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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 10

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: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+120}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-8}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+272}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e+120)
   (* x y)
   (if (<= x -8.2e-8)
     (- 0.918938533204673 x)
     (if (<= x 4.5e-8)
       (fma -0.5 y 0.918938533204673)
       (if (<= x 7.4e+272) (- 0.918938533204673 x) (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e+120) {
		tmp = x * y;
	} else if (x <= -8.2e-8) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 4.5e-8) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (x <= 7.4e+272) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e+120)
		tmp = Float64(x * y);
	elseif (x <= -8.2e-8)
		tmp = Float64(0.918938533204673 - x);
	elseif (x <= 4.5e-8)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (x <= 7.4e+272)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e+120], N[(x * y), $MachinePrecision], If[LessEqual[x, -8.2e-8], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[x, 4.5e-8], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[x, 7.4e+272], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999991e120 or 7.3999999999999998e272 < x

   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 62.4

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

   7. Applied rewrites 62.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 3.5%

       \[\leadsto -0.5 \cdot y \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 62.4%

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

   if -1.59999999999999991e120 < x < -8.20000000000000063e-8 or 4.49999999999999993e-8 < x < 7.3999999999999998e272

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if -8.20000000000000063e-8 < x < 4.49999999999999993e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-9} \lor \neg \left(y \leq 9.8 \cdot 10^{-25}\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y + 0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.9e-9) (not (<= y 9.8e-25)))
   (+ (* (- x 0.5) y) 0.918938533204673)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e-9) || !(y <= 9.8e-25)) {
		tmp = ((x - 0.5) * y) + 0.918938533204673;
	} 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 <= (-1.9d-9)) .or. (.not. (y <= 9.8d-25))) then
        tmp = ((x - 0.5d0) * y) + 0.918938533204673d0
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e-9) || !(y <= 9.8e-25)) {
		tmp = ((x - 0.5) * y) + 0.918938533204673;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.9e-9) or not (y <= 9.8e-25):
		tmp = ((x - 0.5) * y) + 0.918938533204673
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.9e-9) || !(y <= 9.8e-25))
		tmp = Float64(Float64(Float64(x - 0.5) * y) + 0.918938533204673);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.9e-9) || ~((y <= 9.8e-25)))
		tmp = ((x - 0.5) * y) + 0.918938533204673;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.9e-9], N[Not[LessEqual[y, 9.8e-25]], $MachinePrecision]], N[(N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision] + 0.918938533204673), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.90000000000000006e-9 or 9.7999999999999998e-25 < 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. 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 99.0

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

   5. Applied rewrites 99.0%

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

   if -1.90000000000000006e-9 < y < 9.7999999999999998e-25

   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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-9} \lor \neg \left(y \leq 9.8 \cdot 10^{-25}\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y + 0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.55) (not (<= y 1.1)))
   (* (- x 0.5) y)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.55) || !(y <= 1.1)) {
		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 <= (-1.55d0)) .or. (.not. (y <= 1.1d0))) 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 <= -1.55) || !(y <= 1.1)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.55) or not (y <= 1.1):
		tmp = (x - 0.5) * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.55) || !(y <= 1.1))
		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 <= -1.55) || ~((y <= 1.1)))
		tmp = (x - 0.5) * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.55], N[Not[LessEqual[y, 1.1]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000004 or 1.1000000000000001 < 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 99.1

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

   7. Applied rewrites 99.1%

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

   if -1.55000000000000004 < y < 1.1000000000000001

   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 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14 \lor \neg \left(y \leq 1.85\right):\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -14.0) (not (<= y 1.85))) (* -0.5 y) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -14.0) || !(y <= 1.85)) {
		tmp = -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 <= (-14.0d0)) .or. (.not. (y <= 1.85d0))) then
        tmp = (-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 <= -14.0) || !(y <= 1.85)) {
		tmp = -0.5 * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -14.0) or not (y <= 1.85):
		tmp = -0.5 * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -14.0) || !(y <= 1.85))
		tmp = Float64(-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 <= -14.0) || ~((y <= 1.85)))
		tmp = -0.5 * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -14.0], N[Not[LessEqual[y, 1.85]], $MachinePrecision]], N[(-0.5 * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -14 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. 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 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.8%

       \[\leadsto -0.5 \cdot y \]

   if -14 < 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 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14 \lor \neg \left(y \leq 1.85\right):\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+15} \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e+15) (not (<= y 1.1))) (* x y) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+15) || !(y <= 1.1)) {
		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 <= (-1.6d+15)) .or. (.not. (y <= 1.1d0))) 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 <= -1.6e+15) || !(y <= 1.1)) {
		tmp = x * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6e+15) or not (y <= 1.1):
		tmp = x * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6e+15) || !(y <= 1.1))
		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 <= -1.6e+15) || ~((y <= 1.1)))
		tmp = x * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6e+15], N[Not[LessEqual[y, 1.1]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e15 or 1.1000000000000001 < 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 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.4%

       \[\leadsto -0.5 \cdot y \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 39.8%

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

   if -1.6e15 < y < 1.1000000000000001

   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 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+15} \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 10^{-15}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.92) (not (<= x 1e-15))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.92) || !(x <= 1e-15)) {
		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 <= (-0.92d0)) .or. (.not. (x <= 1d-15))) 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 <= -0.92) || !(x <= 1e-15)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.92) or not (x <= 1e-15):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.92) || !(x <= 1e-15))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.92) || ~((x <= 1e-15)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.92], N[Not[LessEqual[x, 1e-15]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 1.0000000000000001e-15 < 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 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.5%

      \[\leadsto -x \]

   if -0.92000000000000004 < x < 1.0000000000000001e-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 y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 46.6

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.3%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 10^{-15}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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. Add Preprocessing

Alternative 9: 49.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 50.2

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

5. Applied rewrites 50.2%

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

Alternative 10: 26.1% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

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

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 50.2

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

5. Applied rewrites 50.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.1%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024324 
(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))