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

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 14

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 85.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, y, -0.5\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 -4e-112)
       (* (fma -0.25 y -0.5) y)
       (if (<= t_0 0.002) (* (fma 0.25 x 0.5) x) (+ (/ 2.0 y) 1.0))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-112) {
		tmp = fma(-0.25, y, -0.5) * y;
	} else if (t_0 <= 0.002) {
		tmp = fma(0.25, x, 0.5) * x;
	} else {
		tmp = (2.0 / y) + 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-112)
		tmp = Float64(fma(-0.25, y, -0.5) * y);
	elseif (t_0 <= 0.002)
		tmp = Float64(fma(0.25, x, 0.5) * x);
	else
		tmp = Float64(Float64(2.0 / y) + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, -4e-112], N[(N[(-0.25 * y + -0.5), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(N[(0.25 * x + 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 / y), $MachinePrecision] + 1.0), $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. Applied rewrites 98.3%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999998e-112

   1. Initial program 99.8%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.6%

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

   if -3.9999999999999998e-112 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2e-3

   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. lower-/.f64 N/A

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

      2. lower--.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.3%

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

   if 2e-3 < (/.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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq -4 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, y, -0.5\right) \cdot y\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, y, -0.5\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 -4e-112)
       (* (fma -0.25 y -0.5) y)
       (if (<= t_0 0.002) (* (fma 0.25 x 0.5) x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-112) {
		tmp = fma(-0.25, y, -0.5) * y;
	} else if (t_0 <= 0.002) {
		tmp = fma(0.25, x, 0.5) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-112)
		tmp = Float64(fma(-0.25, y, -0.5) * y);
	elseif (t_0 <= 0.002)
		tmp = Float64(fma(0.25, x, 0.5) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, -4e-112], N[(N[(-0.25 * y + -0.5), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(N[(0.25 * x + 0.5), $MachinePrecision] * x), $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. Applied rewrites 98.3%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999998e-112

   1. Initial program 99.8%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.6%

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

   if -3.9999999999999998e-112 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2e-3

   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. lower-/.f64 N/A

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

      2. lower--.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.3%

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

   if 2e-3 < (/.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. Applied rewrites 97.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq -4 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, y, -0.5\right) \cdot y\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-112}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 -4e-112)
       (* -0.5 y)
       (if (<= t_0 0.002) (* (fma 0.25 x 0.5) x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-112) {
		tmp = -0.5 * y;
	} else if (t_0 <= 0.002) {
		tmp = fma(0.25, x, 0.5) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-112)
		tmp = Float64(-0.5 * y);
	elseif (t_0 <= 0.002)
		tmp = Float64(fma(0.25, x, 0.5) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, -4e-112], N[(-0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(N[(0.25 * x + 0.5), $MachinePrecision] * x), $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. Applied rewrites 98.3%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999998e-112

   1. Initial program 99.8%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.0%

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

   if -3.9999999999999998e-112 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2e-3

   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. lower-/.f64 N/A

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

      2. lower--.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.3%

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

   if 2e-3 < (/.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. Applied rewrites 97.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq -4 \cdot 10^{-112}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-112}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 -4e-112) (* -0.5 y) (if (<= t_0 2e-27) (* 0.5 x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-112) {
		tmp = -0.5 * y;
	} else if (t_0 <= 2e-27) {
		tmp = 0.5 * x;
	} 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 = (y - x) / ((y + x) - 2.0d0)
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= (-4d-112)) then
        tmp = (-0.5d0) * y
    else if (t_0 <= 2d-27) then
        tmp = 0.5d0 * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= -4e-112) {
		tmp = -0.5 * y;
	} else if (t_0 <= 2e-27) {
		tmp = 0.5 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / ((y + x) - 2.0)
	tmp = 0
	if t_0 <= -0.5:
		tmp = -1.0
	elif t_0 <= -4e-112:
		tmp = -0.5 * y
	elif t_0 <= 2e-27:
		tmp = 0.5 * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-112)
		tmp = Float64(-0.5 * y);
	elseif (t_0 <= 2e-27)
		tmp = Float64(0.5 * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) / ((y + x) - 2.0);
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= -4e-112)
		tmp = -0.5 * y;
	elseif (t_0 <= 2e-27)
		tmp = 0.5 * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, -4e-112], N[(-0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-27], N[(0.5 * x), $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. Applied rewrites 98.3%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -3.9999999999999998e-112

