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

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 12

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, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 - \left(y \cdot 0.5 - x \cdot \left(y + -1\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0 - ((y * 0.5d0) - (x * (y + (-1.0d0))))
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673 - ((y * 0.5) - (x * (y + -1.0)));
}

Python

def code(x, y):
	return 0.918938533204673 - ((y * 0.5) - (x * (y + -1.0)))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673 - ((y * 0.5) - (x * (y + -1.0)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -48000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-67}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-188}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 18500000000:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -48000.0)
     t_0
     (if (<= x -9e-67)
       (- 0.918938533204673 x)
       (if (<= x 1.25e-188)
         (* y -0.5)
         (if (<= x 18500000000.0) (- 0.918938533204673 x) t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -48000.0) {
		tmp = t_0;
	} else if (x <= -9e-67) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 1.25e-188) {
		tmp = y * -0.5;
	} else if (x <= 18500000000.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y + (-1.0d0))
    if (x <= (-48000.0d0)) then
        tmp = t_0
    else if (x <= (-9d-67)) then
        tmp = 0.918938533204673d0 - x
    else if (x <= 1.25d-188) then
        tmp = y * (-0.5d0)
    else if (x <= 18500000000.0d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -48000.0) {
		tmp = t_0;
	} else if (x <= -9e-67) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 1.25e-188) {
		tmp = y * -0.5;
	} else if (x <= 18500000000.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + -1.0)
	tmp = 0
	if x <= -48000.0:
		tmp = t_0
	elif x <= -9e-67:
		tmp = 0.918938533204673 - x
	elif x <= 1.25e-188:
		tmp = y * -0.5
	elif x <= 18500000000.0:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y + -1.0))
	tmp = 0.0
	if (x <= -48000.0)
		tmp = t_0;
	elseif (x <= -9e-67)
		tmp = Float64(0.918938533204673 - x);
	elseif (x <= 1.25e-188)
		tmp = Float64(y * -0.5);
	elseif (x <= 18500000000.0)
		tmp = Float64(0.918938533204673 - x);
	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 <= -48000.0)
		tmp = t_0;
	elseif (x <= -9e-67)
		tmp = 0.918938533204673 - x;
	elseif (x <= 1.25e-188)
		tmp = y * -0.5;
	elseif (x <= 18500000000.0)
		tmp = 0.918938533204673 - x;
	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, -48000.0], t$95$0, If[LessEqual[x, -9e-67], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[x, 1.25e-188], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 18500000000.0], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -48000 or 1.85e10 < 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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 98.8%

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

   7. Simplified 98.8%

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

   if -48000 < x < -9.00000000000000031e-67 or 1.25e-188 < x < 1.85e10

   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 66.3%

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

   7. Simplified 66.3%

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

   if -9.00000000000000031e-67 < x < 1.25e-188

   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 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 67.6%

       \[\leadsto y \cdot \color{blue}{-0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -180000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -180000.0)
     t_0
     (if (<= y -5.6e-11)
       (* x (+ y -1.0))
       (if (<= y 1.4) (- 0.918938533204673 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -180000.0) {
		tmp = t_0;
	} else if (y <= -5.6e-11) {
		tmp = x * (y + -1.0);
	} else if (y <= 1.4) {
		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 <= (-180000.0d0)) then
        tmp = t_0
    else if (y <= (-5.6d-11)) then
        tmp = x * (y + (-1.0d0))
    else if (y <= 1.4d0) 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 <= -180000.0) {
		tmp = t_0;
	} else if (y <= -5.6e-11) {
		tmp = x * (y + -1.0);
	} else if (y <= 1.4) {
		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 <= -180000.0:
		tmp = t_0
	elif y <= -5.6e-11:
		tmp = x * (y + -1.0)
	elif y <= 1.4:
		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 <= -180000.0)
		tmp = t_0;
	elseif (y <= -5.6e-11)
		tmp = Float64(x * Float64(y + -1.0));
	elseif (y <= 1.4)
		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 <= -180000.0)
		tmp = t_0;
	elseif (y <= -5.6e-11)
		tmp = x * (y + -1.0);
	elseif (y <= 1.4)
		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, -180000.0], t$95$0, If[LessEqual[y, -5.6e-11], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e5 or 1.3999999999999999 < 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 98.9%

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

   7. Simplified 98.9%

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

   if -1.8e5 < y < -5.6e-11

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 100.0%

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

   7. Simplified 100.0%

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

   if -5.6e-11 < 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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 97.0%

