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

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x} + -1\\ t_1 := \left(x - y\right) \cdot 0.5\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ y x) -1.0)) (t_1 (* (- x y) 0.5)))
   (if (<= x -2.0)
     t_0
     (if (<= x -9.5e-82)
       t_1
       (if (<= x 1.6e-99)
         1.0
         (if (<= x 4.8e-66) t_1 (if (<= x 7.5e+42) 1.0 t_0)))))))

C

double code(double x, double y) {
	double t_0 = (y / x) + -1.0;
	double t_1 = (x - y) * 0.5;
	double tmp;
	if (x <= -2.0) {
		tmp = t_0;
	} else if (x <= -9.5e-82) {
		tmp = t_1;
	} else if (x <= 1.6e-99) {
		tmp = 1.0;
	} else if (x <= 4.8e-66) {
		tmp = t_1;
	} else if (x <= 7.5e+42) {
		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) :: t_1
    real(8) :: tmp
    t_0 = (y / x) + (-1.0d0)
    t_1 = (x - y) * 0.5d0
    if (x <= (-2.0d0)) then
        tmp = t_0
    else if (x <= (-9.5d-82)) then
        tmp = t_1
    else if (x <= 1.6d-99) then
        tmp = 1.0d0
    else if (x <= 4.8d-66) then
        tmp = t_1
    else if (x <= 7.5d+42) 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 = (y / x) + -1.0;
	double t_1 = (x - y) * 0.5;
	double tmp;
	if (x <= -2.0) {
		tmp = t_0;
	} else if (x <= -9.5e-82) {
		tmp = t_1;
	} else if (x <= 1.6e-99) {
		tmp = 1.0;
	} else if (x <= 4.8e-66) {
		tmp = t_1;
	} else if (x <= 7.5e+42) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / x) + -1.0
	t_1 = (x - y) * 0.5
	tmp = 0
	if x <= -2.0:
		tmp = t_0
	elif x <= -9.5e-82:
		tmp = t_1
	elif x <= 1.6e-99:
		tmp = 1.0
	elif x <= 4.8e-66:
		tmp = t_1
	elif x <= 7.5e+42:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / x) + -1.0)
	t_1 = Float64(Float64(x - y) * 0.5)
	tmp = 0.0
	if (x <= -2.0)
		tmp = t_0;
	elseif (x <= -9.5e-82)
		tmp = t_1;
	elseif (x <= 1.6e-99)
		tmp = 1.0;
	elseif (x <= 4.8e-66)
		tmp = t_1;
	elseif (x <= 7.5e+42)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / x) + -1.0;
	t_1 = (x - y) * 0.5;
	tmp = 0.0;
	if (x <= -2.0)
		tmp = t_0;
	elseif (x <= -9.5e-82)
		tmp = t_1;
	elseif (x <= 1.6e-99)
		tmp = 1.0;
	elseif (x <= 4.8e-66)
		tmp = t_1;
	elseif (x <= 7.5e+42)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -2.0], t$95$0, If[LessEqual[x, -9.5e-82], t$95$1, If[LessEqual[x, 1.6e-99], 1.0, If[LessEqual[x, 4.8e-66], t$95$1, If[LessEqual[x, 7.5e+42], 1.0, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2 or 7.50000000000000041e42 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 75.9%

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

      1. neg-mul-1 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in x around 0 75.9%

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

   if -2 < x < -9.4999999999999996e-82 or 1.6e-99 < x < 4.80000000000000052e-66

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.9%

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

      3. associate--l- 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 97.7%

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

   7. Taylor expanded in y around 0 77.3%

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

   if -9.4999999999999996e-82 < x < 1.6e-99 or 4.80000000000000052e-66 < x < 7.50000000000000041e42

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 58.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{y}{x} + -1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-82}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-66}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