   1. Initial program 99.8%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.0%

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

   if -3.9999999999999998e-112 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.0000000000000001e-27

   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. lower-/.f64 N/A

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

      2. lower--.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.4%

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

   if 2.0000000000000001e-27 < (/.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. Applied rewrites 94.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq -4 \cdot 10^{-112}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -0.5)
     (/ x (- 2.0 x))
     (if (<= t_0 2e-27) (/ (- x y) 2.0) (/ y (- y 2.0))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-27) {
		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 = (y - x) / ((y + x) - 2.0d0)
    if (t_0 <= (-0.5d0)) then
        tmp = x / (2.0d0 - x)
    else if (t_0 <= 2d-27) 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 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-27) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (y - 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / ((y + x) - 2.0)
	tmp = 0
	if t_0 <= -0.5:
		tmp = x / (2.0 - x)
	elif t_0 <= 2e-27:
		tmp = (x - y) / 2.0
	else:
		tmp = y / (y - 2.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (t_0 <= 2e-27)
		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 = (y - x) / ((y + x) - 2.0);
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = x / (2.0 - x);
	elseif (t_0 <= 2e-27)
		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[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-27], 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))) < -0.5

   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. lower-/.f64 N/A

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

      2. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.0000000000000001e-27

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.6%

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

   if 2.0000000000000001e-27 < (/.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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-8}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -5e-8) -1.0 (if (<= t_0 2e-27) (* 0.5 x) 1.0))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -5e-8) {
		tmp = -1.0;
	} else if (t_0 <= 2e-27) {
		tmp = 0.5 * x;
	} 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 = (y - x) / ((y + x) - 2.0d0)
    if (t_0 <= (-5d-8)) then
        tmp = -1.0d0
    else if (t_0 <= 2d-27) then
        tmp = 0.5d0 * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -5e-8) {
		tmp = -1.0;
	} else if (t_0 <= 2e-27) {
		tmp = 0.5 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / ((y + x) - 2.0)
	tmp = 0
	if t_0 <= -5e-8:
		tmp = -1.0
	elif t_0 <= 2e-27:
		tmp = 0.5 * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0))
	tmp = 0.0
	if (t_0 <= -5e-8)
		tmp = -1.0;
	elseif (t_0 <= 2e-27)
		tmp = Float64(0.5 * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) / ((y + x) - 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-8)
		tmp = -1.0;
	elseif (t_0 <= 2e-27)
		tmp = 0.5 * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-8], -1.0, If[LessEqual[t$95$0, 2e-27], N[(0.5 * x), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -4.9999999999999998e-8

   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. Applied rewrites 97.4%

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

   if -4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.0000000000000001e-27

   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. lower-/.f64 N/A

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

      2. lower--.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.0%

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

   if 2.0000000000000001e-27 < (/.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. Applied rewrites 94.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, y, -2\right)}{x} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -0.5)
   (- (/ (fma 2.0 y -2.0) x) 1.0)
   (/ (- x y) (- 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= -0.5) {
		tmp = (fma(2.0, y, -2.0) / x) - 1.0;
	} else {
		tmp = (x - y) / (2.0 - y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0)) <= -0.5)
		tmp = Float64(Float64(fma(2.0, y, -2.0) / x) - 1.0);
	else
		tmp = Float64(Float64(x - y) / Float64(2.0 - y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(2.0 * y + -2.0), $MachinePrecision] / x), $MachinePrecision] - 1.0), $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))) < -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