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

   7. Simplified 97.0%

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

Alternative 4: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -145000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.55:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+208}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -145000.0)
   (* y -0.5)
   (if (<= y 1.55)
     (- 0.918938533204673 x)
     (if (<= y 3.8e+208) (* x y) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -145000.0) {
		tmp = y * -0.5;
	} else if (y <= 1.55) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 3.8e+208) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-145000.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.55d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 3.8d+208) then
        tmp = x * y
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -145000.0) {
		tmp = y * -0.5;
	} else if (y <= 1.55) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 3.8e+208) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -145000.0:
		tmp = y * -0.5
	elif y <= 1.55:
		tmp = 0.918938533204673 - x
	elif y <= 3.8e+208:
		tmp = x * y
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -145000.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.55)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 3.8e+208)
		tmp = Float64(x * y);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -145000.0)
		tmp = y * -0.5;
	elseif (y <= 1.55)
		tmp = 0.918938533204673 - x;
	elseif (y <= 3.8e+208)
		tmp = x * y;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -145000.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.55], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 3.8e+208], N[(x * y), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -145000 or 3.8000000000000002e208 < 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 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 61.1%

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

   if -145000 < y < 1.55000000000000004

   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 92.4%

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

   7. Simplified 92.4%

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

   if 1.55000000000000004 < y < 3.8000000000000002e208

   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 97.7%

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

   7. Simplified 97.7%

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

   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 54.1%

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

   10. Simplified 54.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -145000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.55:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+208}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-190}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e-8)
   (* x y)
   (if (<= x 1.85e-190)
     (* y -0.5)
     (if (<= x 0.72) 0.918938533204673 (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e-8) {
		tmp = x * y;
	} else if (x <= 1.85e-190) {
		tmp = y * -0.5;
	} else if (x <= 0.72) {
		tmp = 0.918938533204673;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.5d-8)) then
        tmp = x * y
    else if (x <= 1.85d-190) then
        tmp = y * (-0.5d0)
    else if (x <= 0.72d0) then
        tmp = 0.918938533204673d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.5e-8) {
		tmp = x * y;
	} else if (x <= 1.85e-190) {
		tmp = y * -0.5;
	} else if (x <= 0.72) {
		tmp = 0.918938533204673;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.5e-8:
		tmp = x * y
	elif x <= 1.85e-190:
		tmp = y * -0.5
	elif x <= 0.72:
		tmp = 0.918938533204673
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e-8)
		tmp = Float64(x * y);
	elseif (x <= 1.85e-190)
		tmp = Float64(y * -0.5);
	elseif (x <= 0.72)
		tmp = 0.918938533204673;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.5e-8)
		tmp = x * y;
	elseif (x <= 1.85e-190)
		tmp = y * -0.5;
	elseif (x <= 0.72)
		tmp = 0.918938533204673;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e-8], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.85e-190], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 0.72], 0.918938533204673, N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.49999999999999935e-8 or 0.71999999999999997 < 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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 50.7%

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

   7. Simplified 50.7%

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

   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 50.0%

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

   10. Simplified 50.0%

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

   if -8.49999999999999935e-8 < x < 1.8500000000000001e-190

   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 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 64.7%

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

   if 1.8500000000000001e-190 < x < 0.71999999999999997

   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 64.5%

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

   7. Simplified 64.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 64.0%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-190}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\left(0.918938533204673 - x\right) + x \cdot y\\ \mathbf{elif}\;x \leq 19000000000:\\ \;\;\;\;\left(0.918938533204673 - x\right) + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.5)
   (+ (- 0.918938533204673 x) (* x y))
   (if (<= x 19000000000.0)
     (+ (- 0.918938533204673 x) (* y -0.5))
     (* x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.5) {
		tmp = (0.918938533204673 - x) + (x * y);
	} else if (x <= 19000000000.0) {
		tmp = (0.918938533204673 - x) + (y * -0.5);
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.5d0)) then
        tmp = (0.918938533204673d0 - x) + (x * y)
    else if (x <= 19000000000.0d0) then
        tmp = (0.918938533204673d0 - x) + (y * (-0.5d0))
    else
        tmp = x * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.5) {
		tmp = (0.918938533204673 - x) + (x * y);
	} else if (x <= 19000000000.0) {
		tmp = (0.918938533204673 - x) + (y * -0.5);
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.5:
		tmp = (0.918938533204673 - x) + (x * y)
	elif x <= 19000000000.0:
		tmp = (0.918938533204673 - x) + (y * -0.5)
	else:
		tmp = x * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.5)
		tmp = Float64(Float64(0.918938533204673 - x) + Float64(x * y));
	elseif (x <= 19000000000.0)
		tmp = Float64(Float64(0.918938533204673 - x) + Float64(y * -0.5));
	else
		tmp = Float64(x * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.5)
		tmp = (0.918938533204673 - x) + (x * y);
	elseif (x <= 19000000000.0)
		tmp = (0.918938533204673 - x) + (y * -0.5);
	else
		tmp = x * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.5], N[(N[(0.918938533204673 - x), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 19000000000.0], N[(N[(0.918938533204673 - x), $MachinePrecision] + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   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

