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

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

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]]

Derivation

1. Initial program 99.9%

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

   1. div-sub 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -6.3 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-158}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+58}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -6.3e+15)
     t_0
     (if (<= y -7.2e-43)
       -1.0
       (if (<= y -4.8e-158) (* y -0.5) (if (<= y 2.5e+58) -1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -6.3e+15) {
		tmp = t_0;
	} else if (y <= -7.2e-43) {
		tmp = -1.0;
	} else if (y <= -4.8e-158) {
		tmp = y * -0.5;
	} else if (y <= 2.5e+58) {
		tmp = -1.0;
	} else {
		tmp = t_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 = 1.0d0 - (x / y)
    if (y <= (-6.3d+15)) then
        tmp = t_0
    else if (y <= (-7.2d-43)) then
        tmp = -1.0d0
    else if (y <= (-4.8d-158)) then
        tmp = y * (-0.5d0)
    else if (y <= 2.5d+58) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -6.3e+15) {
		tmp = t_0;
	} else if (y <= -7.2e-43) {
		tmp = -1.0;
	} else if (y <= -4.8e-158) {
		tmp = y * -0.5;
	} else if (y <= 2.5e+58) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -6.3e+15:
		tmp = t_0
	elif y <= -7.2e-43:
		tmp = -1.0
	elif y <= -4.8e-158:
		tmp = y * -0.5
	elif y <= 2.5e+58:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -6.3e+15)
		tmp = t_0;
	elseif (y <= -7.2e-43)
		tmp = -1.0;
	elseif (y <= -4.8e-158)
		tmp = Float64(y * -0.5);
	elseif (y <= 2.5e+58)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -6.3e+15)
		tmp = t_0;
	elseif (y <= -7.2e-43)
		tmp = -1.0;
	elseif (y <= -4.8e-158)
		tmp = y * -0.5;
	elseif (y <= 2.5e+58)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.3e+15], t$95$0, If[LessEqual[y, -7.2e-43], -1.0, If[LessEqual[y, -4.8e-158], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 2.5e+58], -1.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.3e15 or 2.49999999999999993e58 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 73.8%

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

      1. neg-mul-1 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   7. Simplified 73.8%

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

   if -6.3e15 < y < -7.1999999999999998e-43 or -4.80000000000000015e-158 < y < 2.49999999999999993e58

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 63.8%

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

   if -7.1999999999999998e-43 < y < -4.80000000000000015e-158

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

   4. Simplified 58.9%

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

   5. Taylor expanded in y around 0 58.9%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.3 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-158}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+58}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 3: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -1.22 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x} + -1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-158}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+60}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -1.22e+16)
     t_0
     (if (<= y -3.2e-112)
       (+ (/ y x) -1.0)
       (if (<= y -4.8e-158) (* y -0.5) (if (<= y 2.5e+60) -1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -1.22e+16) {
		tmp = t_0;
	} else if (y <= -3.2e-112) {
		tmp = (y / x) + -1.0;
	} else if (y <= -4.8e-158) {
		tmp = y * -0.5;
	} else if (y <= 2.5e+60) {
		tmp = -1.0;
	} else {
		tmp = t_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 = 1.0d0 - (x / y)
    if (y <= (-1.22d+16)) then
        tmp = t_0
    else if (y <= (-3.2d-112)) then
        tmp = (y / x) + (-1.0d0)
    else if (y <= (-4.8d-158)) then
        tmp = y * (-0.5d0)
    else if (y <= 2.5d+60) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -1.22e+16) {
		tmp = t_0;
	} else if (y <= -3.2e-112) {
		tmp = (y / x) + -1.0;
	} else if (y <= -4.8e-158) {
		tmp = y * -0.5;
	} else if (y <= 2.5e+60) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -1.22e+16:
		tmp = t_0
	elif y <= -3.2e-112:
		tmp = (y / x) + -1.0
	elif y <= -4.8e-158:
		tmp = y * -0.5
	elif y <= 2.5e+60:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -1.22e+16)
		tmp = t_0;
	elseif (y <= -3.2e-112)
		tmp = Float64(Float64(y / x) + -1.0);
	elseif (y <= -4.8e-158)
		tmp = Float64(y * -0.5);
	elseif (y <= 2.5e+60)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -1.22e+16)
		tmp = t_0;
	elseif (y <= -3.2e-112)
		tmp = (y / x) + -1.0;
	elseif (y <= -4.8e-158)
		tmp = y * -0.5;
	elseif (y <= 2.5e+60)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.22e+16], t$95$0, If[LessEqual[y, -3.2e-112], N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, -4.8e-158], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 2.5e+60], -1.0, t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.22e16 or 2.49999999999999987e60 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 73.8%