Alternative 3: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot 0.5\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{+43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) 0.5)))
   (if (<= x -2.5)
     -1.0
     (if (<= x -1.45e-83)
       t_0
       (if (<= x 7.2e-100)
         1.0
         (if (<= x 4.5e-66) t_0 (if (<= x 1e+43) 1.0 -1.0)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) * 0.5;
	double tmp;
	if (x <= -2.5) {
		tmp = -1.0;
	} else if (x <= -1.45e-83) {
		tmp = t_0;
	} else if (x <= 7.2e-100) {
		tmp = 1.0;
	} else if (x <= 4.5e-66) {
		tmp = t_0;
	} else if (x <= 1e+43) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) * 0.5d0
    if (x <= (-2.5d0)) then
        tmp = -1.0d0
    else if (x <= (-1.45d-83)) then
        tmp = t_0
    else if (x <= 7.2d-100) then
        tmp = 1.0d0
    else if (x <= 4.5d-66) then
        tmp = t_0
    else if (x <= 1d+43) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) * 0.5;
	double tmp;
	if (x <= -2.5) {
		tmp = -1.0;
	} else if (x <= -1.45e-83) {
		tmp = t_0;
	} else if (x <= 7.2e-100) {
		tmp = 1.0;
	} else if (x <= 4.5e-66) {
		tmp = t_0;
	} else if (x <= 1e+43) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * 0.5
	tmp = 0
	if x <= -2.5:
		tmp = -1.0
	elif x <= -1.45e-83:
		tmp = t_0
	elif x <= 7.2e-100:
		tmp = 1.0
	elif x <= 4.5e-66:
		tmp = t_0
	elif x <= 1e+43:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * 0.5)
	tmp = 0.0
	if (x <= -2.5)
		tmp = -1.0;
	elseif (x <= -1.45e-83)
		tmp = t_0;
	elseif (x <= 7.2e-100)
		tmp = 1.0;
	elseif (x <= 4.5e-66)
		tmp = t_0;
	elseif (x <= 1e+43)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) * 0.5;
	tmp = 0.0;
	if (x <= -2.5)
		tmp = -1.0;
	elseif (x <= -1.45e-83)
		tmp = t_0;
	elseif (x <= 7.2e-100)
		tmp = 1.0;
	elseif (x <= 4.5e-66)
		tmp = t_0;
	elseif (x <= 1e+43)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -2.5], -1.0, If[LessEqual[x, -1.45e-83], t$95$0, If[LessEqual[x, 7.2e-100], 1.0, If[LessEqual[x, 4.5e-66], t$95$0, If[LessEqual[x, 1e+43], 1.0, -1.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5 or 1.00000000000000001e43 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.9%

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

   if -2.5 < x < -1.45e-83 or 7.1999999999999997e-100 < x < 4.4999999999999998e-66

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.9%

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

      3. associate--l- 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 97.7%

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

   7. Taylor expanded in y around 0 77.3%

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

   if -1.45e-83 < x < 7.1999999999999997e-100 or 4.4999999999999998e-66 < x < 1.00000000000000001e43

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 58.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-83}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-66}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10^{+43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1}{2 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-5)
   (/ (- x y) (- 2.0 x))
   (if (<= x 7.5e+42) (* (- x y) (/ 1.0 (- 2.0 y))) (+ (/ y x) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-5) {
		tmp = (x - y) / (2.0 - x);
	} else if (x <= 7.5e+42) {
		tmp = (x - y) * (1.0 / (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 <= (-1.6d-5)) then
        tmp = (x - y) / (2.0d0 - x)
    else if (x <= 7.5d+42) then
        tmp = (x - y) * (1.0d0 / (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 <= -1.6e-5) {
		tmp = (x - y) / (2.0 - x);
	} else if (x <= 7.5e+42) {
		tmp = (x - y) * (1.0 / (2.0 - y));
	} else {
		tmp = (y / x) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.6e-5:
		tmp = (x - y) / (2.0 - x)
	elif x <= 7.5e+42:
		tmp = (x - y) * (1.0 / (2.0 - y))
	else:
		tmp = (y / x) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-5)
		tmp = Float64(Float64(x - y) / Float64(2.0 - x));
	elseif (x <= 7.5e+42)
		tmp = Float64(Float64(x - y) * Float64(1.0 / 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 <= -1.6e-5)
		tmp = (x - y) / (2.0 - x);
	elseif (x <= 7.5e+42)
		tmp = (x - y) * (1.0 / (2.0 - y));
	else
		tmp = (y / x) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e-5], N[(N[(x - y), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+42], N[(N[(x - y), $MachinePrecision] * N[(1.0 / N[(2.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999993e-5

