Data.Colour.RGB:hslsv from colour-2.3.3, C

Percentage Accurate: 100.0% → 100.0%

Time: 10.8s

Alternatives: 16

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{2 - \left(x + y\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 2.0 (+ x y))))

C

double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (2.0d0 - (x + y))
end function

Java

public static double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Python

def code(x, y):
	return (x - y) / (2.0 - (x + y))

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (2.0 - (x + y));
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{2 - \left(x + y\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 2.0 (+ x y))))

C

double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (2.0d0 - (x + y))
end function

Java

public static double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Python

def code(x, y):
	return (x - y) / (2.0 - (x + y))

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (2.0 - (x + y));
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{2 - \left(x + y\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 2.0 (+ x y))))

C

double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (2.0d0 - (x + y))
end function

Java

public static double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

Python

def code(x, y):
	return (x - y) / (2.0 - (x + y))

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (2.0 - (x + y));
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{2 - \left(x + y\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 85.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, 0.25, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, -0.25, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 -4e-186)
       (* x (fma x 0.25 0.5))
       (if (<= t_0 1e-13) (* y (fma y -0.25 -0.5)) (+ 1.0 (/ 2.0 y)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-186) {
		tmp = x * fma(x, 0.25, 0.5);
	} else if (t_0 <= 1e-13) {
		tmp = y * fma(y, -0.25, -0.5);
	} else {
		tmp = 1.0 + (2.0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-186)
		tmp = Float64(x * fma(x, 0.25, 0.5));
	elseif (t_0 <= 1e-13)
		tmp = Float64(y * fma(y, -0.25, -0.5));
	else
		tmp = Float64(1.0 + Float64(2.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, -4e-186], N[(x * N[(x * 0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], N[(y * N[(y * -0.25 + -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -0.5

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 96.1%

      \[\leadsto \color{blue}{-1} \]

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999996e-186

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

      2. --lowering--.f64 64.0

         \[\leadsto \frac{x}{\color{blue}{2 - x}} \]

   5. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

   6. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot x\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{4} \cdot x + \frac{1}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \frac{1}{4}} + \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 61.6

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 0.25, 0.5\right)} \]

   8. Simplified 61.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 0.25, 0.5\right)} \]

   if -3.9999999999999996e-186 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1e-13

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 64.5

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

   5. Simplified 64.5%

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

   6. Taylor expanded in y around 0

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

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

         \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto y \cdot \left(y \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{2}}\right) \]

      5. accelerator-lowering-fma.f64 64.5

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, -0.25, -0.5\right)} \]

   8. Simplified 64.5%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, -0.25, -0.5\right)} \]

   if 1e-13 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 97.1

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

   5. Simplified 97.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 97.1

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

   8. Simplified 97.1%

      \[\leadsto \color{blue}{1 + \frac{2}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, 0.25, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, -0.25, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 -4e-186)
       (* x (fma x 0.25 0.5))
       (if (<= t_0 1e-13) (* y (fma y -0.25 -0.5)) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-186) {
		tmp = x * fma(x, 0.25, 0.5);
	} else if (t_0 <= 1e-13) {
		tmp = y * fma(y, -0.25, -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-186)
		tmp = Float64(x * fma(x, 0.25, 0.5));
	elseif (t_0 <= 1e-13)
		tmp = Float64(y * fma(y, -0.25, -0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, -4e-186], N[(x * N[(x * 0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], N[(y * N[(y * -0.25 + -0.5), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -0.5

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 96.1%

      \[\leadsto \color{blue}{-1} \]

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999996e-186

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

      2. --lowering--.f64 64.0

         \[\leadsto \frac{x}{\color{blue}{2 - x}} \]

   5. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

   6. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot x\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{4} \cdot x + \frac{1}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \frac{1}{4}} + \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 61.6

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 0.25, 0.5\right)} \]

   8. Simplified 61.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 0.25, 0.5\right)} \]

   if -3.9999999999999996e-186 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1e-13

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 64.5

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

   5. Simplified 64.5%

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

   6. Taylor expanded in y around 0

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

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

         \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto y \cdot \left(y \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{2}}\right) \]

      5. accelerator-lowering-fma.f64 64.5

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, -0.25, -0.5\right)} \]

