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

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

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

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

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

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. associate-+r- N/A

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

   8. unsub-neg N/A

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

   9. unsub-neg N/A

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

   10. distribute-rgt-out-- N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. *-rgt-identity N/A

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

   14. rgt-mult-inverse N/A

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

   15. associate-*r/ N/A

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

   16. *-rgt-identity N/A

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

   17. associate-/l* N/A

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

   18. associate-*l/ N/A

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

   19. neg-mul-1 N/A

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

   20. distribute-neg-frac2 N/A

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

   21. mul-1-neg N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-10}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+248}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.3e-10)
   (- 0.918938533204673 x)
   (if (<= x 2.1e-5)
     (fma -0.5 y 0.918938533204673)
     (if (<= x 1.06e+248) (- 0.918938533204673 x) (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.3e-10) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 2.1e-5) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (x <= 1.06e+248) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.3e-10)
		tmp = Float64(0.918938533204673 - x);
	elseif (x <= 2.1e-5)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (x <= 1.06e+248)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.3e-10], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[x, 2.1e-5], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[x, 1.06e+248], N[(0.918938533204673 - x), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.30000000000000014e-10 or 2.09999999999999988e-5 < x < 1.06e248

   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 58.6

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

   5. Applied rewrites 58.6%

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

   if -4.30000000000000014e-10 < x < 2.09999999999999988e-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 99.1

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

   5. Applied rewrites 99.1%

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

   if 1.06e248 < 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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r- N/A

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

      8. unsub-neg N/A

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

      9. unsub-neg N/A

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

      10. distribute-rgt-out-- N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-*l/ N/A

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

      19. neg-mul-1 N/A

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

      20. distribute-neg-frac2 N/A

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

      21. mul-1-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 83.1

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 83.1%

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

Alternative 3: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, -x\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-10}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;\left(x - 0.5\right) \cdot y + 0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6e-15)
   (fma (- x 0.5) y (- x))
   (if (<= y 4.5e-10)
     (- 0.918938533204673 x)
     (+ (* (- x 0.5) y) 0.918938533204673))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.6e-15) {
		tmp = fma((x - 0.5), y, -x);
	} else if (y <= 4.5e-10) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = ((x - 0.5) * y) + 0.918938533204673;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.6e-15)
		tmp = fma(Float64(x - 0.5), y, Float64(-x));
	elseif (y <= 4.5e-10)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(Float64(Float64(x - 0.5) * y) + 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.6e-15], N[(N[(x - 0.5), $MachinePrecision] * y + (-x)), $MachinePrecision], If[LessEqual[y, 4.5e-10], N[(0.918938533204673 - x), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision] + 0.918938533204673), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6e-15

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r- N/A

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

      8. unsub-neg N/A

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

      9. unsub-neg N/A

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

      10. distribute-rgt-out-- N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-*l/ N/A

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

      19. neg-mul-1 N/A

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

      20. distribute-neg-frac2 N/A

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

      21. mul-1-neg N/A

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

   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

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

   8. Applied rewrites 99.7%

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

   if -1.6e-15 < y < 4.5e-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 y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 4.5e-10 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 97.8

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

   5. Applied rewrites 97.8%

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

Alternative 4: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x - 0.5, y, -x\right)\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- x 0.5) y (- x))))
   (if (<= y -1.6e-15) t_0 (if (<= y 6.2e-6) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((x - 0.5), y, -x);
	double tmp;
	if (y <= -1.6e-15) {
		tmp = t_0;
	} else if (y <= 6.2e-6) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(x - 0.5), y, Float64(-x))
	tmp = 0.0
	if (y <= -1.6e-15)
		tmp = t_0;
	elseif (y <= 6.2e-6)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - 0.5), $MachinePrecision] * y + (-x)), $MachinePrecision]}, If[LessEqual[y, -1.6e-15], t$95$0, If[LessEqual[y, 6.2e-6], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e-15 or 6.1999999999999999e-6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r- N/A

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

      8. unsub-neg N/A

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

      9. unsub-neg N/A

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

      10. distribute-rgt-out-- N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-*l/ N/A

