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

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 9

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: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x} + -1\\ \mathbf{if}\;x \leq -6.7 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-267}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+89}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ y x) -1.0)))
   (if (<= x -6.7e+22)
     t_0
     (if (<= x -4.8e-65)
       1.0
       (if (<= x -5.8e-267) (* (- x y) 0.5) (if (<= x 5.2e+89) 1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = (y / x) + -1.0;
	double tmp;
	if (x <= -6.7e+22) {
		tmp = t_0;
	} else if (x <= -4.8e-65) {
		tmp = 1.0;
	} else if (x <= -5.8e-267) {
		tmp = (x - y) * 0.5;
	} else if (x <= 5.2e+89) {
		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 = (y / x) + (-1.0d0)
    if (x <= (-6.7d+22)) then
        tmp = t_0
    else if (x <= (-4.8d-65)) then
        tmp = 1.0d0
    else if (x <= (-5.8d-267)) then
        tmp = (x - y) * 0.5d0
    else if (x <= 5.2d+89) 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 tmp;
	if (x <= -6.7e+22) {
		tmp = t_0;
	} else if (x <= -4.8e-65) {
		tmp = 1.0;
	} else if (x <= -5.8e-267) {
		tmp = (x - y) * 0.5;
	} else if (x <= 5.2e+89) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / x) + -1.0
	tmp = 0
	if x <= -6.7e+22:
		tmp = t_0
	elif x <= -4.8e-65:
		tmp = 1.0
	elif x <= -5.8e-267:
		tmp = (x - y) * 0.5
	elif x <= 5.2e+89:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / x) + -1.0)
	tmp = 0.0
	if (x <= -6.7e+22)
		tmp = t_0;
	elseif (x <= -4.8e-65)
		tmp = 1.0;
	elseif (x <= -5.8e-267)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 5.2e+89)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / x) + -1.0;
	tmp = 0.0;
	if (x <= -6.7e+22)
		tmp = t_0;
	elseif (x <= -4.8e-65)
		tmp = 1.0;
	elseif (x <= -5.8e-267)
		tmp = (x - y) * 0.5;
	elseif (x <= 5.2e+89)
		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]}, If[LessEqual[x, -6.7e+22], t$95$0, If[LessEqual[x, -4.8e-65], 1.0, If[LessEqual[x, -5.8e-267], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.2e+89], 1.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7000000000000002e22 or 5.2000000000000001e89 < x

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.8%

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

      3. associate--r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 85.1%

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

   5. Taylor expanded in x around 0 85.2%

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

   if -6.7000000000000002e22 < x < -4.8000000000000003e-65 or -5.80000000000000043e-267 < x < 5.2000000000000001e89

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 60.5%

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

   if -4.8000000000000003e-65 < x < -5.80000000000000043e-267

   1. Initial program 100.0%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

      3. associate--r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in y around 0 65.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x} + -1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-267}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+89}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} + -1\\ \end{array} \]

Alternative 3: 62.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-265}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10^{+73}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e+23)
   -1.0
   (if (<= x -2.1e-65)
     1.0
     (if (<= x -9.8e-265) (* (- x y) 0.5) (if (<= x 1e+73) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e+23) {
		tmp = -1.0;
	} else if (x <= -2.1e-65) {
		tmp = 1.0;
	} else if (x <= -9.8e-265) {
		tmp = (x - y) * 0.5;
	} else if (x <= 1e+73) {
		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 <= (-1.5d+23)) then
        tmp = -1.0d0
    else if (x <= (-2.1d-65)) then
        tmp = 1.0d0
    else if (x <= (-9.8d-265)) then
        tmp = (x - y) * 0.5d0
    else if (x <= 1d+73) 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 <= -1.5e+23) {
		tmp = -1.0;
	} else if (x <= -2.1e-65) {
		tmp = 1.0;
	} else if (x <= -9.8e-265) {
		tmp = (x - y) * 0.5;
	} else if (x <= 1e+73) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e+23:
		tmp = -1.0
	elif x <= -2.1e-65:
		tmp = 1.0
	elif x <= -9.8e-265:
		tmp = (x - y) * 0.5
	elif x <= 1e+73:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e+23)
		tmp = -1.0;
	elseif (x <= -2.1e-65)
		tmp = 1.0;
	elseif (x <= -9.8e-265)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 1e+73)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.5e+23)
		tmp = -1.0;
	elseif (x <= -2.1e-65)
		tmp = 1.0;
	elseif (x <= -9.8e-265)
		tmp = (x - y) * 0.5;
	elseif (x <= 1e+73)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e+23], -1.0, If[LessEqual[x, -2.1e-65], 1.0, If[LessEqual[x, -9.8e-265], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1e+73], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e23 or 9.99999999999999983e72 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.1%

