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

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 10

Speedup: 1.0×

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

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

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]

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%

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

Alternative 2: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -0.085:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-186}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-170}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -0.085)
     t_0
     (if (<= x -7e-186)
       (* y -0.5)
       (if (<= x 6.4e-170)
         0.918938533204673
         (if (<= x 0.5) (* y -0.5) t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -0.085) {
		tmp = t_0;
	} else if (x <= -7e-186) {
		tmp = y * -0.5;
	} else if (x <= 6.4e-170) {
		tmp = 0.918938533204673;
	} else if (x <= 0.5) {
		tmp = 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 <= (-0.085d0)) then
        tmp = t_0
    else if (x <= (-7d-186)) then
        tmp = y * (-0.5d0)
    else if (x <= 6.4d-170) then
        tmp = 0.918938533204673d0
    else if (x <= 0.5d0) then
        tmp = 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 <= -0.085) {
		tmp = t_0;
	} else if (x <= -7e-186) {
		tmp = y * -0.5;
	} else if (x <= 6.4e-170) {
		tmp = 0.918938533204673;
	} else if (x <= 0.5) {
		tmp = y * -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + -1.0)
	tmp = 0
	if x <= -0.085:
		tmp = t_0
	elif x <= -7e-186:
		tmp = y * -0.5
	elif x <= 6.4e-170:
		tmp = 0.918938533204673
	elif x <= 0.5:
		tmp = 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 <= -0.085)
		tmp = t_0;
	elseif (x <= -7e-186)
		tmp = Float64(y * -0.5);
	elseif (x <= 6.4e-170)
		tmp = 0.918938533204673;
	elseif (x <= 0.5)
		tmp = 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 <= -0.085)
		tmp = t_0;
	elseif (x <= -7e-186)
		tmp = y * -0.5;
	elseif (x <= 6.4e-170)
		tmp = 0.918938533204673;
	elseif (x <= 0.5)
		tmp = 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, -0.085], t$95$0, If[LessEqual[x, -7e-186], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 6.4e-170], 0.918938533204673, If[LessEqual[x, 0.5], N[(y * -0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0850000000000000061 or 0.5 < x

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

      10. distribute-lft-out N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

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

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

      18. *-lft-identity N/A

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

      19. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 98.5%

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

   7. Simplified 98.5%

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

   if -0.0850000000000000061 < x < -6.99999999999999978e-186 or 6.3999999999999999e-170 < 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 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 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.4%

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

   10. Simplified 63.4%

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

   if -6.99999999999999978e-186 < x < 6.3999999999999999e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 61.8%

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

Alternative 3: 49.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1400:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-189}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-168}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1400.0)
   (* x y)
   (if (<= x -1.8e-189)
     (* y -0.5)
     (if (<= x 1.35e-168)
       0.918938533204673
       (if (<= x 0.5) (* y -0.5) (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1400.0) {
		tmp = x * y;
	} else if (x <= -1.8e-189) {
		tmp = y * -0.5;
	} else if (x <= 1.35e-168) {
		tmp = 0.918938533204673;
	} else if (x <= 0.5) {
		tmp = y * -0.5;
	} 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 <= (-1400.0d0)) then
        tmp = x * y
    else if (x <= (-1.8d-189)) then
        tmp = y * (-0.5d0)
    else if (x <= 1.35d-168) then
        tmp = 0.918938533204673d0
    else if (x <= 0.5d0) then
        tmp = y * (-0.5d0)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1400.0) {
		tmp = x * y;
	} else if (x <= -1.8e-189) {
		tmp = y * -0.5;
	} else if (x <= 1.35e-168) {
		tmp = 0.918938533204673;
	} else if (x <= 0.5) {
		tmp = y * -0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1400.0:
		tmp = x * y
	elif x <= -1.8e-189:
		tmp = y * -0.5
	elif x <= 1.35e-168:
		tmp = 0.918938533204673
	elif x <= 0.5:
		tmp = y * -0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1400.0)
		tmp = Float64(x * y);
	elseif (x <= -1.8e-189)
		tmp = Float64(y * -0.5);
	elseif (x <= 1.35e-168)
		tmp = 0.918938533204673;
	elseif (x <= 0.5)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1400.0)
		tmp = x * y;
	elseif (x <= -1.8e-189)
		tmp = y * -0.5;
	elseif (x <= 1.35e-168)
		tmp = 0.918938533204673;
	elseif (x <= 0.5)
		tmp = y * -0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1400.0], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.8e-189], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 1.35e-168], 0.918938533204673, If[LessEqual[x, 0.5], N[(y * -0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1400 or 0.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 49.6%

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

   5. Simplified 49.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 48.9%

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

   8. Simplified 48.9%

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

   if -1400 < x < -1.80000000000000008e-189 or 1.35000000000000008e-168 < 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 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 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 62.5%

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

   10. Simplified 62.5%

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

   if -1.80000000000000008e-189 < x < 1.35000000000000008e-168

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 61.8%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1400:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-189}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-168}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+198}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -23:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.8e+198)
   (* x y)
   (if (<= y -23.0)
     (* y -0.5)
     (if (<= y 1.85) (- 0.918938533204673 x) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.8e+198) {
		tmp = x * y;
	} else if (y <= -23.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.8d+198)) then
        tmp = x * y
    else if (y <= (-23.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.8e+198) {
		tmp = x * y;
	} else if (y <= -23.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.8e+198:
		tmp = x * y
	elif y <= -23.0:
		tmp = y * -0.5
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.8e+198)
		tmp = Float64(x * y);
	elseif (y <= -23.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.8e+198)
		tmp = x * y;
	elseif (y <= -23.0)
		tmp = y * -0.5;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.8e+198], N[(x * y), $MachinePrecision], If[LessEqual[y, -23.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.8000000000000002e198