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

   7. Simplified 98.3%

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

   if -0.5 < x < 1.9e10

   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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.5%

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

   7. Simplified 99.5%

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

   if 1.9e10 < 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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\left(0.918938533204673 - x\right) + x \cdot y\\ \mathbf{elif}\;x \leq 19000000000:\\ \;\;\;\;\left(0.918938533204673 - x\right) + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -38000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 18500000000:\\ \;\;\;\;\left(0.918938533204673 - x\right) + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -38000.0)
     t_0
     (if (<= x 18500000000.0) (+ (- 0.918938533204673 x) (* y -0.5)) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -38000.0) {
		tmp = t_0;
	} else if (x <= 18500000000.0) {
		tmp = (0.918938533204673 - x) + (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 <= (-38000.0d0)) then
        tmp = t_0
    else if (x <= 18500000000.0d0) then
        tmp = (0.918938533204673d0 - x) + (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 <= -38000.0) {
		tmp = t_0;
	} else if (x <= 18500000000.0) {
		tmp = (0.918938533204673 - x) + (y * -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + -1.0)
	tmp = 0
	if x <= -38000.0:
		tmp = t_0
	elif x <= 18500000000.0:
		tmp = (0.918938533204673 - x) + (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 <= -38000.0)
		tmp = t_0;
	elseif (x <= 18500000000.0)
		tmp = Float64(Float64(0.918938533204673 - x) + 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 <= -38000.0)
		tmp = t_0;
	elseif (x <= 18500000000.0)
		tmp = (0.918938533204673 - x) + (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, -38000.0], t$95$0, If[LessEqual[x, 18500000000.0], N[(N[(0.918938533204673 - x), $MachinePrecision] + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -38000 or 1.85e10 < 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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 98.8%

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

   7. Simplified 98.8%

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

   if -38000 < x < 1.85e10

   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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -38000:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 18500000000:\\ \;\;\;\;\left(0.918938533204673 - x\right) + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.52:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -0.75)
     t_0
     (if (<= x 0.52) (+ 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 <= -0.75) {
		tmp = t_0;
	} else if (x <= 0.52) {
		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 <= (-0.75d0)) then
        tmp = t_0
    else if (x <= 0.52d0) 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 <= -0.75) {
		tmp = t_0;
	} else if (x <= 0.52) {
		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 <= -0.75:
		tmp = t_0
	elif x <= 0.52:
		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 <= -0.75)
		tmp = t_0;
	elseif (x <= 0.52)
		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 <= -0.75)
		tmp = t_0;
	elseif (x <= 0.52)
		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, -0.75], t$95$0, If[LessEqual[x, 0.52], N[(0.918938533204673 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.75 or 0.52000000000000002 < 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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]
   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 97.9%

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

   7. Simplified 97.9%

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

   if -0.75 < x < 0.52000000000000002

   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 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 0.52:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 0.95:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.75) (* y -0.5) (if (<= y 0.95) 0.918938533204673 (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.75) {
		tmp = y * -0.5;
	} else if (y <= 0.95) {
		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 <= (-0.75d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 0.95d0) 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 <= -0.75) {
		tmp = y * -0.5;
	} else if (y <= 0.95) {
		tmp = 0.918938533204673;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.75:
		tmp = y * -0.5
	elif y <= 0.95:
		tmp = 0.918938533204673
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.75)
		tmp = Float64(y * -0.5);
	elseif (y <= 0.95)
		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 <= -0.75)
		tmp = y * -0.5;
	elseif (y <= 0.95)
		tmp = 0.918938533204673;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.75], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 0.95], 0.918938533204673, N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.75 or 0.94999999999999996 < 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 99.9%

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

   3. Simplified 99.9%

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

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

   10. Simplified 53.3%

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

   if -0.75 < y < 0.94999999999999996

   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 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 50.3%

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

Alternative 10: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative N/A

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

   2. associate-+l- N/A

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

   3. --lowering--.f64 N/A

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

   4. --lowering--.f64 N/A

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

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

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

   6. +-lowering-+.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 11: 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 12: 26.3% 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 y around 0

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

   1. --lowering--.f64 45.8%

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

7. Simplified 45.8%

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

8. Taylor expanded in x around 0

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

10. Simplified 24.5%

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

Reproduce

herbie shell --seed 2024155 
(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))