   8. Simplified 64.5%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, -0.25, -0.5\right)} \]

   if 1e-13 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, 0.25, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 -4e-186)
       (* x (fma x 0.25 0.5))
       (if (<= t_0 1e-13) (* y -0.5) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-186) {
		tmp = x * fma(x, 0.25, 0.5);
	} else if (t_0 <= 1e-13) {
		tmp = y * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-186)
		tmp = Float64(x * fma(x, 0.25, 0.5));
	elseif (t_0 <= 1e-13)
		tmp = Float64(y * -0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, -4e-186], N[(x * N[(x * 0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], N[(y * -0.5), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -0.5

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 96.1%

      \[\leadsto \color{blue}{-1} \]

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999996e-186

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

      2. --lowering--.f64 64.0

         \[\leadsto \frac{x}{\color{blue}{2 - x}} \]

   5. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

   6. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot x\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{4} \cdot x + \frac{1}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \frac{1}{4}} + \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 61.6

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 0.25, 0.5\right)} \]

   8. Simplified 61.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 0.25, 0.5\right)} \]

   if -3.9999999999999996e-186 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1e-13

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 64.5

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

   5. Simplified 64.5%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 64.0

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

   8. Simplified 64.0%

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

   if 1e-13 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 -4e-186) (* x 0.5) (if (<= t_0 1e-13) (* y -0.5) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-186) {
		tmp = x * 0.5;
	} else if (t_0 <= 1e-13) {
		tmp = y * -0.5;
	} else {
		tmp = 1.0;
	}
	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) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= (-4d-186)) then
        tmp = x * 0.5d0
    else if (t_0 <= 1d-13) then
        tmp = y * (-0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-186) {
		tmp = x * 0.5;
	} else if (t_0 <= 1e-13) {
		tmp = y * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -1.0
	elif t_0 <= -4e-186:
		tmp = x * 0.5
	elif t_0 <= 1e-13:
		tmp = y * -0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-186)
		tmp = Float64(x * 0.5);
	elseif (t_0 <= 1e-13)
		tmp = Float64(y * -0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-186)
		tmp = x * 0.5;
	elseif (t_0 <= 1e-13)
		tmp = y * -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, -4e-186], N[(x * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], N[(y * -0.5), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -0.5

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 96.1%

      \[\leadsto \color{blue}{-1} \]

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999996e-186

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 64.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{2 - y}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 60.2

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

   8. Simplified 60.2%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 60.2

          \[\leadsto \color{blue}{x \cdot 0.5} \]

   11. Simplified 60.2%

       \[\leadsto \color{blue}{x \cdot 0.5} \]

   if -3.9999999999999996e-186 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1e-13

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 64.5

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

   5. Simplified 64.5%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 64.0

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

   8. Simplified 64.0%

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

   if 1e-13 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - \left(x + y\right)\\ t_1 := \frac{x - y}{t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x + y\right) + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 2.0 (+ x y))) (t_1 (/ (- x y) t_0)))
   (if (<= t_1 -2e-7)
     (/ x t_0)
     (if (<= t_1 2e-11) (/ (- x y) 2.0) (/ y (+ (+ x y) -2.0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	double t_1 = (x - y) / t_0;
	double tmp;
	if (t_1 <= -2e-7) {
		tmp = x / t_0;
	} else if (t_1 <= 2e-11) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / ((x + y) + -2.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 - (x + y)
    t_1 = (x - y) / t_0
    if (t_1 <= (-2d-7)) then
        tmp = x / t_0
    else if (t_1 <= 2d-11) then
        tmp = (x - y) / 2.0d0
    else
        tmp = y / ((x + y) + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	double t_1 = (x - y) / t_0;
	double tmp;
	if (t_1 <= -2e-7) {
		tmp = x / t_0;
	} else if (t_1 <= 2e-11) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / ((x + y) + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 - (x + y)
	t_1 = (x - y) / t_0
	tmp = 0
	if t_1 <= -2e-7:
		tmp = x / t_0
	elif t_1 <= 2e-11:
		tmp = (x - y) / 2.0
	else:
		tmp = y / ((x + y) + -2.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 - Float64(x + y))
	t_1 = Float64(Float64(x - y) / t_0)
	tmp = 0.0
	if (t_1 <= -2e-7)
		tmp = Float64(x / t_0);
	elseif (t_1 <= 2e-11)
		tmp = Float64(Float64(x - y) / 2.0);
	else
		tmp = Float64(y / Float64(Float64(x + y) + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 - (x + y);
	t_1 = (x - y) / t_0;
	tmp = 0.0;
	if (t_1 <= -2e-7)
		tmp = x / t_0;
	elseif (t_1 <= 2e-11)
		tmp = (x - y) / 2.0;
	else
		tmp = y / ((x + y) + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-7], N[(x / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / N[(N[(x + y), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -1.9999999999999999e-7