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   if -0.5 < (/.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. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, y, -2\right)}{x} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -0.5)
   (/ (- x y) (- 2.0 x))
   (/ (- x y) (- 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= -0.5) {
		tmp = (x - y) / (2.0 - x);
	} 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) :: tmp
    if (((y - x) / ((y + x) - 2.0d0)) <= (-0.5d0)) then
        tmp = (x - y) / (2.0d0 - x)
    else
        tmp = (x - y) / (2.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= -0.5) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = (x - y) / (2.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / ((y + x) - 2.0)) <= -0.5:
		tmp = (x - y) / (2.0 - x)
	else:
		tmp = (x - y) / (2.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0)) <= -0.5)
		tmp = Float64(Float64(x - y) / Float64(2.0 - x));
	else
		tmp = Float64(Float64(x - y) / Float64(2.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y - x) / ((y + x) - 2.0)) <= -0.5)
		tmp = (x - y) / (2.0 - x);
	else
		tmp = (x - y) / (2.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(x - y), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $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))) < -0.5

   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. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -0.5 < (/.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. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) 2e-27)
   (/ (- x y) (- 2.0 x))
   (/ y (- y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= 2e-27) {
		tmp = (x - y) / (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 (((y - x) / ((y + x) - 2.0d0)) <= 2d-27) then
        tmp = (x - y) / (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 (((y - x) / ((y + x) - 2.0)) <= 2e-27) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = y / (y - 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / ((y + x) - 2.0)) <= 2e-27:
		tmp = (x - y) / (2.0 - x)
	else:
		tmp = y / (y - 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0)) <= 2e-27)
		tmp = Float64(Float64(x - y) / 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 (((y - x) / ((y + x) - 2.0)) <= 2e-27)
		tmp = (x - y) / (2.0 - x);
	else
		tmp = y / (y - 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], 2e-27], N[(N[(x - y), $MachinePrecision] / 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))) < 2.0000000000000001e-27

   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. lower--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2.0000000000000001e-27 < (/.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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq 10^{-48}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) 1e-48) (/ x (- 2.0 x)) (/ y (- y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= 1e-48) {
		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 (((y - x) / ((y + x) - 2.0d0)) <= 1d-48) 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 (((y - x) / ((y + x) - 2.0)) <= 1e-48) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (y - 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / ((y + x) - 2.0)) <= 1e-48:
		tmp = x / (2.0 - x)
	else:
		tmp = y / (y - 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0)) <= 1e-48)
		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 (((y - x) / ((y + x) - 2.0)) <= 1e-48)
		tmp = x / (2.0 - x);
	else
		tmp = y / (y - 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], 1e-48], 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))) < 9.9999999999999997e-49

   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. lower-/.f64 N/A

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

      2. lower--.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if 9.9999999999999997e-49 < (/.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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq 10^{-48}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq 0.002:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) 0.002) (/ x (- 2.0 x)) (+ (/ 2.0 y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= 0.002) {
		tmp = x / (2.0 - x);
	} else {
		tmp = (2.0 / y) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((y - x) / ((y + x) - 2.0d0)) <= 0.002d0) then
        tmp = x / (2.0d0 - x)
    else
        tmp = (2.0d0 / y) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= 0.002) {
		tmp = x / (2.0 - x);
	} else {
		tmp = (2.0 / y) + 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / ((y + x) - 2.0)) <= 0.002:
		tmp = x / (2.0 - x)
	else:
		tmp = (2.0 / y) + 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0)) <= 0.002)
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = Float64(Float64(2.0 / y) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y - x) / ((y + x) - 2.0)) <= 0.002)
		tmp = x / (2.0 - x);
	else
		tmp = (2.0 / y) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], 0.002], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2e-3

   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. lower-/.f64 N/A

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

      2. lower--.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if 2e-3 < (/.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. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.1%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq 0.002:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -5e-310) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= -5e-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 (((y - x) / ((y + x) - 2.0d0)) <= (-5d-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 (((y - x) / ((y + x) - 2.0)) <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / ((y + x) - 2.0)) <= -5e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0)) <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y - x) / ((y + x) - 2.0)) <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], -5e-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))) < -4.999999999999985e-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. Applied rewrites 78.8%

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

   if -4.999999999999985e-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. Applied rewrites 73.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.3% 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. Applied rewrites 35.7%

   \[\leadsto \color{blue}{-1} \]
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 2024298 
(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))))