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

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. --lowering--.f64 N/A

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

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

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

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

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

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

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

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

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

   7. --lowering--.f64 N/A

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

   8. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5)
     (/ x (- 2.0 x))
     (if (<= t_0 1e-10) (* -0.5 (- y x)) (/ y (+ y -2.0))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 1e-10) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = x / (2.0d0 - x)
    else if (t_0 <= 1d-10) then
        tmp = (-0.5d0) * (y - x)
    else
        tmp = y / (y + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 1e-10) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = x / (2.0 - x)
	elif t_0 <= 1e-10:
		tmp = -0.5 * (y - x)
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (t_0 <= 1e-10)
		tmp = Float64(-0.5 * Float64(y - x));
	else
		tmp = Float64(y / Float64(y + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = x / (2.0 - x);
	elseif (t_0 <= 1e-10)
		tmp = -0.5 * (y - x);
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-10], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. --lowering--.f64 99.2

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

   5. Simplified 99.2%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.00000000000000004e-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. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 98.3%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 98.3

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

   10. Applied egg-rr 98.3%

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

   if 1.00000000000000004e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

      15. metadata-eval 99.2

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

   5. Simplified 99.2%

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

4. Final simplification 99.0%

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

Alternative 3: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5)
     (/ x (- 2.0 x))
     (if (<= t_0 0.004) (* -0.5 (- y x)) (- 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 0.004) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = 1.0 - (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 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = x / (2.0d0 - x)
    else if (t_0 <= 0.004d0) then
        tmp = (-0.5d0) * (y - x)
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = x / (2.0 - x);
	} else if (t_0 <= 0.004) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = x / (2.0 - x)
	elif t_0 <= 0.004:
		tmp = -0.5 * (y - x)
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(x / Float64(2.0 - x));
	elseif (t_0 <= 0.004)
		tmp = Float64(-0.5 * Float64(y - x));
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = x / (2.0 - x);
	elseif (t_0 <= 0.004)
		tmp = -0.5 * (y - x);
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.004], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. --lowering--.f64 99.2

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

   5. Simplified 99.2%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 96.3%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 96.3

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

   10. Applied egg-rr 96.3%

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

   if 0.0040000000000000001 < (/.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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. /-lowering-/.f64 96.8

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

   8. Simplified 96.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq -0.5:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{elif}\;\frac{x - y}{2 - \left(x + y\right)} \leq 0.004:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5)
     -1.0
     (if (<= t_0 0.004) (* -0.5 (- y x)) (- 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 0.004) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = 1.0 - (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 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= 0.004d0) then
        tmp = (-0.5d0) * (y - x)
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 0.004) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -1.0
	elif t_0 <= 0.004:
		tmp = -0.5 * (y - x)
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 0.004)
		tmp = Float64(-0.5 * Float64(y - x));
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 0.004)
		tmp = -0.5 * (y - x);
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 0.004], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.3%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 96.3%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 96.3

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

   10. Applied egg-rr 96.3%

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

   if 0.0040000000000000001 < (/.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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. /-lowering-/.f64 96.8

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

   8. Simplified 96.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{x - y}{2 - \left(x + y\right)} \leq 0.004:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5) -1.0 (if (<= t_0 0.004) (* -0.5 (- y x)) 1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 0.004) {
		tmp = -0.5 * (y - 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 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= 0.004d0) then
        tmp = (-0.5d0) * (y - x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 0.004) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -1.0
	elif t_0 <= 0.004:
		tmp = -0.5 * (y - x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 0.004)
		tmp = Float64(-0.5 * Float64(y - x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 0.004)
		tmp = -0.5 * (y - x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 0.004], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.3%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 96.3%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 96.3

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

   10. Applied egg-rr 96.3%

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

   if 0.0040000000000000001 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 96.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{x - y}{2 - \left(x + y\right)} \leq 0.004:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5) -1.0 (if (<= t_0 0.004) (* y -0.5) 1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 0.004) {
		tmp = y * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= 0.004d0) then
        tmp = y * (-0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 0.004) {
		tmp = y * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -1.0
	elif t_0 <= 0.004:
		tmp = y * -0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 0.004)
		tmp = Float64(y * -0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 0.004)
		tmp = y * -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 0.004], N[(y * -0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.3%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 96.3%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 50.3

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

   11. Simplified 50.3%

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

   if 0.0040000000000000001 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 96.8%

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

Alternative 7: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 10^{-10}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -5e-18) -1.0 (if (<= t_0 1e-10) (* x 0.5) 1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -5e-18) {
		tmp = -1.0;
	} else if (t_0 <= 1e-10) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-5d-18)) then
        tmp = -1.0d0
    else if (t_0 <= 1d-10) then
        tmp = x * 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -5e-18) {
		tmp = -1.0;
	} else if (t_0 <= 1e-10) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -5e-18:
		tmp = -1.0
	elif t_0 <= 1e-10:
		tmp = x * 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -5e-18)
		tmp = -1.0;
	elseif (t_0 <= 1e-10)
		tmp = Float64(x * 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -5e-18)
		tmp = -1.0;
	elseif (t_0 <= 1e-10)
		tmp = x * 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-18], -1.0, If[LessEqual[t$95$0, 1e-10], N[(x * 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -5.00000000000000036e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.5%

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

   if -5.00000000000000036e-18 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.00000000000000004e-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. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 53.8

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

   5. Simplified 53.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 53.2

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

   8. Simplified 53.2%

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

   if 1.00000000000000004e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 94.2%

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

Alternative 8: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (2.0 - (x + y))) <= -0.5:
		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(x - y) / Float64(2.0 - Float64(x + y))) <= -0.5)
		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 (((x - y) / (2.0 - (x + y))) <= -0.5)
		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[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], 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))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. --lowering--.f64 99.2

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

   5. Simplified 99.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

Alternative 9: 74.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -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. Simplified 71.1%

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

   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. Simplified 74.0%

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

Alternative 10: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 11: 38.3% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf

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

5. Simplified 35.8%

   \[\leadsto \color{blue}{-1} \]
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 2024198 
(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))))