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 99.5%

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

   if -1.9999999999999999e-7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999988e-11

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x - y}{\color{blue}{2 - x}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{x - y}{\color{blue}{2}} \]
   7. Step-by-step derivation

   8. Simplified 98.5%

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

   if 1.99999999999999988e-11 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 98.1

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

   5. Simplified 98.1%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 98.1

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

   7. Applied egg-rr 98.1%

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

Alternative 7: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - \left(x + y\right)\\ t_1 := \frac{x - y}{t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 2.0 (+ x y))) (t_1 (/ (- x y) t_0)))
   (if (<= t_1 -2e-7)
     (/ x t_0)
     (if (<= t_1 2e-11) (/ (- x y) 2.0) (/ y (+ y -2.0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	double t_1 = (x - y) / t_0;
	double tmp;
	if (t_1 <= -2e-7) {
		tmp = x / t_0;
	} else if (t_1 <= 2e-11) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (y + -2.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 - (x + y)
    t_1 = (x - y) / t_0
    if (t_1 <= (-2d-7)) then
        tmp = x / t_0
    else if (t_1 <= 2d-11) then
        tmp = (x - y) / 2.0d0
    else
        tmp = y / (y + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	double t_1 = (x - y) / t_0;
	double tmp;
	if (t_1 <= -2e-7) {
		tmp = x / t_0;
	} else if (t_1 <= 2e-11) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 - (x + y)
	t_1 = (x - y) / t_0
	tmp = 0
	if t_1 <= -2e-7:
		tmp = x / t_0
	elif t_1 <= 2e-11:
		tmp = (x - y) / 2.0
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 - Float64(x + y))
	t_1 = Float64(Float64(x - y) / t_0)
	tmp = 0.0
	if (t_1 <= -2e-7)
		tmp = Float64(x / t_0);
	elseif (t_1 <= 2e-11)
		tmp = Float64(Float64(x - y) / 2.0);
	else
		tmp = Float64(y / Float64(y + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 - (x + y);
	t_1 = (x - y) / t_0;
	tmp = 0.0;
	if (t_1 <= -2e-7)
		tmp = x / t_0;
	elseif (t_1 <= 2e-11)
		tmp = (x - y) / 2.0;
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-7], N[(x / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -1.9999999999999999e-7

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 99.5%

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

   if -1.9999999999999999e-7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999988e-11

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x - y}{\color{blue}{2 - x}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{x - y}{\color{blue}{2}} \]
   7. Step-by-step derivation

   8. Simplified 98.5%

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

   if 1.99999999999999988e-11 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 98.1

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

   5. Simplified 98.1%

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

Alternative 8: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -2e-7)
     (/ x (- 2.0 x))
     (if (<= t_0 2e-11) (/ (- x y) 2.0) (/ y (+ y -2.0))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -2e-7) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-11) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (y + -2.0);
	}
	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) / (2.0d0 - (x + y))
    if (t_0 <= (-2d-7)) then
        tmp = x / (2.0d0 - x)
    else if (t_0 <= 2d-11) then
        tmp = (x - y) / 2.0d0
    else
        tmp = y / (y + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -2e-7) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-11) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -2e-7:
		tmp = x / (2.0 - x)
	elif t_0 <= 2e-11:
		tmp = (x - y) / 2.0
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -2e-7)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (t_0 <= 2e-11)
		tmp = Float64(Float64(x - y) / 2.0);
	else
		tmp = Float64(y / Float64(y + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -2e-7)
		tmp = x / (2.0 - x);
	elseif (t_0 <= 2e-11)
		tmp = (x - y) / 2.0;
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-7], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-11], N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -1.9999999999999999e-7

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

      2. --lowering--.f64 99.4

         \[\leadsto \frac{x}{\color{blue}{2 - x}} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

   if -1.9999999999999999e-7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999988e-11

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x - y}{\color{blue}{2 - x}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{x - y}{\color{blue}{2}} \]
   7. Step-by-step derivation

   8. Simplified 98.5%

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

   if 1.99999999999999988e-11 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 98.1