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

      19. neg-mul-1 N/A

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

      20. distribute-neg-frac2 N/A

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

      21. mul-1-neg N/A

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

   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

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

   8. Applied rewrites 98.9%

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

   if -1.6e-15 < y < 6.1999999999999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 5: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45)
   (* (- x 0.5) y)
   (if (<= y 1.85) (- 0.918938533204673 x) (fma y x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45) {
		tmp = (x - 0.5) * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = fma(y, x, (-0.5 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45)
		tmp = Float64(Float64(x - 0.5) * y);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = fma(y, x, Float64(-0.5 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45], N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(y * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.44999999999999996

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r- N/A

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

      8. unsub-neg N/A

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

      9. unsub-neg N/A

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

      10. distribute-rgt-out-- N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-*l/ N/A

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

      19. neg-mul-1 N/A

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

      20. distribute-neg-frac2 N/A

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

      21. mul-1-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 99.1

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

   8. Applied rewrites 99.1%

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

   if -1.44999999999999996 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 1.8500000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r- N/A

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

      8. unsub-neg N/A

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

      9. unsub-neg N/A

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

      10. distribute-rgt-out-- N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-*l/ N/A

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

      19. neg-mul-1 N/A

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

      20. distribute-neg-frac2 N/A

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

      21. mul-1-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 97.4

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

   8. Applied rewrites 97.4%

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

   10. Applied rewrites 97.5%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 0.5\right) \cdot y\\ \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x 0.5) y)))
   (if (<= y -1.45) t_0 (if (<= y 1.85) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (x - 0.5) * y;
	double tmp;
	if (y <= -1.45) {
		tmp = t_0;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - 0.5d0) * y
    if (y <= (-1.45d0)) then
        tmp = t_0
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - 0.5) * y;
	double tmp;
	if (y <= -1.45) {
		tmp = t_0;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - 0.5) * y
	tmp = 0
	if y <= -1.45:
		tmp = t_0
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - 0.5) * y)
	tmp = 0.0
	if (y <= -1.45)
		tmp = t_0;
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - 0.5) * y;
	tmp = 0.0;
	if (y <= -1.45)
		tmp = t_0;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.45], t$95$0, If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999996 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r- N/A

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

      8. unsub-neg N/A

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

      9. unsub-neg N/A

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

      10. distribute-rgt-out-- N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-*l/ N/A

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

      19. neg-mul-1 N/A

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

      20. distribute-neg-frac2 N/A

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

      21. mul-1-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 98.3

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

   8. Applied rewrites 98.3%

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

   if -1.44999999999999996 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 7: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -400:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -400.0)
   (* -0.5 y)
   (if (<= y 1.85) (- 0.918938533204673 x) (* -0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -400.0) {
		tmp = -0.5 * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-400.0d0)) then
        tmp = (-0.5d0) * y
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -400.0) {
		tmp = -0.5 * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -400.0:
		tmp = -0.5 * y
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -400.0)
		tmp = Float64(-0.5 * y);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -400.0)
		tmp = -0.5 * y;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -400.0], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -400 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r- N/A

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

      8. unsub-neg N/A

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

      9. unsub-neg N/A

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

      10. distribute-rgt-out-- N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-*l/ N/A

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

      19. neg-mul-1 N/A

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

      20. distribute-neg-frac2 N/A

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

      21. mul-1-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 55.2%

       \[\leadsto -0.5 \cdot y \]

   if -400 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 8: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1050000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1050000000000.0)
   (* y x)
   (if (<= y 1.85) (- 0.918938533204673 x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1050000000000.0) {
		tmp = y * x;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1050000000000.0d0)) then
        tmp = y * x
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1050000000000.0) {
		tmp = y * x;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1050000000000.0:
		tmp = y * x
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1050000000000.0)
		tmp = Float64(y * x);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1050000000000.0)
		tmp = y * x;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1050000000000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e12 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. associate-+r- N/A

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

      8. unsub-neg N/A

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

      9. unsub-neg N/A

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

      10. distribute-rgt-out-- N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-*l/ N/A

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

      19. neg-mul-1 N/A

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

      20. distribute-neg-frac2 N/A

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

      21. mul-1-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 98.4

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 44.3%

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

   if -1.05e12 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 9: 49.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.92d0)) then
        tmp = -x
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= -0.92:
		tmp = -x
	elif x <= 0.92:
		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 <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 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 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 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.8%

      \[\leadsto -x \]

   if -0.92000000000000004 < 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 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 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.0%

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

Alternative 10: 50.2% 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 51.8

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

5. Applied rewrites 51.8%

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

Alternative 11: 26.3% 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 51.8

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

5. Applied rewrites 51.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.8%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(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))