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

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 11

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: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- (+ y x) 2.0))))
   (if (<= t_0 -5e-14)
     (/ x (- 2.0 x))
     (if (<= t_0 2e-11) (/ (- 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 <= -5e-14) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-11) {
		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 <= (-5d-14)) then
        tmp = x / (2.0d0 - x)
    else if (t_0 <= 2d-11) 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 <= -5e-14) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 2e-11) {
		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 <= -5e-14:
		tmp = x / (2.0 - x)
	elif t_0 <= 2e-11:
		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 <= -5e-14)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (t_0 <= 2e-11)
		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 <= -5e-14)
		tmp = x / (2.0 - x);
	elseif (t_0 <= 2e-11)
		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, -5e-14], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-11], N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / N[(-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))) < -5.0000000000000002e-14

   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 98.3

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

   5. Applied rewrites 98.3%

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

   if -5.0000000000000002e-14 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999988e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. 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 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{\left(y + x\right) - 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 2e-11) (* (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 <= 2e-11) {
		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 <= 2e-11)
		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, 2e-11], N[(N[(0.25 * x + 0.5), $MachinePrecision] * 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 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 98.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 57.1

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

   5. Applied rewrites 57.1%

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

   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 57.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.25, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.4× speedup

Math

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

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-11) (* 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-11) {
		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-11) 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-11) {
		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-11:
		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-11)
		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-11)
		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-11], 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 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 98.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -0.5)
   (/ (- x y) (- (- y) 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) / (-y - 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) / (-y - 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) / (-y - 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) / (-y - 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(Float64(-y) - 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) / (-y - 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[((-y) - 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. 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 100.0

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. associate--r+ N/A

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

      18. lift--.f64 N/A

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

      19. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   9. Applied rewrites 99.8%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. clear-num N/A

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

       5. lower-/.f64 N/A

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

       6. lift--.f64 99.8

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

   11. Applied rewrites 99.8%

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

   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 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 98.5%

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

Alternative 6: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) -5e-14)
   (/ 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)) <= -5e-14) {
		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)) <= (-5d-14)) 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)) <= -5e-14) {
		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)) <= -5e-14:
		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)) <= -5e-14)
		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)) <= -5e-14)
		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], -5e-14], 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))) < -5.0000000000000002e-14

   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 98.3

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

   5. Applied rewrites 98.3%

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

   if -5.0000000000000002e-14 < (/.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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- (+ y x) 2.0)) 2e-33) (/ 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-33) {
		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-33) 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-33) {
		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-33:
		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-33)
		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-33)
		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-33], 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))) < 2.0000000000000001e-33

   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 84.1

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

   5. Applied rewrites 84.1%

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

   if 2.0000000000000001e-33 < (/.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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{\left(y + x\right) - 2} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y):
	tmp = 0
	if ((y - x) / ((y + x) - 2.0)) <= 2e-11:
		tmp = x / (2.0 - x)
	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)) <= 2e-11)
		tmp = Float64(x / Float64(2.0 - x));
	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)) <= 2e-11)
		tmp = x / (2.0 - x);
	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], 2e-11], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 82.2

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

   5. Applied rewrites 82.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.3%

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

4. Final simplification 87.0%

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

Alternative 9: 73.8% 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 78.0%

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

   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 72.4%

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

4. Final simplification 75.0%

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

Alternative 10: 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 11: 38.2% 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 37.1%

   \[\leadsto \color{blue}{-1} \]
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 2024270 
(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))))