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 69.5%

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

   8. Simplified 69.5%

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

   if -5.8000000000000002e198 < y < -23 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

      10. distribute-lft-out N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

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

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

      18. *-lft-identity N/A

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

      19. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 56.5%

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

   10. Simplified 56.5%

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

   if -23 < y < 1.8500000000000001

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

      10. distribute-lft-out N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

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

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

      18. *-lft-identity N/A

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

      19. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 96.3%

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

   7. Simplified 96.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+198}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -23:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -1350:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 760000:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -1350.0)
     t_0
     (if (<= x 760000.0) (+ 0.918938533204673 (* y (+ x -0.5))) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -1350.0) {
		tmp = t_0;
	} else if (x <= 760000.0) {
		tmp = 0.918938533204673 + (y * (x + -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 <= (-1350.0d0)) then
        tmp = t_0
    else if (x <= 760000.0d0) then
        tmp = 0.918938533204673d0 + (y * (x + (-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 <= -1350.0) {
		tmp = t_0;
	} else if (x <= 760000.0) {
		tmp = 0.918938533204673 + (y * (x + -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + -1.0)
	tmp = 0
	if x <= -1350.0:
		tmp = t_0
	elif x <= 760000.0:
		tmp = 0.918938533204673 + (y * (x + -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 <= -1350.0)
		tmp = t_0;
	elseif (x <= 760000.0)
		tmp = Float64(0.918938533204673 + Float64(y * Float64(x + -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 <= -1350.0)
		tmp = t_0;
	elseif (x <= 760000.0)
		tmp = 0.918938533204673 + (y * (x + -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, -1350.0], t$95$0, If[LessEqual[x, 760000.0], N[(0.918938533204673 + N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1350 or 7.6e5 < x

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

      10. distribute-lft-out N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

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

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

      18. *-lft-identity N/A

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

      19. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 99.1%

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

   7. Simplified 99.1%

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

   if -1350 < x < 7.6e5

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

      10. distribute-lft-out N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

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

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

      18. *-lft-identity N/A

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

      19. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1350:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 760000:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -0.65)
     t_0
     (if (<= x 0.92) (+ 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 <= -0.65) {
		tmp = t_0;
	} else if (x <= 0.92) {
		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 <= (-0.65d0)) then
        tmp = t_0
    else if (x <= 0.92d0) 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 <= -0.65) {
		tmp = t_0;
	} else if (x <= 0.92) {
		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 <= -0.65:
		tmp = t_0
	elif x <= 0.92:
		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 <= -0.65)
		tmp = t_0;
	elseif (x <= 0.92)
		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 <= -0.65)
		tmp = t_0;
	elseif (x <= 0.92)
		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, -0.65], t$95$0, If[LessEqual[x, 0.92], N[(0.918938533204673 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022 or 0.92000000000000004 < x

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

      10. distribute-lft-out N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

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

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

      18. *-lft-identity N/A

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

      19. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 98.5%

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

   7. Simplified 98.5%

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

   if -0.650000000000000022 < x < 0.92000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot y\right)}, \frac{918938533204673}{1000000000000000}\right) \]
   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 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -1.4) t_0 (if (<= y 1.8) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.4) {
		tmp = t_0;
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x + (-0.5d0))
    if (y <= (-1.4d0)) then
        tmp = t_0
    else if (y <= 1.8d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.4) {
		tmp = t_0;
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x + -0.5)
	tmp = 0
	if y <= -1.4:
		tmp = t_0
	elif y <= 1.8:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -0.5))
	tmp = 0.0
	if (y <= -1.4)
		tmp = t_0;
	elseif (y <= 1.8)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x + -0.5);
	tmp = 0.0;
	if (y <= -1.4)
		tmp = t_0;
	elseif (y <= 1.8)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4], t$95$0, If[LessEqual[y, 1.8], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999 or 1.80000000000000004 < 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 99.0%

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

   7. Simplified 99.0%

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

   if -1.3999999999999999 < y < 1.80000000000000004

   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 96.3%

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

   7. Simplified 96.3%

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

Alternative 8: 50.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 620000000:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8)
   (* y -0.5)
   (if (<= y 620000000.0) 0.918938533204673 (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8) {
		tmp = y * -0.5;
	} else if (y <= 620000000.0) {
		tmp = 0.918938533204673;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.8d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 620000000.0d0) then
        tmp = 0.918938533204673d0
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8) {
		tmp = y * -0.5;
	} else if (y <= 620000000.0) {
		tmp = 0.918938533204673;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8:
		tmp = y * -0.5
	elif y <= 620000000.0:
		tmp = 0.918938533204673
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8)
		tmp = Float64(y * -0.5);
	elseif (y <= 620000000.0)
		tmp = 0.918938533204673;
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8)
		tmp = y * -0.5;
	elseif (y <= 620000000.0)
		tmp = 0.918938533204673;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 620000000.0], 0.918938533204673, N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000004 or 6.2e8 < 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 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 54.1%

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

   10. Simplified 54.1%

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

   if -1.80000000000000004 < y < 6.2e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 49.0%

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

Alternative 9: 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 10: 26.2% 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. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. *-lowering-*.f64 75.3%

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

5. Simplified 75.3%

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

6. Taylor expanded in y around 0

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

8. Simplified 26.5%

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

Reproduce

herbie shell --seed 2024158 
(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))