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

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 13

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, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = (y + x) - 2.0;
	return (y / t_0) - (x / 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 = (y + x) - 2.0d0
    code = (y / t_0) - (x / t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ t_1 := \frac{x}{2 - x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-156}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))) (t_1 (/ x (- 2.0 x))))
   (if (<= t_0 -2e-15)
     t_1
     (if (<= t_0 2e-156) (* -0.5 y) (if (<= t_0 2e-10) t_1 1.0)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double t_1 = x / (2.0 - x);
	double tmp;
	if (t_0 <= -2e-15) {
		tmp = t_1;
	} else if (t_0 <= 2e-156) {
		tmp = -0.5 * y;
	} else if (t_0 <= 2e-10) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (y - x) / ((y + x) - 2.0d0)
    t_1 = x / (2.0d0 - x)
    if (t_0 <= (-2d-15)) then
        tmp = t_1
    else if (t_0 <= 2d-156) then
        tmp = (-0.5d0) * y
    else if (t_0 <= 2d-10) then
        tmp = t_1
    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 t_1 = x / (2.0 - x);
	double tmp;
	if (t_0 <= -2e-15) {
		tmp = t_1;
	} else if (t_0 <= 2e-156) {
		tmp = -0.5 * y;
	} else if (t_0 <= 2e-10) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / ((y + x) - 2.0)
	t_1 = x / (2.0 - x)
	tmp = 0
	if t_0 <= -2e-15:
		tmp = t_1
	elif t_0 <= 2e-156:
		tmp = -0.5 * y
	elif t_0 <= 2e-10:
		tmp = t_1
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(Float64(y + x) - 2.0))
	t_1 = Float64(x / Float64(2.0 - x))
	tmp = 0.0
	if (t_0 <= -2e-15)
		tmp = t_1;
	elseif (t_0 <= 2e-156)
		tmp = Float64(-0.5 * y);
	elseif (t_0 <= 2e-10)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) / ((y + x) - 2.0);
	t_1 = x / (2.0 - x);
	tmp = 0.0;
	if (t_0 <= -2e-15)
		tmp = t_1;
	elseif (t_0 <= 2e-156)
		tmp = -0.5 * y;
	elseif (t_0 <= 2e-10)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-15], t$95$1, If[LessEqual[t$95$0, 2e-156], N[(-0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-10], t$95$1, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -2.0000000000000002e-15 or 2.00000000000000008e-156 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000007e-10

   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 96.7

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

   5. Applied rewrites 96.7%

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

   if -2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000008e-156

   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 5.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{2 - y}} \]
   7. 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. lower-/.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. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 62.5

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 62.5%

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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-156}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -2e-15)
     -1.0
     (if (<= t_0 2e-156)
       (* -0.5 y)
       (if (<= t_0 2e-10) (* (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 <= -2e-15) {
		tmp = -1.0;
	} else if (t_0 <= 2e-156) {
		tmp = -0.5 * y;
	} else if (t_0 <= 2e-10) {
		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 <= -2e-15)
		tmp = -1.0;
	elseif (t_0 <= 2e-156)
		tmp = Float64(-0.5 * y);
	elseif (t_0 <= 2e-10)
		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, -2e-15], -1.0, If[LessEqual[t$95$0, 2e-156], N[(-0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-10], 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))) < -2.0000000000000002e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.8%

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

   if -2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000008e-156

   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 5.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{2 - y}} \]
   7. 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. lower-/.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. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 62.5

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 62.5%

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

   if 2.00000000000000008e-156 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000007e-10

   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 78.9

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

   5. Applied rewrites 78.9%

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

   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 78.8%

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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-156}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -2e-15)
     -1.0
     (if (<= t_0 2e-156) (* -0.5 y) (if (<= t_0 2e-10) (* 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 <= -2e-15) {
		tmp = -1.0;
	} else if (t_0 <= 2e-156) {
		tmp = -0.5 * y;
	} else if (t_0 <= 2e-10) {
		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 <= (-2d-15)) then
        tmp = -1.0d0
    else if (t_0 <= 2d-156) then
        tmp = (-0.5d0) * y
    else if (t_0 <= 2d-10) 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 <= -2e-15) {
		tmp = -1.0;
	} else if (t_0 <= 2e-156) {
		tmp = -0.5 * y;
	} else if (t_0 <= 2e-10) {
		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 <= -2e-15:
		tmp = -1.0
	elif t_0 <= 2e-156:
		tmp = -0.5 * y
	elif t_0 <= 2e-10:
		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 <= -2e-15)
		tmp = -1.0;
	elseif (t_0 <= 2e-156)
		tmp = Float64(-0.5 * y);
	elseif (t_0 <= 2e-10)
		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 <= -2e-15)
		tmp = -1.0;
	elseif (t_0 <= 2e-156)
		tmp = -0.5 * y;
	elseif (t_0 <= 2e-10)
		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, -2e-15], -1.0, If[LessEqual[t$95$0, 2e-156], N[(-0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-10], 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))) < -2.0000000000000002e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.8%

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

   if -2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000008e-156

   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 5.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{2 - y}} \]
   7. 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. lower-/.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. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 62.5

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 62.5%

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

   if 2.00000000000000008e-156 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000007e-10

   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 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.1%