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

      1. neg-mul-1 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   7. Simplified 73.8%

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

   if -1.22e16 < y < -3.19999999999999993e-112

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 59.7%

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

      1. mul-1-neg 59.7%

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

   4. Simplified 59.7%

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

   5. Taylor expanded in x around 0 59.7%

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

   if -3.19999999999999993e-112 < y < -4.80000000000000015e-158

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. associate-*r/ 67.8%

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

      2. neg-mul-1 67.8%

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

   4. Simplified 67.8%

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

   5. Taylor expanded in y around 0 67.8%

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

   if -4.80000000000000015e-158 < y < 2.49999999999999987e60

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 63.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+16}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x} + -1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-158}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+60}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 4: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y}{-x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.6e+55)
   (/ (- x y) (- x))
   (if (<= x 4e+31) (/ (- x y) (- 2.0 y)) (+ (/ y x) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.6e+55) {
		tmp = (x - y) / -x;
	} else if (x <= 4e+31) {
		tmp = (x - y) / (2.0 - y);
	} else {
		tmp = (y / x) + -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 <= (-8.6d+55)) then
        tmp = (x - y) / -x
    else if (x <= 4d+31) then
        tmp = (x - y) / (2.0d0 - y)
    else
        tmp = (y / x) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.6e+55) {
		tmp = (x - y) / -x;
	} else if (x <= 4e+31) {
		tmp = (x - y) / (2.0 - y);
	} else {
		tmp = (y / x) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.6e+55:
		tmp = (x - y) / -x
	elif x <= 4e+31:
		tmp = (x - y) / (2.0 - y)
	else:
		tmp = (y / x) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.6e+55)
		tmp = Float64(Float64(x - y) / Float64(-x));
	elseif (x <= 4e+31)
		tmp = Float64(Float64(x - y) / Float64(2.0 - y));
	else
		tmp = Float64(Float64(y / x) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.6e+55)
		tmp = (x - y) / -x;
	elseif (x <= 4e+31)
		tmp = (x - y) / (2.0 - y);
	else
		tmp = (y / x) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.6e+55], N[(N[(x - y), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[x, 4e+31], N[(N[(x - y), $MachinePrecision] / N[(2.0 - y), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5999999999999998e55

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 83.5%

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

      1. mul-1-neg 83.5%

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

   4. Simplified 83.5%

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

   if -8.5999999999999998e55 < x < 3.9999999999999999e31

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 93.1%

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

   if 3.9999999999999999e31 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 75.9%

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

      1. mul-1-neg 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x around 0 75.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y}{-x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

Alternative 5: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-158}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+53}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.4e+15)
   1.0
   (if (<= y -9e-43)
     -1.0
     (if (<= y -4.8e-158) (* y -0.5) (if (<= y 1.6e+53) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.4e+15) {
		tmp = 1.0;
	} else if (y <= -9e-43) {
		tmp = -1.0;
	} else if (y <= -4.8e-158) {
		tmp = y * -0.5;
	} else if (y <= 1.6e+53) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.4d+15)) then
        tmp = 1.0d0
    else if (y <= (-9d-43)) then
        tmp = -1.0d0
    else if (y <= (-4.8d-158)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.6d+53) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.4e+15) {
		tmp = 1.0;
	} else if (y <= -9e-43) {
		tmp = -1.0;
	} else if (y <= -4.8e-158) {
		tmp = y * -0.5;
	} else if (y <= 1.6e+53) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.4e+15:
		tmp = 1.0
	elif y <= -9e-43:
		tmp = -1.0
	elif y <= -4.8e-158:
		tmp = y * -0.5
	elif y <= 1.6e+53:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.4e+15)
		tmp = 1.0;
	elseif (y <= -9e-43)
		tmp = -1.0;
	elseif (y <= -4.8e-158)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.6e+53)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.4e+15)
		tmp = 1.0;
	elseif (y <= -9e-43)
		tmp = -1.0;
	elseif (y <= -4.8e-158)
		tmp = y * -0.5;
	elseif (y <= 1.6e+53)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.4e+15], 1.0, If[LessEqual[y, -9e-43], -1.0, If[LessEqual[y, -4.8e-158], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.6e+53], -1.0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4e15 or 1.6e53 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 73.0%

