y <= 3.2e+161)
		tmp = x * y;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+74], N[(x * y), $MachinePrecision], If[LessEqual[y, -75.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.4], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 3.2e+161], N[(x * y), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.80000000000000017e74 or 1.3999999999999999 < y < 3.20000000000000002e161

   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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.9%

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

   if -4.80000000000000017e74 < y < -75 or 3.20000000000000002e161 < 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

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.6%

      \[\leadsto y \cdot -0.5 \]

   if -75 < y < 1.3999999999999999

   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 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+74}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -75:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+161}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1300000000:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, 0.918938533204673\right) - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -6e+24)
     t_0
     (if (<= y 1300000000.0) (- (fma y -0.5 0.918938533204673) x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -6e+24) {
		tmp = t_0;
	} else if (y <= 1300000000.0) {
		tmp = fma(y, -0.5, 0.918938533204673) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -0.5))
	tmp = 0.0
	if (y <= -6e+24)
		tmp = t_0;
	elseif (y <= 1300000000.0)
		tmp = Float64(fma(y, -0.5, 0.918938533204673) - x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e+24], t$95$0, If[LessEqual[y, 1300000000.0], N[(N[(y * -0.5 + 0.918938533204673), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.9999999999999999e24 or 1.3e9 < 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

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -5.9999999999999999e24 < y < 1.3e9

   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%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \mathbf{elif}\;y \leq 1300000000:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, 0.918938533204673\right) - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1300000000:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -6e+24)
     t_0
     (if (<= y 1300000000.0) (- 0.918938533204673 (fma y 0.5 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -6e+24) {
		tmp = t_0;
	} else if (y <= 1300000000.0) {
		tmp = 0.918938533204673 - fma(y, 0.5, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -0.5))
	tmp = 0.0
	if (y <= -6e+24)
		tmp = t_0;
	elseif (y <= 1300000000.0)
		tmp = Float64(0.918938533204673 - fma(y, 0.5, x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e+24], t$95$0, If[LessEqual[y, 1300000000.0], N[(0.918938533204673 - N[(y * 0.5 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.9999999999999999e24 or 1.3e9 < 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

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -5.9999999999999999e24 < y < 1.3e9

   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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \mathbf{elif}\;y \leq 1300000000:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.7:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -1.3) t_0 (if (<= y 1.7) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.3) {
		tmp = t_0;
	} else if (y <= 1.7) {
		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 = y * (x + (-0.5d0))
    if (y <= (-1.3d0)) then
        tmp = t_0
    else if (y <= 1.7d0) 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 = y * (x + -0.5);
	double tmp;
	if (y <= -1.3) {
		tmp = t_0;
	} else if (y <= 1.7) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x + -0.5)
	tmp = 0
	if y <= -1.3:
		tmp = t_0
	elif y <= 1.7:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -0.5))
	tmp = 0.0
	if (y <= -1.3)
		tmp = t_0;
	elseif (y <= 1.7)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x + -0.5);
	tmp = 0.0;
	if (y <= -1.3)
		tmp = t_0;
	elseif (y <= 1.7)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.3], t$95$0, If[LessEqual[y, 1.7], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000004 or 1.69999999999999996 < 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

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -1.30000000000000004 < y < 1.69999999999999996

   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 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \mathbf{elif}\;y \leq 1.7:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+24) (* x y) (if (<= y 1.4) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+24) {
		tmp = x * y;
	} else if (y <= 1.4) {
		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 <= (-6d+24)) then
        tmp = x * y
    else if (y <= 1.4d0) 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 <= -6e+24) {
		tmp = x * y;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6e+24:
		tmp = x * y
	elif y <= 1.4:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+24)
		tmp = Float64(x * y);
	elseif (y <= 1.4)
		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 <= -6e+24)
		tmp = x * y;
	elseif (y <= 1.4)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+24], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.4], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.9999999999999999e24 or 1.3999999999999999 < 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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 53.7%

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

   if -5.9999999999999999e24 < y < 1.3999999999999999

   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 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2300000000000:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.92) (- x) (if (<= x 2300000000000.0) 0.918938533204673 (- x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = -x;
	} else if (x <= 2300000000000.0) {
		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 <= 2300000000000.0d0) 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 <= 2300000000000.0) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.92:
		tmp = -x
	elif x <= 2300000000000.0:
		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 <= 2300000000000.0)
		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 <= 2300000000000.0)
		tmp = 0.918938533204673;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.92], (-x), If[LessEqual[x, 2300000000000.0], 0.918938533204673, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 2.3e12 < 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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 44.5%

      \[\leadsto -x \]

   if -0.92000000000000004 < x < 2.3e12

   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 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.7%

      \[\leadsto 0.918938533204673 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 50.4% 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 46.7

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

5. Applied rewrites 46.7%

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

Alternative 11: 26.7% 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 46.7

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

5. Applied rewrites 46.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 24.1%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(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))