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

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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 99.9%

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

Alternative 2: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ t_1 := \frac{x}{2 - x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-122}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.98:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))) (t_1 (/ x (- 2.0 x))))
   (if (<= t_0 -5e-86)
     t_1
     (if (<= t_0 2e-122) (* -0.5 y) (if (<= t_0 0.98) t_1 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double t_1 = x / (2.0 - x);
	double tmp;
	if (t_0 <= -5e-86) {
		tmp = t_1;
	} else if (t_0 <= 2e-122) {
		tmp = -0.5 * y;
	} else if (t_0 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - y) / (2.0d0 - (x + y))
    t_1 = x / (2.0d0 - x)
    if (t_0 <= (-5d-86)) then
        tmp = t_1
    else if (t_0 <= 2d-122) then
        tmp = (-0.5d0) * y
    else if (t_0 <= 0.98d0) then
        tmp = t_1
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double t_1 = x / (2.0 - x);
	double tmp;
	if (t_0 <= -5e-86) {
		tmp = t_1;
	} else if (t_0 <= 2e-122) {
		tmp = -0.5 * y;
	} else if (t_0 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	t_1 = x / (2.0 - x)
	tmp = 0
	if t_0 <= -5e-86:
		tmp = t_1
	elif t_0 <= 2e-122:
		tmp = -0.5 * y
	elif t_0 <= 0.98:
		tmp = t_1
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	t_1 = Float64(x / Float64(2.0 - x))
	tmp = 0.0
	if (t_0 <= -5e-86)
		tmp = t_1;
	elseif (t_0 <= 2e-122)
		tmp = Float64(-0.5 * y);
	elseif (t_0 <= 0.98)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	t_1 = x / (2.0 - x);
	tmp = 0.0;
	if (t_0 <= -5e-86)
		tmp = t_1;
	elseif (t_0 <= 2e-122)
		tmp = -0.5 * y;
	elseif (t_0 <= 0.98)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-86], t$95$1, If[LessEqual[t$95$0, 2e-122], N[(-0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, 0.98], t$95$1, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -4.9999999999999999e-86 or 2.00000000000000012e-122 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 0.97999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -4.9999999999999999e-86 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000012e-122

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 N/A

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

      11. metadata-eval 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.8%

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

   if 0.97999999999999998 < (/.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 96.6%

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

Alternative 3: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = ((y + y) / x) - 1.0;
	} else if (t_0 <= 2e-10) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (-2.0 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = ((y + y) / x) - 1.0d0
    else if (t_0 <= 2d-10) then
        tmp = (x - y) / 2.0d0
    else
        tmp = y / ((-2.0d0) + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = ((y + y) / x) - 1.0;
	} else if (t_0 <= 2e-10) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (-2.0 + y);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = ((y + y) / x) - 1.0;
	elseif (t_0 <= 2e-10)
		tmp = (x - y) / 2.0;
	else
		tmp = y / (-2.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(N[(y + y), $MachinePrecision] / x), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e-10], N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / N[(-2.0 + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. div-sub N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. div-add-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-fma.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 98.3%

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

   13. Applied rewrites 98.3%

       \[\leadsto \frac{y + y}{x} - 1 \]

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000007e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 95.4%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 N/A

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

      11. metadata-eval 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 4: 97.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2e-10) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (-2.0 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2d-10) then
        tmp = (x - y) / 2.0d0
    else
        tmp = y / ((-2.0d0) + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2e-10) {
		tmp = (x - y) / 2.0;
	} else {
		tmp = y / (-2.0 + y);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 2e-10)
		tmp = (x - y) / 2.0;
	else
		tmp = y / (-2.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2e-10], N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / N[(-2.0 + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < -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 97.9%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000007e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 95.4%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 N/A

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

      11. metadata-eval 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 5: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, y, -0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5) -1.0 (if (<= t_0 2e-6) (* (fma -0.25 y -0.5) y) 1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2e-6) {
		tmp = fma(-0.25, y, -0.5) * y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 2e-6)
		tmp = Float64(fma(-0.25, y, -0.5) * y);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2e-6], N[(N[(-0.25 * y + -0.5), $MachinePrecision] * y), $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 97.9%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999991e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 N/A

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

      11. metadata-eval 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 54.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 54.6%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.2%

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

Alternative 6: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.5) -1.0 (if (<= t_0 2e-6) (* -0.5 y) 1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2e-6) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2d-6) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2e-6) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (2.0 - (x + y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -1.0
	elif t_0 <= 2e-6:
		tmp = -0.5 * y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 2e-6)
		tmp = Float64(-0.5 * y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (2.0 - (x + y));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 2e-6)
		tmp = -0.5 * y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2e-6], N[(-0.5 * y), $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 97.9%