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 74.0%

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

   if -1.59999999999999993e-5 < x < 7.50000000000000041e42

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

      3. associate--l- 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 97.5%

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

   if 7.50000000000000041e42 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 79.6%

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

      1. neg-mul-1 79.6%

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

   6. Simplified 79.6%

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

   7. Taylor expanded in x around 0 79.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1}{2 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

Alternative 5: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1}{2 - y}\\ \mathbf{elif}\;y \leq 4200000000:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(2 - x\right) - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e-7)
   (* (- x y) (/ 1.0 (- 2.0 y)))
   (if (<= y 4200000000.0)
     (/ (- x y) (- 2.0 x))
     (+ 1.0 (/ (- (- 2.0 x) x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e-7) {
		tmp = (x - y) * (1.0 / (2.0 - y));
	} else if (y <= 4200000000.0) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = 1.0 + (((2.0 - x) - x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.5d-7)) then
        tmp = (x - y) * (1.0d0 / (2.0d0 - y))
    else if (y <= 4200000000.0d0) then
        tmp = (x - y) / (2.0d0 - x)
    else
        tmp = 1.0d0 + (((2.0d0 - x) - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e-7) {
		tmp = (x - y) * (1.0 / (2.0 - y));
	} else if (y <= 4200000000.0) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = 1.0 + (((2.0 - x) - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e-7:
		tmp = (x - y) * (1.0 / (2.0 - y))
	elif y <= 4200000000.0:
		tmp = (x - y) / (2.0 - x)
	else:
		tmp = 1.0 + (((2.0 - x) - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e-7)
		tmp = Float64(Float64(x - y) * Float64(1.0 / Float64(2.0 - y)));
	elseif (y <= 4200000000.0)
		tmp = Float64(Float64(x - y) / Float64(2.0 - x));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(2.0 - x) - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.5e-7)
		tmp = (x - y) * (1.0 / (2.0 - y));
	elseif (y <= 4200000000.0)
		tmp = (x - y) / (2.0 - x);
	else
		tmp = 1.0 + (((2.0 - x) - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e-7], N[(N[(x - y), $MachinePrecision] * N[(1.0 / N[(2.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4200000000.0], N[(N[(x - y), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(2.0 - x), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000001e-7

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.6%

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

      3. associate--l- 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 79.8%

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

   if -9.5000000000000001e-7 < y < 4.2e9

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.5%

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

   if 4.2e9 < y

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 70.9%

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

      1. associate--l+ 70.9%

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

      2. associate-*r/ 70.9%

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

      3. neg-mul-1 70.9%

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

      4. associate-*r/ 70.9%

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

      5. div-sub 70.9%

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

      6. cancel-sign-sub-inv 70.9%

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

      7. metadata-eval 70.9%

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

      8. *-lft-identity 70.9%

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

      9. +-commutative 70.9%

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

      10. unsub-neg 70.9%

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

   6. Simplified 70.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1}{2 - y}\\ \mathbf{elif}\;y \leq 4200000000:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(2 - x\right) - x}{y}\\ \end{array} \]