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

   if -8.4e15 < y < -9.0000000000000005e-43 or -4.80000000000000015e-158 < y < 1.6e53

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 63.8%

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

   if -9.0000000000000005e-43 < y < -4.80000000000000015e-158

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

   4. Simplified 58.9%

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

   5. Taylor expanded in y around 0 58.9%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-158}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+53}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{2}{y}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-158}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+47}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e+16)
   (+ 1.0 (/ 2.0 y))
   (if (<= y -5.6e-41)
     -1.0
     (if (<= y -3.3e-158) (* y -0.5) (if (<= y 4e+47) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+16) {
		tmp = 1.0 + (2.0 / y);
	} else if (y <= -5.6e-41) {
		tmp = -1.0;
	} else if (y <= -3.3e-158) {
		tmp = y * -0.5;
	} else if (y <= 4e+47) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d+16)) then
        tmp = 1.0d0 + (2.0d0 / y)
    else if (y <= (-5.6d-41)) then
        tmp = -1.0d0
    else if (y <= (-3.3d-158)) then
        tmp = y * (-0.5d0)
    else if (y <= 4d+47) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+16) {
		tmp = 1.0 + (2.0 / y);
	} else if (y <= -5.6e-41) {
		tmp = -1.0;
	} else if (y <= -3.3e-158) {
		tmp = y * -0.5;
	} else if (y <= 4e+47) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e+16:
		tmp = 1.0 + (2.0 / y)
	elif y <= -5.6e-41:
		tmp = -1.0
	elif y <= -3.3e-158:
		tmp = y * -0.5
	elif y <= 4e+47:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e+16)
		tmp = Float64(1.0 + Float64(2.0 / y));
	elseif (y <= -5.6e-41)
		tmp = -1.0;
	elseif (y <= -3.3e-158)
		tmp = Float64(y * -0.5);
	elseif (y <= 4e+47)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e+16)
		tmp = 1.0 + (2.0 / y);
	elseif (y <= -5.6e-41)
		tmp = -1.0;
	elseif (y <= -3.3e-158)
		tmp = y * -0.5;
	elseif (y <= 4e+47)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e+16], N[(1.0 + N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.6e-41], -1.0, If[LessEqual[y, -3.3e-158], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 4e+47], -1.0, 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.4e16

   1. Initial program 99.9%

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

   2. Taylor expanded in y around -inf 71.9%

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

   3. Taylor expanded in x around 0 70.7%

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

      1. associate-*r/ 70.7%

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

      2. metadata-eval 70.7%

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

   5. Simplified 70.7%

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

   if -1.4e16 < y < -5.6000000000000003e-41 or -3.3000000000000002e-158 < y < 4.0000000000000002e47

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 63.8%

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

   if -5.6000000000000003e-41 < y < -3.3000000000000002e-158

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

   4. Simplified 58.9%

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

   5. Taylor expanded in y around 0 58.9%

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

   if 4.0000000000000002e47 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 76.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{2}{y}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-158}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+47}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+16} \lor \neg \left(y \leq 8.2 \cdot 10^{+50}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.15e+16) (not (<= y 8.2e+50)))
   (- 1.0 (/ x y))
   (/ x (- 2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.15e+16) || !(y <= 8.2e+50)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (2.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.15d+16)) .or. (.not. (y <= 8.2d+50))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / (2.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.15e+16) || !(y <= 8.2e+50)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (2.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.15e+16) or not (y <= 8.2e+50):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (2.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.15e+16) || !(y <= 8.2e+50))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(2.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.15e+16) || ~((y <= 8.2e+50)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (2.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.15e+16], N[Not[LessEqual[y, 8.2e+50]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e16 or 8.2000000000000002e50 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 73.8%

