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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

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, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

   9. associate-+l- N/A

      \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

   10. neg-sub0 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+59} \lor \neg \left(y \leq 3 \cdot 10^{+128}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3e-5)
   (fma -0.5 y 0.918938533204673)
   (if (<= y 1.8)
     (- 0.918938533204673 x)
     (if (or (<= y 1.75e+59) (not (<= y 3e+128))) (* x y) (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.3e-5) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 1.75e+59) || !(y <= 3e+128)) {
		tmp = x * y;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.3e-5)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (y <= 1.8)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 1.75e+59) || !(y <= 3e+128))
		tmp = Float64(x * y);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.3e-5], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[y, 1.8], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 1.75e+59], N[Not[LessEqual[y, 3e+128]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.29999999999999992e-5

   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 57.7

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

   5. Applied rewrites 57.7%

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

   if -1.29999999999999992e-5 < y < 1.80000000000000004

   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 98.0

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

   5. Applied rewrites 98.0%

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

   if 1.80000000000000004 < y < 1.75e59 or 2.9999999999999998e128 < y

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 94.6

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.5%

       \[\leadsto -0.5 \cdot y \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 64.5%

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

   if 1.75e59 < y < 2.9999999999999998e128

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 91.0%

       \[\leadsto -0.5 \cdot y \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+59} \lor \neg \left(y \leq 3 \cdot 10^{+128}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -210:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+59} \lor \neg \left(y \leq 3 \cdot 10^{+128}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -210.0)
   (* -0.5 y)
   (if (<= y 1.8)
     (- 0.918938533204673 x)
     (if (or (<= y 1.75e+59) (not (<= y 3e+128))) (* x y) (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -210.0) {
		tmp = -0.5 * y;
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 1.75e+59) || !(y <= 3e+128)) {
		tmp = x * y;
	} 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 <= (-210.0d0)) then
        tmp = (-0.5d0) * y
    else if (y <= 1.8d0) then
        tmp = 0.918938533204673d0 - x
    else if ((y <= 1.75d+59) .or. (.not. (y <= 3d+128))) then
        tmp = x * y
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -210.0) {
		tmp = -0.5 * y;
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 1.75e+59) || !(y <= 3e+128)) {
		tmp = x * y;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -210.0:
		tmp = -0.5 * y
	elif y <= 1.8:
		tmp = 0.918938533204673 - x
	elif (y <= 1.75e+59) or not (y <= 3e+128):
		tmp = x * y
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -210.0)
		tmp = Float64(-0.5 * y);
	elseif (y <= 1.8)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 1.75e+59) || !(y <= 3e+128))
		tmp = Float64(x * y);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -210.0)
		tmp = -0.5 * y;
	elseif (y <= 1.8)
		tmp = 0.918938533204673 - x;
	elseif ((y <= 1.75e+59) || ~((y <= 3e+128)))
		tmp = x * y;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -210.0], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 1.8], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 1.75e+59], N[Not[LessEqual[y, 3e+128]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -210 or 1.75e59 < y < 2.9999999999999998e128

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 98.1

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 63.2%

       \[\leadsto -0.5 \cdot y \]

   if -210 < y < 1.80000000000000004

   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 98.0

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

   5. Applied rewrites 98.0%

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

   if 1.80000000000000004 < y < 1.75e59 or 2.9999999999999998e128 < y

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 94.6

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.5%

       \[\leadsto -0.5 \cdot y \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 64.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -210:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+59} \lor \neg \left(y \leq 3 \cdot 10^{+128}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10000000000 \lor \neg \left(x \leq 38000\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x - 0.5\right) \cdot y + 0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -10000000000.0) (not (<= x 38000.0)))
   (* (- y 1.0) x)
   (+ (* (- x 0.5) y) 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -10000000000.0) || !(x <= 38000.0)) {
		tmp = (y - 1.0) * x;
	} else {
		tmp = ((x - 0.5) * y) + 0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-10000000000.0d0)) .or. (.not. (x <= 38000.0d0))) then
        tmp = (y - 1.0d0) * x
    else
        tmp = ((x - 0.5d0) * y) + 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -10000000000.0) || !(x <= 38000.0)) {
		tmp = (y - 1.0) * x;
	} else {
		tmp = ((x - 0.5) * y) + 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -10000000000.0) or not (x <= 38000.0):
		tmp = (y - 1.0) * x
	else:
		tmp = ((x - 0.5) * y) + 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -10000000000.0) || !(x <= 38000.0))
		tmp = Float64(Float64(y - 1.0) * x);
	else
		tmp = Float64(Float64(Float64(x - 0.5) * y) + 0.918938533204673);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -10000000000.0) || ~((x <= 38000.0)))
		tmp = (y - 1.0) * x;
	else
		tmp = ((x - 0.5) * y) + 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -10000000000.0], N[Not[LessEqual[x, 38000.0]], $MachinePrecision]], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision] + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e10 or 38000 < 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%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.1%

