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

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 9

Speedup: 1.2×

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, 1.2× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y (+ x -0.5)) (- 0.918938533204673 x)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * (x + (-0.5d0))) + (0.918938533204673d0 - x)
end function

Java

public static double code(double x, double y) {
	return (y * (x + -0.5)) + (0.918938533204673 - x);
}

Python

def code(x, y):
	return (y * (x + -0.5)) + (0.918938533204673 - x)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (y * (x + -0.5)) + (0.918938533204673 - x);
end

Wolfram

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. associate-+l+ N/A

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

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

   17. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

   18. *-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

   19. --lowering--.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 73.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -920:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -920.0)
   (* y -0.5)
   (if (<= y 1.4e-7)
     (- 0.918938533204673 x)
     (if (<= y 3.6e+99) (* x (+ y -1.0)) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -920.0) {
		tmp = y * -0.5;
	} else if (y <= 1.4e-7) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 3.6e+99) {
		tmp = x * (y + -1.0);
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-920.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.4d-7) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 3.6d+99) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -920.0) {
		tmp = y * -0.5;
	} else if (y <= 1.4e-7) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 3.6e+99) {
		tmp = x * (y + -1.0);
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -920.0:
		tmp = y * -0.5
	elif y <= 1.4e-7:
		tmp = 0.918938533204673 - x
	elif y <= 3.6e+99:
		tmp = x * (y + -1.0)
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -920.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.4e-7)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 3.6e+99)
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -920.0)
		tmp = y * -0.5;
	elseif (y <= 1.4e-7)
		tmp = 0.918938533204673 - x;
	elseif (y <= 3.6e+99)
		tmp = x * (y + -1.0);
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -920.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.4e-7], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 3.6e+99], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -920 or 3.6000000000000002e99 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x - \frac{1}{2}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \frac{-1}{2}\right)\right) \]

      4. +-lowering-+.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.2%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{2}}\right) \]

   10. Simplified 63.2%

       \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -920 < y < 1.4000000000000001e-7

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]
   6. Step-by-step derivation

      1. --lowering--.f64 98.2%

         \[\leadsto \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right) \]

   7. Simplified 98.2%

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

   if 1.4000000000000001e-7 < y < 3.6000000000000002e99

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y + -1\right)\right) \]

      4. +-lowering-+.f64 62.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   7. Simplified 62.2%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+98}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -16.0)
   (* y -0.5)
   (if (<= y 1.4)
     (- 0.918938533204673 x)
     (if (<= y 5e+98) (* y x) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -16.0) {
		tmp = y * -0.5;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 5e+98) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-16.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.4d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 5d+98) then
        tmp = y * x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -16.0) {
		tmp = y * -0.5;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 5e+98) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -16.0:
		tmp = y * -0.5
	elif y <= 1.4:
		tmp = 0.918938533204673 - x
	elif y <= 5e+98:
		tmp = y * x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -16.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.4)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 5e+98)
		tmp = Float64(y * x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -16.0)
		tmp = y * -0.5;
	elseif (y <= 1.4)
		tmp = 0.918938533204673 - x;
	elseif (y <= 5e+98)
		tmp = y * x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -16.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.4], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 5e+98], N[(y * x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -16 or 4.9999999999999998e98 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x - \frac{1}{2}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \frac{-1}{2}\right)\right) \]

      4. +-lowering-+.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.2%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{2}}\right) \]

   10. Simplified 63.2%

       \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -16 < y < 1.3999999999999999

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]
   6. Step-by-step derivation

      1. --lowering--.f64 97.8%

         \[\leadsto \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right) \]

   7. Simplified 97.8%

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

   if 1.3999999999999999 < y < 4.9999999999999998e98

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \color{blue}{\frac{918938533204673}{1000000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 88.7%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 52.1%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 52.1%