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

   if -0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 1.99999999999999991e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 N/A

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

      11. metadata-eval 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.3%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.2%

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

Alternative 7: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{2 - \left(x + y\right)}\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 2.0 (+ x y)))))
   (if (<= t_0 -0.01) -1.0 (if (<= t_0 2e-5) (* 0.5 x) 1.0))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = -1.0;
	} else if (t_0 <= 2e-5) {
		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 = (x - y) / (2.0d0 - (x + y))
    if (t_0 <= (-0.01d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2d-5) 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 = (x - y) / (2.0 - (x + y));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = -1.0;
	} else if (t_0 <= 2e-5) {
		tmp = 0.5 * 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 <= -0.01:
		tmp = -1.0
	elif t_0 <= 2e-5:
		tmp = 0.5 * 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 <= -0.01)
		tmp = -1.0;
	elseif (t_0 <= 2e-5)
		tmp = Float64(0.5 * 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 <= -0.01)
		tmp = -1.0;
	elseif (t_0 <= 2e-5)
		tmp = 0.5 * 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, -0.01], -1.0, If[LessEqual[t$95$0, 2e-5], 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))) < -0.0100000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.3%

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

   if -0.0100000000000000002 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 2.00000000000000016e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.6%

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

   if 2.00000000000000016e-5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.1%

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

Alternative 8: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq 0.98:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x, 2\right)}{y} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 2.0 (+ x y))) 0.98)
   (/ (- x y) (- 2.0 x))
   (+ (/ (fma -2.0 x 2.0) y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= 0.98) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = (fma(-2.0, x, 2.0) / y) + 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(2.0 - Float64(x + y))) <= 0.98)
		tmp = Float64(Float64(x - y) / Float64(2.0 - x));
	else
		tmp = Float64(Float64(fma(-2.0, x, 2.0) / y) + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.98], N[(N[(x - y), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * x + 2.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y))) < 0.97999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.97999999999999998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-sub N/A

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

      6. lower-+.f64 N/A

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

   5. Applied rewrites 99.2%

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

Alternative 9: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x - y) / (2.0d0 - (x + y))) <= (-0.5d0)) then
        tmp = ((y + y) / x) - 1.0d0
    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 = ((y + y) / x) - 1.0;
	} 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 = ((y + y) / x) - 1.0
	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(Float64(Float64(y + y) / x) - 1.0);
	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 = ((y + y) / x) - 1.0;
	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[(N[(N[(y + y), $MachinePrecision] / x), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / N[(2.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. div-sub N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. div-add-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-fma.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 98.3%

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

   13. Applied rewrites 98.3%

       \[\leadsto \frac{y + y}{x} - 1 \]

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 10: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{2 - \left(x + y\right)} \leq 0.98:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-2 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 2.0 (+ x y))) 0.98)
   (/ (- x y) (- 2.0 x))
   (/ y (+ -2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= 0.98) {
		tmp = (x - y) / (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 (((x - y) / (2.0d0 - (x + y))) <= 0.98d0) then
        tmp = (x - y) / (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 (((x - y) / (2.0 - (x + y))) <= 0.98) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = y / (-2.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (2.0 - (x + y))) <= 0.98:
		tmp = (x - y) / (2.0 - x)
	else:
		tmp = 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))) <= 0.98)
		tmp = Float64(Float64(x - y) / 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 (((x - y) / (2.0 - (x + y))) <= 0.98)
		tmp = (x - y) / (2.0 - x);
	else
		tmp = 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], 0.98], N[(N[(x - y), $MachinePrecision] / 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))) < 0.97999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.97999999999999998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 2 binary64) (+.f64 x y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 N/A

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

      11. metadata-eval 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 11: 85.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (2.0 - (x + y))) <= -0.5) {
		tmp = -1.0;
	} 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) :: tmp
    if (((x - y) / (2.0d0 - (x + y))) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = 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 = -1.0;
	} else {
		tmp = y / (-2.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (2.0 - (x + y))) <= -0.5:
		tmp = -1.0
	else:
		tmp = 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 = -1.0;
	else
		tmp = Float64(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 = -1.0;
	else
		tmp = 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], -1.0, 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))) < -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 97.9%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 N/A

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

      11. metadata-eval 81.0

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

   5. Applied rewrites 81.0%

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

Alternative 12: 74.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 75.2%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 75.0%

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

Alternative 13: 37.4% 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 99.9%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 39.9%

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

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024337 
(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))))