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

   5. Simplified 98.1%

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

Alternative 9: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, -0.25, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -4e-186)
     (/ x (- 2.0 x))
     (if (<= t_0 1e-13) (* y (fma y -0.25 -0.5)) (+ 1.0 (/ 2.0 y))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -4e-186) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 1e-13) {
		tmp = y * fma(y, -0.25, -0.5);
	} else {
		tmp = 1.0 + (2.0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -4e-186)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (t_0 <= 1e-13)
		tmp = Float64(y * fma(y, -0.25, -0.5));
	else
		tmp = Float64(1.0 + Float64(2.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-186], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], N[(y * N[(y * -0.25 + -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999996e-186

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

      2. --lowering--.f64 90.4

         \[\leadsto \frac{x}{\color{blue}{2 - x}} \]

   5. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

   if -3.9999999999999996e-186 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1e-13

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 64.5

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

   5. Simplified 64.5%

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

   6. Taylor expanded in y around 0

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

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

         \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto y \cdot \left(y \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{2}}\right) \]

      5. accelerator-lowering-fma.f64 64.5

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, -0.25, -0.5\right)} \]

   8. Simplified 64.5%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, -0.25, -0.5\right)} \]

   if 1e-13 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 97.1

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

   5. Simplified 97.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 97.1

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

   8. Simplified 97.1%

      \[\leadsto \color{blue}{1 + \frac{2}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5) -1.0 (if (<= t_0 2e-11) (* x 0.5) 1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2e-11) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0;
	}
	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) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2d-11) then
        tmp = x * 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2e-11) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -1.0
	elif t_0 <= 2e-11:
		tmp = x * 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 2e-11)
		tmp = Float64(x * 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 2e-11)
		tmp = x * 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2e-11], N[(x * 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -0.5

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 96.1%

      \[\leadsto \color{blue}{-1} \]

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999988e-11

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 50.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{2 - y}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 48.5

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

   8. Simplified 48.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 48.5

          \[\leadsto \color{blue}{x \cdot 0.5} \]

   11. Simplified 48.5%

       \[\leadsto \color{blue}{x \cdot 0.5} \]

   if 1.99999999999999988e-11 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 97.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{2 - \left(x + x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 2.0 (+ x y))) 2e-11)
   (/ (- x y) (- 2.0 x))
   (+ 1.0 (/ (- 2.0 (+ x x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= 2e-11) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = 1.0 + ((2.0 - (x + 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 - y) / (2.0d0 - (x + y))) <= 2d-11) then
        tmp = (x - y) / (2.0d0 - x)
    else
        tmp = 1.0d0 + ((2.0d0 - (x + x)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= 2e-11) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = 1.0 + ((2.0 - (x + x)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (2.0 - (x + y))) <= 2e-11:
		tmp = (x - y) / (2.0 - x)
	else:
		tmp = 1.0 + ((2.0 - (x + x)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(2.0 - Float64(x + y))) <= 2e-11)
		tmp = Float64(Float64(x - y) / Float64(2.0 - x));
	else
		tmp = Float64(1.0 + Float64(Float64(2.0 - Float64(x + x)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x - y) / (2.0 - (x + y))) <= 2e-11)
		tmp = (x - y) / (2.0 - x);
	else
		tmp = 1.0 + ((2.0 - (x + x)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[(x - y), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(2.0 - N[(x + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999988e-11

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x - y}{\color{blue}{2 - x}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

   if 1.99999999999999988e-11 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

   5. Simplified 98.7%

      \[\leadsto \color{blue}{1 + \frac{2 - \left(x + x\right)}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - \left(x + y\right)\\ \mathbf{if}\;\frac{x - y}{t\_0} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 2.0 (+ x y))))
   (if (<= (/ (- x y) t_0) -2e-7) (/ x t_0) (/ (- x y) (- 2.0 y)))))

C

double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	double tmp;
	if (((x - y) / t_0) <= -2e-7) {
		tmp = x / t_0;
	} else {
		tmp = (x - y) / (2.0 - y);
	}
	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 = 2.0d0 - (x + y)
    if (((x - y) / t_0) <= (-2d-7)) then
        tmp = x / t_0
    else
        tmp = (x - y) / (2.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	double tmp;
	if (((x - y) / t_0) <= -2e-7) {
		tmp = x / t_0;
	} else {
		tmp = (x - y) / (2.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 - (x + y)
	tmp = 0
	if ((x - y) / t_0) <= -2e-7:
		tmp = x / t_0
	else:
		tmp = (x - y) / (2.0 - y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 - Float64(x + y))
	tmp = 0.0
	if (Float64(Float64(x - y) / t_0) <= -2e-7)
		tmp = Float64(x / t_0);
	else
		tmp = Float64(Float64(x - y) / Float64(2.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 - (x + y);
	tmp = 0.0;
	if (((x - y) / t_0) <= -2e-7)
		tmp = x / t_0;
	else
		tmp = (x - y) / (2.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x - y), $MachinePrecision] / t$95$0), $MachinePrecision], -2e-7], N[(x / t$95$0), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / N[(2.0 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -1.9999999999999999e-7