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

Alternative 4: 49.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+102}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.85)
   (* y -0.5)
   (if (<= y 1.85) 0.918938533204673 (if (<= y 2.5e+102) (* y x) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.85) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673;
	} else if (y <= 2.5e+102) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.85d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0
    else if (y <= 2.5d+102) then
        tmp = y * x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.85) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673;
	} else if (y <= 2.5e+102) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.85:
		tmp = y * -0.5
	elif y <= 1.85:
		tmp = 0.918938533204673
	elif y <= 2.5e+102:
		tmp = y * x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.85)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.85)
		tmp = 0.918938533204673;
	elseif (y <= 2.5e+102)
		tmp = Float64(y * x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.85)
		tmp = y * -0.5;
	elseif (y <= 1.85)
		tmp = 0.918938533204673;
	elseif (y <= 2.5e+102)
		tmp = y * x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.85], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.85], 0.918938533204673, If[LessEqual[y, 2.5e+102], N[(y * x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8500000000000001 or 2.5e102 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x - \frac{1}{2}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \frac{-1}{2}\right)\right) \]

      4. +-lowering-+.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.2%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{2}}\right) \]

   10. Simplified 63.2%

       \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -1.8500000000000001 < y < 1.8500000000000001

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \color{blue}{\frac{918938533204673}{1000000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 55.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000}} \]
   9. Step-by-step derivation

   10. Simplified 54.3%

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

   if 1.8500000000000001 < y < 2.5e102

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \color{blue}{\frac{918938533204673}{1000000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 88.7%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 52.1%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 52.1%

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

Alternative 5: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -2400000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1950000:\\ \;\;\;\;y \cdot \left(x + -0.5\right) + 0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -2400000.0)
     t_0
     (if (<= x 1950000.0) (+ (* y (+ x -0.5)) 0.918938533204673) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -2400000.0) {
		tmp = t_0;
	} else if (x <= 1950000.0) {
		tmp = (y * (x + -0.5)) + 0.918938533204673;
	} 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 * (y + (-1.0d0))
    if (x <= (-2400000.0d0)) then
        tmp = t_0
    else if (x <= 1950000.0d0) then
        tmp = (y * (x + (-0.5d0))) + 0.918938533204673d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -2400000.0) {
		tmp = t_0;
	} else if (x <= 1950000.0) {
		tmp = (y * (x + -0.5)) + 0.918938533204673;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + -1.0)
	tmp = 0
	if x <= -2400000.0:
		tmp = t_0
	elif x <= 1950000.0:
		tmp = (y * (x + -0.5)) + 0.918938533204673
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y + -1.0))
	tmp = 0.0
	if (x <= -2400000.0)
		tmp = t_0;
	elseif (x <= 1950000.0)
		tmp = Float64(Float64(y * Float64(x + -0.5)) + 0.918938533204673);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y + -1.0);
	tmp = 0.0;
	if (x <= -2400000.0)
		tmp = t_0;
	elseif (x <= 1950000.0)
		tmp = (y * (x + -0.5)) + 0.918938533204673;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2400000.0], t$95$0, If[LessEqual[x, 1950000.0], N[(N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e6 or 1.95e6 < x

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y + -1\right)\right) \]

      4. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   7. Simplified 99.9%

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

   if -2.4e6 < x < 1.95e6

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \color{blue}{\frac{918938533204673}{1000000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 98.8%