Alternative 6: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-74}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-62}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 10^{+43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.0)
   -1.0
   (if (<= x -5.6e-74)
     (* x 0.5)
     (if (<= x 3.9e-100)
       1.0
       (if (<= x 3.1e-62) (* x 0.5) (if (<= x 1e+43) 1.0 -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.0) {
		tmp = -1.0;
	} else if (x <= -5.6e-74) {
		tmp = x * 0.5;
	} else if (x <= 3.9e-100) {
		tmp = 1.0;
	} else if (x <= 3.1e-62) {
		tmp = x * 0.5;
	} else if (x <= 1e+43) {
		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 <= (-2.0d0)) then
        tmp = -1.0d0
    else if (x <= (-5.6d-74)) then
        tmp = x * 0.5d0
    else if (x <= 3.9d-100) then
        tmp = 1.0d0
    else if (x <= 3.1d-62) then
        tmp = x * 0.5d0
    else if (x <= 1d+43) 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 <= -2.0) {
		tmp = -1.0;
	} else if (x <= -5.6e-74) {
		tmp = x * 0.5;
	} else if (x <= 3.9e-100) {
		tmp = 1.0;
	} else if (x <= 3.1e-62) {
		tmp = x * 0.5;
	} else if (x <= 1e+43) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.0:
		tmp = -1.0
	elif x <= -5.6e-74:
		tmp = x * 0.5
	elif x <= 3.9e-100:
		tmp = 1.0
	elif x <= 3.1e-62:
		tmp = x * 0.5
	elif x <= 1e+43:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.0)
		tmp = -1.0;
	elseif (x <= -5.6e-74)
		tmp = Float64(x * 0.5);
	elseif (x <= 3.9e-100)
		tmp = 1.0;
	elseif (x <= 3.1e-62)
		tmp = Float64(x * 0.5);
	elseif (x <= 1e+43)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = -1.0;
	elseif (x <= -5.6e-74)
		tmp = x * 0.5;
	elseif (x <= 3.9e-100)
		tmp = 1.0;
	elseif (x <= 3.1e-62)
		tmp = x * 0.5;
	elseif (x <= 1e+43)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.0], -1.0, If[LessEqual[x, -5.6e-74], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 3.9e-100], 1.0, If[LessEqual[x, 3.1e-62], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1e+43], 1.0, -1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2 or 1.00000000000000001e43 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.9%

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

   if -2 < x < -5.59999999999999976e-74 or 3.89999999999999977e-100 < x < 3.0999999999999999e-62

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 72.4%

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

   5. Taylor expanded in x around 0 72.4%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 72.4%

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

   7. Simplified 72.4%

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

   if -5.59999999999999976e-74 < x < 3.89999999999999977e-100 or 3.0999999999999999e-62 < x < 1.00000000000000001e43

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 58.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-74}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-62}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 10^{+43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e-7)
   (/ y (+ y -2.0))
   (if (<= y 6e+35) (/ (- x y) (- 2.0 x)) (- 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e-7) {
		tmp = y / (y + -2.0);
	} else if (y <= 6e+35) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.5d-7)) then
        tmp = y / (y + (-2.0d0))
    else if (y <= 6d+35) then
        tmp = (x - y) / (2.0d0 - x)
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e-7) {
		tmp = y / (y + -2.0);
	} else if (y <= 6e+35) {
		tmp = (x - y) / (2.0 - x);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e-7:
		tmp = y / (y + -2.0)
	elif y <= 6e+35:
		tmp = (x - y) / (2.0 - x)
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e-7)
		tmp = Float64(y / Float64(y + -2.0));
	elseif (y <= 6e+35)
		tmp = Float64(Float64(x - y) / Float64(2.0 - x));
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.5e-7)
		tmp = y / (y + -2.0);
	elseif (y <= 6e+35)
		tmp = (x - y) / (2.0 - x);
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e-7], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+35], N[(N[(x - y), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000001e-7

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. neg-mul-1 79.1%

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

   6. Simplified 79.1%

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

      1. frac-2neg 79.1%

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

      2. div-inv 79.0%

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

      3. remove-double-neg 79.0%

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

      4. sub-neg 79.0%

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

      5. distribute-neg-in 79.0%

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

      6. metadata-eval 79.0%

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

      7. remove-double-neg 79.0%

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

   8. Applied egg-rr 79.0%

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

      1. associate-*r/ 79.1%

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

      2. *-rgt-identity 79.1%

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

      3. +-commutative 79.1%

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

   10. Simplified 79.1%

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

   if -9.5000000000000001e-7 < y < 5.99999999999999981e35

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 96.4%

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

   if 5.99999999999999981e35 < y

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.7%

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

      3. associate--l- 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around inf 71.1%