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 99.5%

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

   if -1.9999999999999999e-7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{x - y}{\color{blue}{2 - y}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 98.6

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

   5. Simplified 98.6%

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

Alternative 13: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x + y\right) + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 2.0 (+ x y))) 2e-11)
   (/ (- x y) (- 2.0 x))
   (/ y (+ (+ x y) -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= 2e-11) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = y / ((x + y) + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x - y) / (2.0d0 - (x + y))) <= 2d-11) then
        tmp = (x - y) / (2.0d0 - x)
    else
        tmp = y / ((x + y) + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= 2e-11) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = y / ((x + y) + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (2.0 - (x + y))) <= 2e-11:
		tmp = (x - y) / (2.0 - x)
	else:
		tmp = y / ((x + y) + -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(2.0 - Float64(x + y))) <= 2e-11)
		tmp = Float64(Float64(x - y) / Float64(2.0 - x));
	else
		tmp = Float64(y / Float64(Float64(x + y) + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x - y) / (2.0 - (x + y))) <= 2e-11)
		tmp = (x - y) / (2.0 - x);
	else
		tmp = y / ((x + y) + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[(x - y), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(x + y), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999988e-11

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x - y}{\color{blue}{2 - x}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 99.3

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

   5. Simplified 99.3%

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

   if 1.99999999999999988e-11 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 98.1

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

   5. Simplified 98.1%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-lowering-+.f64 98.1

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

   7. Applied egg-rr 98.1%

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

Alternative 14: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq -4 \cdot 10^{-186}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 2.0 (+ x y))) -4e-186)
   (/ x (- 2.0 x))
   (/ y (+ y -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= -4e-186) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x - y) / (2.0d0 - (x + y))) <= (-4d-186)) then
        tmp = x / (2.0d0 - x)
    else
        tmp = y / (y + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= -4e-186) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (2.0 - (x + y))) <= -4e-186:
		tmp = x / (2.0 - x)
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(2.0 - Float64(x + y))) <= -4e-186)
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = Float64(y / Float64(y + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x - y) / (2.0 - (x + y))) <= -4e-186)
		tmp = x / (2.0 - x);
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-186], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999996e-186

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

      2. --lowering--.f64 90.4

         \[\leadsto \frac{x}{\color{blue}{2 - x}} \]

   5. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{x}{2 - x}} \]

   if -3.9999999999999996e-186 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 89.3

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

   5. Simplified 89.3%

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

Alternative 15: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 2.0 (+ x y))) -2e-310) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= -2e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x - y) / (2.0d0 - (x + y))) <= (-2d-310)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= -2e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (2.0 - (x + y))) <= -2e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(2.0 - Float64(x + y))) <= -2e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x - y) / (2.0 - (x + y))) <= -2e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -1.999999999999994e-310

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 70.7%

      \[\leadsto \color{blue}{-1} \]

   if -1.999999999999994e-310 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 77.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 37.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{2 - \left(x + y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 35.9%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - \left(x + y\right)\\ \frac{x}{t\_0} - \frac{y}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 2.0 (+ x y)))) (- (/ x t_0) (/ y t_0))))

C

double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	return (x / t_0) - (y / t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = 2.0d0 - (x + y)
    code = (x / t_0) - (y / t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 - (x + y);
	return (x / t_0) - (y / t_0);
}

Python

def code(x, y):
	t_0 = 2.0 - (x + y)
	return (x / t_0) - (y / t_0)

Julia

function code(x, y)
	t_0 = Float64(2.0 - Float64(x + y))
	return Float64(Float64(x / t_0) - Float64(y / t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = 2.0 - (x + y);
	tmp = (x / t_0) - (y / t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]}, N[(N[(x / t$95$0), $MachinePrecision] - N[(y / t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, C"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ x (- 2 (+ x y))) (/ y (- 2 (+ x y)))))

  (/ (- x y) (- 2.0 (+ x y))))