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

      1. neg-mul-1 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   7. Simplified 73.8%

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

   if -1.15e16 < y < 8.2000000000000002e50

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+16} \lor \neg \left(y \leq 8.2 \cdot 10^{+50}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 - x}\\ \end{array} \]

Alternative 8: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+56} \lor \neg \left(x \leq 3.05 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{y}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+56) (not (<= x 3.05e+31)))
   (+ (/ y x) -1.0)
   (/ y (+ y -2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+56) || !(x <= 3.05e+31)) {
		tmp = (y / x) + -1.0;
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1d+56)) .or. (.not. (x <= 3.05d+31))) then
        tmp = (y / x) + (-1.0d0)
    else
        tmp = y / (y + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1e+56) || !(x <= 3.05e+31)) {
		tmp = (y / x) + -1.0;
	} else {
		tmp = y / (y + -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1e+56) or not (x <= 3.05e+31):
		tmp = (y / x) + -1.0
	else:
		tmp = y / (y + -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1e+56) || !(x <= 3.05e+31))
		tmp = Float64(Float64(y / x) + -1.0);
	else
		tmp = Float64(y / Float64(y + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1e+56) || ~((x <= 3.05e+31)))
		tmp = (y / x) + -1.0;
	else
		tmp = y / (y + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1e+56], N[Not[LessEqual[x, 3.05e+31]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000009e56 or 3.05000000000000005e31 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 79.8%

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

      1. mul-1-neg 79.8%

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

   4. Simplified 79.8%

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

   5. Taylor expanded in x around 0 79.8%

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

   if -1.00000000000000009e56 < x < 3.05000000000000005e31

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

      1. expm1-log1p-u 74.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-y}{2 - y}\right)\right)} \]

      2. expm1-udef 52.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-y}{2 - y}\right)} - 1} \]

      3. add-sqr-sqrt 28.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{2 - y}\right)} - 1 \]

      4. sqrt-unprod 17.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{2 - y}\right)} - 1 \]

      5. sqr-neg 17.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{2 - y}\right)} - 1 \]

      6. sqrt-unprod 1.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{2 - y}\right)} - 1 \]

      7. add-sqr-sqrt 2.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{2 - y}\right)} - 1 \]

      8. frac-2neg 2.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-y}{-\left(2 - y\right)}}\right)} - 1 \]

      9. add-sqr-sqrt 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{-\left(2 - y\right)}\right)} - 1 \]

      10. sqrt-unprod 12.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{-\left(2 - y\right)}\right)} - 1 \]

      11. sqr-neg 12.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{-\left(2 - y\right)}\right)} - 1 \]

      12. sqrt-unprod 24.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{-\left(2 - y\right)}\right)} - 1 \]

      13. add-sqr-sqrt 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{-\left(2 - y\right)}\right)} - 1 \]

      14. sub-neg 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{-\color{blue}{\left(2 + \left(-y\right)\right)}}\right)} - 1 \]

      15. distribute-neg-in 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\color{blue}{\left(-2\right) + \left(-\left(-y\right)\right)}}\right)} - 1 \]

      16. metadata-eval 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\color{blue}{-2} + \left(-\left(-y\right)\right)}\right)} - 1 \]

      17. remove-double-neg 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{-2 + \color{blue}{y}}\right)} - 1 \]

   6. Applied egg-rr 52.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{-2 + y}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 74.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{-2 + y}\right)\right)} \]

      2. expm1-log1p 74.3%

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

      3. +-commutative 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+56} \lor \neg \left(x \leq 3.05 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{y}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -2}\\ \end{array} \]