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

   7. Taylor expanded in y around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. unsub-neg 71.3%

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

   9. Simplified 71.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - y}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 8: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -27000000000000 \lor \neg \left(y \leq 4.2 \cdot 10^{+35}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -27000000000000.0) (not (<= y 4.2e+35)))
   (- 1.0 (/ x y))
   (/ x (- 2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -27000000000000.0) || !(y <= 4.2e+35)) {
		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 <= (-27000000000000.0d0)) .or. (.not. (y <= 4.2d+35))) 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 <= -27000000000000.0) || !(y <= 4.2e+35)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (2.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -27000000000000.0) or not (y <= 4.2e+35):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (2.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -27000000000000.0) || !(y <= 4.2e+35))
		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 <= -27000000000000.0) || ~((y <= 4.2e+35)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (2.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -27000000000000.0], N[Not[LessEqual[y, 4.2e+35]], $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 < -2.7e13 or 4.1999999999999998e35 < y

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.7%

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

      3. associate--l- 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around inf 75.1%

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

   7. Taylor expanded in y around 0 75.3%

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

      1. mul-1-neg 75.3%

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

      2. unsub-neg 75.3%

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

   9. Simplified 75.3%

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

   if -2.7e13 < y < 4.1999999999999998e35

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 79.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -27000000000000 \lor \neg \left(y \leq 4.2 \cdot 10^{+35}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 - x}\\ \end{array} \]

Alternative 9: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e-8)
   (/ y (+ y -2.0))
   (if (<= y 6e+35) (/ x (- 2.0 x)) (- 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e-8) {
		tmp = y / (y + -2.0);
	} else if (y <= 6e+35) {
		tmp = x / (2.0 - x);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.5d-8)) then
        tmp = y / (y + (-2.0d0))
    else if (y <= 6d+35) then
        tmp = x / (2.0d0 - x)
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.5e-8) {
		tmp = y / (y + -2.0);
	} else if (y <= 6e+35) {
		tmp = x / (2.0 - x);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e-8:
		tmp = y / (y + -2.0)
	elif y <= 6e+35:
		tmp = x / (2.0 - x)
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e-8)
		tmp = Float64(y / Float64(y + -2.0));
	elseif (y <= 6e+35)
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.5e-8)
		tmp = y / (y + -2.0);
	elseif (y <= 6e+35)
		tmp = x / (2.0 - x);
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e-8], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+35], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.49999999999999935e-8

   1. Initial program 99.9%

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

      1. associate--r+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. neg-mul-1 79.1%

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

   6. Simplified 79.1%

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

      1. frac-2neg 79.1%

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

      2. div-inv 79.0%

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

      3. remove-double-neg 79.0%

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

      4. sub-neg 79.0%

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

      5. distribute-neg-in 79.0%

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

      6. metadata-eval 79.0%

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

      7. remove-double-neg 79.0%

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

   8. Applied egg-rr 79.0%

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

      1. associate-*r/ 79.1%

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

      2. *-rgt-identity 79.1%

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

      3. +-commutative 79.1%

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

   10. Simplified 79.1%

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

   if -8.49999999999999935e-8 < y < 5.99999999999999981e35

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 82.3%

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

   if 5.99999999999999981e35 < y

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.7%

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

      3. associate--l- 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around inf 71.1%

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

   7. Taylor expanded in y around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. unsub-neg 71.3%

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

   9. Simplified 71.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 10: 62.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 10^{+43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-5) -1.0 (if (<= x 1e+43) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-5) {
		tmp = -1.0;
	} else if (x <= 1e+43) {
		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 <= (-1.6d-5)) then
        tmp = -1.0d0
    else if (x <= 1d+43) 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 <= -1.6e-5) {
		tmp = -1.0;
	} else if (x <= 1e+43) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.6e-5:
		tmp = -1.0
	elif x <= 1e+43:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-5)
		tmp = -1.0;
	elseif (x <= 1e+43)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e-5)
		tmp = -1.0;
	elseif (x <= 1e+43)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e-5], -1.0, If[LessEqual[x, 1e+43], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999993e-5 or 1.00000000000000001e43 < x

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.4%

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

   if -1.59999999999999993e-5 < x < 1.00000000000000001e43

   1. Initial program 100.0%

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

      1. associate--r+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 50.7%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 10^{+43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11: 38.2% 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 100.0%

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

   1. associate--r+ 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 41.0%

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

5. Final simplification 41.0%

   \[\leadsto -1 \]

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 2023268 
(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))))