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

Alternative 6: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -0.72)
     t_0
     (if (<= x 0.85) (+ 0.918938533204673 (* y -0.5)) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -0.72) {
		tmp = t_0;
	} else if (x <= 0.85) {
		tmp = 0.918938533204673 + (y * -0.5);
	} 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 * (y + (-1.0d0))
    if (x <= (-0.72d0)) then
        tmp = t_0
    else if (x <= 0.85d0) then
        tmp = 0.918938533204673d0 + (y * (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -0.72) {
		tmp = t_0;
	} else if (x <= 0.85) {
		tmp = 0.918938533204673 + (y * -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + -1.0)
	tmp = 0
	if x <= -0.72:
		tmp = t_0
	elif x <= 0.85:
		tmp = 0.918938533204673 + (y * -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y + -1.0))
	tmp = 0.0
	if (x <= -0.72)
		tmp = t_0;
	elseif (x <= 0.85)
		tmp = Float64(0.918938533204673 + Float64(y * -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y + -1.0);
	tmp = 0.0;
	if (x <= -0.72)
		tmp = t_0;
	elseif (x <= 0.85)
		tmp = 0.918938533204673 + (y * -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.72], t$95$0, If[LessEqual[x, 0.85], N[(0.918938533204673 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.71999999999999997 or 0.849999999999999978 < x

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y + -1\right)\right) \]

      4. +-lowering-+.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   7. Simplified 98.9%

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

   if -0.71999999999999997 < x < 0.849999999999999978

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot y\right)}, \frac{918938533204673}{1000000000000000}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \frac{-1}{2}\right), \frac{918938533204673}{1000000000000000}\right) \]

      2. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{-1}{2}\right), \frac{918938533204673}{1000000000000000}\right) \]

   5. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.42:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -1.42) t_0 (if (<= y 1.15) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.42) {
		tmp = t_0;
	} else if (y <= 1.15) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x + (-0.5d0))
    if (y <= (-1.42d0)) then
        tmp = t_0
    else if (y <= 1.15d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.42) {
		tmp = t_0;
	} else if (y <= 1.15) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x + -0.5)
	tmp = 0
	if y <= -1.42:
		tmp = t_0
	elif y <= 1.15:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -0.5))
	tmp = 0.0
	if (y <= -1.42)
		tmp = t_0;
	elseif (y <= 1.15)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x + -0.5);
	tmp = 0.0;
	if (y <= -1.42)
		tmp = t_0;
	elseif (y <= 1.15)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.42], t$95$0, If[LessEqual[y, 1.15], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4199999999999999 or 1.1499999999999999 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x - \frac{1}{2}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \frac{-1}{2}\right)\right) \]

      4. +-lowering-+.f64 96.2%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right) \]

   7. Simplified 96.2%

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

   if -1.4199999999999999 < y < 1.1499999999999999

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]
   6. Step-by-step derivation

      1. --lowering--.f64 97.8%

         \[\leadsto \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right) \]

   7. Simplified 97.8%

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

Alternative 8: 49.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1.8500000000000001 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x - \frac{1}{2}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \frac{-1}{2}\right)\right) \]

      4. +-lowering-+.f64 96.2%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right) \]

   7. Simplified 96.2%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 56.7%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{2}}\right) \]

   10. Simplified 56.7%

       \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -1.8500000000000001 < y < 1.8500000000000001

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

      17. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

      19. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \color{blue}{\frac{918938533204673}{1000000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 55.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000}} \]
   9. Step-by-step derivation

   10. Simplified 54.3%

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

Alternative 9: 27.1% accurate, 11.0× 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. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. associate-+l+ N/A

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

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right) + y \cdot x\right), \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)}\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + y \cdot x\right), \left(\color{blue}{x} \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{+.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + x\right)\right), \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + \frac{918938533204673}{1000000000000000}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(x \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{918938533204673}{1000000000000000}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \color{blue}{x \cdot \left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{x}\right)\right) \]

   17. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - \color{blue}{1 \cdot x}\right)\right) \]

   18. *-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\frac{918938533204673}{1000000000000000} - x\right)\right) \]

   19. --lowering--.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\frac{918938533204673}{1000000000000000}, \color{blue}{x}\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \color{blue}{\frac{918938533204673}{1000000000000000}}\right) \]
6. Step-by-step derivation

7. Simplified 74.6%

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

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000}} \]
9. Step-by-step derivation

10. Simplified 30.7%

    \[\leadsto \color{blue}{0.918938533204673} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(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))