Alternative 9: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y}{-x}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e+55)
   (/ (- x y) (- x))
   (if (<= x 5.1e+31) (/ y (+ y -2.0)) (+ (/ y x) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e+55) {
		tmp = (x - y) / -x;
	} else if (x <= 5.1e+31) {
		tmp = y / (y + -2.0);
	} else {
		tmp = (y / x) + -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 <= (-5.8d+55)) then
        tmp = (x - y) / -x
    else if (x <= 5.1d+31) then
        tmp = y / (y + (-2.0d0))
    else
        tmp = (y / x) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.8e+55) {
		tmp = (x - y) / -x;
	} else if (x <= 5.1e+31) {
		tmp = y / (y + -2.0);
	} else {
		tmp = (y / x) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.8e+55:
		tmp = (x - y) / -x
	elif x <= 5.1e+31:
		tmp = y / (y + -2.0)
	else:
		tmp = (y / x) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8e+55)
		tmp = Float64(Float64(x - y) / Float64(-x));
	elseif (x <= 5.1e+31)
		tmp = Float64(y / Float64(y + -2.0));
	else
		tmp = Float64(Float64(y / x) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.8e+55)
		tmp = (x - y) / -x;
	elseif (x <= 5.1e+31)
		tmp = y / (y + -2.0);
	else
		tmp = (y / x) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8e+55], N[(N[(x - y), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[x, 5.1e+31], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999997e55

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 83.5%

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

      1. mul-1-neg 83.5%

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

   4. Simplified 83.5%

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

   if -5.7999999999999997e55 < x < 5.0999999999999997e31

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

      1. expm1-log1p-u 74.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-y}{2 - y}\right)\right)} \]

      2. expm1-udef 52.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-y}{2 - y}\right)} - 1} \]

      3. add-sqr-sqrt 28.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{2 - y}\right)} - 1 \]

      4. sqrt-unprod 17.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{2 - y}\right)} - 1 \]

      5. sqr-neg 17.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{2 - y}\right)} - 1 \]

      6. sqrt-unprod 1.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{2 - y}\right)} - 1 \]

      7. add-sqr-sqrt 2.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{2 - y}\right)} - 1 \]

      8. frac-2neg 2.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-y}{-\left(2 - y\right)}}\right)} - 1 \]

      9. add-sqr-sqrt 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{-\left(2 - y\right)}\right)} - 1 \]

      10. sqrt-unprod 12.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{-\left(2 - y\right)}\right)} - 1 \]

      11. sqr-neg 12.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{-\left(2 - y\right)}\right)} - 1 \]

      12. sqrt-unprod 24.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{-\left(2 - y\right)}\right)} - 1 \]

      13. add-sqr-sqrt 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{-\left(2 - y\right)}\right)} - 1 \]

      14. sub-neg 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{-\color{blue}{\left(2 + \left(-y\right)\right)}}\right)} - 1 \]

      15. distribute-neg-in 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\color{blue}{\left(-2\right) + \left(-\left(-y\right)\right)}}\right)} - 1 \]

      16. metadata-eval 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\color{blue}{-2} + \left(-\left(-y\right)\right)}\right)} - 1 \]

      17. remove-double-neg 52.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{-2 + \color{blue}{y}}\right)} - 1 \]

   6. Applied egg-rr 52.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{-2 + y}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 74.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{-2 + y}\right)\right)} \]

      2. expm1-log1p 74.3%

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

      3. +-commutative 74.3%

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

   8. Simplified 74.3%

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

   if 5.0999999999999997e31 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 75.9%

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

      1. mul-1-neg 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x around 0 75.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y}{-x}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 11: 61.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+52}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.15e+16) 1.0 (if (<= y 2e+52) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+16) {
		tmp = 1.0;
	} else if (y <= 2e+52) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.15d+16)) then
        tmp = 1.0d0
    else if (y <= 2d+52) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+16) {
		tmp = 1.0;
	} else if (y <= 2e+52) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.15e+16:
		tmp = 1.0
	elif y <= 2e+52:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.15e+16)
		tmp = 1.0;
	elseif (y <= 2e+52)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.15e+16)
		tmp = 1.0;
	elseif (y <= 2e+52)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.15e+16], 1.0, If[LessEqual[y, 2e+52], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e16 or 2e52 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 73.0%

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

   if -1.15e16 < y < 2e52

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 59.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+52}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 38.3% accurate, 9.0× 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. Taylor expanded in x around inf 43.6%

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

3. Final simplification 43.6%

   \[\leadsto -1 \]

Developer target: 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 2023320 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, C"
  :precision binary64

  :herbie-target
  (- (/ x (- 2.0 (+ x y))) (/ y (- 2.0 (+ x y))))

  (/ (- x y) (- 2.0 (+ x y))))