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

   if -1.5e23 < x < -2.10000000000000003e-65 or -9.79999999999999999e-265 < x < 9.99999999999999983e72

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 60.7%

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

   if -2.10000000000000003e-65 < x < -9.79999999999999999e-265

   1. Initial program 100.0%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

      3. associate--r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in y around 0 65.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(x - y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-265}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10^{+73}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+23} \lor \neg \left(x \leq 5 \cdot 10^{+97}\right):\\ \;\;\;\;\frac{x - y}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.45e+23) (not (<= x 5e+97)))
   (/ (- x y) (- x))
   (/ (- x y) (- 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.45e+23) || !(x <= 5e+97)) {
		tmp = (x - y) / -x;
	} else {
		tmp = (x - y) / (2.0 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.45d+23)) .or. (.not. (x <= 5d+97))) then
        tmp = (x - y) / -x
    else
        tmp = (x - y) / (2.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.45e+23) || !(x <= 5e+97)) {
		tmp = (x - y) / -x;
	} else {
		tmp = (x - y) / (2.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.45e+23) or not (x <= 5e+97):
		tmp = (x - y) / -x
	else:
		tmp = (x - y) / (2.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.45e+23) || !(x <= 5e+97))
		tmp = Float64(Float64(x - y) / Float64(-x));
	else
		tmp = Float64(Float64(x - y) / Float64(2.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.45e+23) || ~((x <= 5e+97)))
		tmp = (x - y) / -x;
	else
		tmp = (x - y) / (2.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.45e+23], N[Not[LessEqual[x, 5e+97]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / N[(2.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000006e23 or 4.99999999999999999e97 < x

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.8%

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

      3. associate--r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 85.7%

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

      1. expm1-log1p-u 83.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{x} \cdot \left(x - y\right)\right)\right)} \]

      2. expm1-udef 83.1%

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

      3. *-commutative 83.1%

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

      4. frac-2neg 83.1%

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

      5. metadata-eval 83.1%

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

      6. un-div-inv 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. expm1-def 83.2%

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

      2. expm1-log1p 85.8%

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

   8. Simplified 85.8%

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

   if -1.45000000000000006e23 < x < 4.99999999999999999e97

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. associate--r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 91.5%

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

   5. Taylor expanded in x around 0 91.6%

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

      1. +-commutative 91.6%

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

      2. mul-1-neg 91.6%

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

      3. sub-neg 91.6%

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

      4. div-sub 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+23} \lor \neg \left(x \leq 5 \cdot 10^{+97}\right):\\ \;\;\;\;\frac{x - y}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{2 - y}\\ \end{array} \]