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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x + x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -2e-15)
     (/ x (- 2.0 x))
     (if (<= t_0 2e-10) (/ (- x y) 2.0) (- 1.0 (/ (+ x x) y))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -2e-15) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-10) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = 1.0 - ((x + x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y - x) / ((y + x) - 2.0d0)
    if (t_0 <= (-2d-15)) then
        tmp = x / (2.0d0 - x)
    else if (t_0 <= 2d-10) then
        tmp = (x - y) / 2.0d0
    else
        tmp = 1.0d0 - ((x + x) / y)
    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 <= -2e-15) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-10) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = 1.0 - ((x + x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / ((y + x) - 2.0)
	tmp = 0
	if t_0 <= -2e-15:
		tmp = x / (2.0 - x)
	elif t_0 <= 2e-10:
		tmp = (x - y) / 2.0
	else:
		tmp = 1.0 - ((x + x) / y)
	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 <= -2e-15)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (t_0 <= 2e-10)
		tmp = Float64(Float64(x - y) / 2.0);
	else
		tmp = Float64(1.0 - Float64(Float64(x + x) / y));
	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 <= -2e-15)
		tmp = x / (2.0 - x);
	elseif (t_0 <= 2e-10)
		tmp = (x - y) / 2.0;
	else
		tmp = 1.0 - ((x + x) / y);
	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, -2e-15], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-10], N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 - N[(N[(x + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -2.0000000000000002e-15

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if -2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000007e-10

   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 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 99.4%

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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   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. distribute-lft-out-- N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. *-lft-identity N/A

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

      13. distribute-rgt-out-- N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. associate-*l/ N/A

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

      18. *-commutative N/A

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

      19. mul-1-neg N/A

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

      20. unsub-neg N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

      \[\leadsto 1 - \frac{x + x}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x + x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -2e-15)
     (/ x (- 2.0 x))
     (if (<= t_0 2e-10) (/ (- x y) 2.0) (/ y (+ -2.0 y))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / ((y + x) - 2.0);
	double tmp;
	if (t_0 <= -2e-15) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-10) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = 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 = (y - x) / ((y + x) - 2.0d0)
    if (t_0 <= (-2d-15)) then
        tmp = x / (2.0d0 - x)
    else if (t_0 <= 2d-10) then
        tmp = (x - y) / 2.0d0
    else
        tmp = y / ((-2.0d0) + y)
    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 <= -2e-15) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-10) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (-2.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / ((y + x) - 2.0)
	tmp = 0
	if t_0 <= -2e-15:
		tmp = x / (2.0 - x)
	elif t_0 <= 2e-10:
		tmp = (x - y) / 2.0
	else:
		tmp = y / (-2.0 + y)
	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 <= -2e-15)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (t_0 <= 2e-10)
		tmp = Float64(Float64(x - y) / 2.0);
	else
		tmp = Float64(y / Float64(-2.0 + y));
	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 <= -2e-15)
		tmp = x / (2.0 - x);
	elseif (t_0 <= 2e-10)
		tmp = (x - y) / 2.0;
	else
		tmp = y / (-2.0 + y);
	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, -2e-15], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-10], N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / 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))) < -2.0000000000000002e-15

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if -2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000007e-10

   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 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 99.4%

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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.4× speedup

Math

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

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 2e-10) (* 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 <= 2e-10) {
		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 <= 2d-10) 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 <= 2e-10) {
		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 <= 2e-10:
		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 <= 2e-10)
		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 <= 2e-10)
		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, 2e-10], 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))) < -0.5

   1. Initial program 99.9%

      \[\frac{x - y}{2 - \left(x + y\right)} \]
   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))) < 2.00000000000000007e-10

   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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.3%

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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.9%

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

4. Final simplification 86.5%

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

Alternative 8: 98.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -0.5)
   (- (/ (fma y 2.0 -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(y, 2.0, -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(y, 2.0, -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[(y * 2.0 + -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 99.9%

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

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   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%

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

Alternative 9: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -2e-15)
   (/ x (- 2.0 x))
   (/ (- x y) (- 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= -2e-15) {
		tmp = x / (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)) <= (-2d-15)) then
        tmp = x / (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)) <= -2e-15) {
		tmp = x / (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)) <= -2e-15:
		tmp = x / (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)) <= -2e-15)
		tmp = Float64(x / 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)) <= -2e-15)
		tmp = x / (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], -2e-15], N[(x / 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))) < -2.0000000000000002e-15

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if -2.0000000000000002e-15 < (/.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 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -2e-15) (/ x (- 2.0 x)) (/ y (+ -2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= -2e-15) {
		tmp = x / (2.0 - x);
	} else {
		tmp = 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)) <= (-2d-15)) then
        tmp = x / (2.0d0 - x)
    else
        tmp = 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)) <= -2e-15) {
		tmp = x / (2.0 - x);
	} else {
		tmp = y / (-2.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / ((y + x) - 2.0)) <= -2e-15:
		tmp = x / (2.0 - x)
	else:
		tmp = 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)) <= -2e-15)
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = Float64(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)) <= -2e-15)
		tmp = x / (2.0 - x);
	else
		tmp = 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], -2e-15], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(y / 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))) < -2.0000000000000002e-15

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if -2.0000000000000002e-15 < (/.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. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval 81.0

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

   5. Applied rewrites 81.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -1e-310) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / ((y + x) - 2.0)) <= -1e-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)) <= (-1d-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)) <= -1e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / ((y + x) - 2.0)) <= -1e-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)) <= -1e-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)) <= -1e-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], -1e-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))) < -9.999999999999969e-311

   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 70.9%

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

   if -9.999999999999969e-311 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 83.0%

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

4. Final simplification 76.2%

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

Alternative 12: 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 13: 37.8% 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 40.9%

   \[\leadsto \color{blue}{-1} \]
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 2024235 
(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))))