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

   if -1e10 < x < 38000

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      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 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000000 \lor \neg \left(x \leq 38000\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x - 0.5\right) \cdot y + 0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.72) (not (<= x 0.92)))
   (* (- y 1.0) x)
   (fma -0.5 y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.92)) {
		tmp = (y - 1.0) * x;
	} else {
		tmp = fma(-0.5, y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.72) || !(x <= 0.92))
		tmp = Float64(Float64(y - 1.0) * x);
	else
		tmp = fma(-0.5, y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.72], N[Not[LessEqual[x, 0.92]], $MachinePrecision]], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(-0.5 * y + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.71999999999999997 or 0.92000000000000004 < 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%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 98.6%

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

   if -0.71999999999999997 < x < 0.92000000000000004

   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 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.5) (not (<= y 1.8)))
   (* (- x 0.5) y)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.5) || !(y <= 1.8)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.5d0)) .or. (.not. (y <= 1.8d0))) then
        tmp = (x - 0.5d0) * y
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.5) || !(y <= 1.8)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.5) or not (y <= 1.8):
		tmp = (x - 0.5) * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.5) || !(y <= 1.8))
		tmp = Float64(Float64(x - 0.5) * y);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.5) || ~((y <= 1.8)))
		tmp = (x - 0.5) * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.5], N[Not[LessEqual[y, 1.8]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5 or 1.80000000000000004 < y

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 96.6

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

   7. Applied rewrites 96.6%

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

   if -1.5 < y < 1.80000000000000004

   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -340000000 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -340000000.0) (not (<= y 1.8)))
   (* x y)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -340000000.0) || !(y <= 1.8)) {
		tmp = x * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-340000000.0d0)) .or. (.not. (y <= 1.8d0))) then
        tmp = x * y
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -340000000.0) || !(y <= 1.8)) {
		tmp = x * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -340000000.0) or not (y <= 1.8):
		tmp = x * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -340000000.0) || !(y <= 1.8))
		tmp = Float64(x * y);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -340000000.0) || ~((y <= 1.8)))
		tmp = x * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -340000000.0], N[Not[LessEqual[y, 1.8]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e8 or 1.80000000000000004 < y

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y - 1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 49.2%

       \[\leadsto -0.5 \cdot y \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 49.8%

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

   if -3.4e8 < y < 1.80000000000000004

   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 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -340000000 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4150000000 \lor \neg \left(x \leq 1.2 \cdot 10^{-5}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4150000000.0) (not (<= x 1.2e-5))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4150000000.0) || !(x <= 1.2e-5)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-4150000000.0d0)) .or. (.not. (x <= 1.2d-5))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4150000000.0) || !(x <= 1.2e-5)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4150000000.0) or not (x <= 1.2e-5):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4150000000.0) || !(x <= 1.2e-5))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4150000000.0) || ~((x <= 1.2e-5)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4150000000.0], N[Not[LessEqual[x, 1.2e-5]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -4.15e9 or 1.2e-5 < 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 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 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.2%

      \[\leadsto -x \]

   if -4.15e9 < x < 1.2e-5

   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 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.2%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4150000000 \lor \neg \left(x \leq 1.2 \cdot 10^{-5}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

6. Final simplification 100.0%

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

Alternative 10: 51.4% 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 54.6

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

5. Applied rewrites 54.6%

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

6. Final simplification 54.6%

   \[\leadsto 0.918938533204673 - x \]
7. Add Preprocessing

Alternative 11: 26.6% 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 54.6

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

5. Applied rewrites 54.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 31.1%

   \[\leadsto 0.918938533204673 \]

9. Final simplification 31.1%

   \[\leadsto 0.918938533204673 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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))