Alternative 5: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-69}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+72}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+23)
   -1.0
   (if (<= x -3.3e-65)
     1.0
     (if (<= x -1e-69) (* x 0.5) (if (<= x 7e+72) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+23) {
		tmp = -1.0;
	} else if (x <= -3.3e-65) {
		tmp = 1.0;
	} else if (x <= -1e-69) {
		tmp = x * 0.5;
	} else if (x <= 7e+72) {
		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 <= (-2d+23)) then
        tmp = -1.0d0
    else if (x <= (-3.3d-65)) then
        tmp = 1.0d0
    else if (x <= (-1d-69)) then
        tmp = x * 0.5d0
    else if (x <= 7d+72) 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 <= -2e+23) {
		tmp = -1.0;
	} else if (x <= -3.3e-65) {
		tmp = 1.0;
	} else if (x <= -1e-69) {
		tmp = x * 0.5;
	} else if (x <= 7e+72) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e+23:
		tmp = -1.0
	elif x <= -3.3e-65:
		tmp = 1.0
	elif x <= -1e-69:
		tmp = x * 0.5
	elif x <= 7e+72:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+23)
		tmp = -1.0;
	elseif (x <= -3.3e-65)
		tmp = 1.0;
	elseif (x <= -1e-69)
		tmp = Float64(x * 0.5);
	elseif (x <= 7e+72)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+23)
		tmp = -1.0;
	elseif (x <= -3.3e-65)
		tmp = 1.0;
	elseif (x <= -1e-69)
		tmp = x * 0.5;
	elseif (x <= 7e+72)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+23], -1.0, If[LessEqual[x, -3.3e-65], 1.0, If[LessEqual[x, -1e-69], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 7e+72], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999998e23 or 7.0000000000000002e72 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.1%

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

   if -1.9999999999999998e23 < x < -3.3000000000000001e-65 or -9.9999999999999996e-70 < x < 7.0000000000000002e72

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 54.7%

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

   if -3.3000000000000001e-65 < x < -9.9999999999999996e-70

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-69}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+72}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.2e-7) 1.0 (if (<= y 4e+46) (/ x (- 2.0 x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.2e-7) {
		tmp = 1.0;
	} else if (y <= 4e+46) {
		tmp = x / (2.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.2d-7)) then
        tmp = 1.0d0
    else if (y <= 4d+46) then
        tmp = x / (2.0d0 - x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.2e-7) {
		tmp = 1.0;
	} else if (y <= 4e+46) {
		tmp = x / (2.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.2e-7:
		tmp = 1.0
	elif y <= 4e+46:
		tmp = x / (2.0 - x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.2e-7)
		tmp = 1.0;
	elseif (y <= 4e+46)
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.2e-7)
		tmp = 1.0;
	elseif (y <= 4e+46)
		tmp = x / (2.0 - x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.2e-7], 1.0, If[LessEqual[y, 4e+46], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999999e-7 or 4e46 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 77.1%

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

   if -6.1999999999999999e-7 < y < 4e46

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e-7) (/ y (+ y -2.0)) (if (<= y 3.7e+41) (/ x (- 2.0 x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e-7) {
		tmp = y / (y + -2.0);
	} else if (y <= 3.7e+41) {
		tmp = x / (2.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.2d-7)) then
        tmp = y / (y + (-2.0d0))
    else if (y <= 3.7d+41) then
        tmp = x / (2.0d0 - x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.2e-7) {
		tmp = y / (y + -2.0);
	} else if (y <= 3.7e+41) {
		tmp = x / (2.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e-7:
		tmp = y / (y + -2.0)
	elif y <= 3.7e+41:
		tmp = x / (2.0 - x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e-7)
		tmp = Float64(y / Float64(y + -2.0));
	elseif (y <= 3.7e+41)
		tmp = Float64(x / Float64(2.0 - x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e-7)
		tmp = y / (y + -2.0);
	elseif (y <= 3.7e+41)
		tmp = x / (2.0 - x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e-7], N[(y / N[(y + -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+41], N[(x / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2000000000000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. distribute-neg-frac 74.1%

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

   4. Simplified 74.1%

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

      1. frac-2neg 74.1%

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

      2. div-inv 74.0%

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

      3. remove-double-neg 74.0%

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

      4. sub-neg 74.0%

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

      5. distribute-neg-in 74.0%

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

      6. metadata-eval 74.0%

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

      7. remove-double-neg 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.1%

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

      2. *-rgt-identity 74.1%

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

      3. +-commutative 74.1%

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

   8. Simplified 74.1%

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

   if -3.2000000000000001e-7 < y < 3.69999999999999981e41

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.2%

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

   if 3.69999999999999981e41 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 87.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 62.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+73}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+23) -1.0 (if (<= x 1.35e+73) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+23) {
		tmp = -1.0;
	} else if (x <= 1.35e+73) {
		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 <= (-1d+23)) then
        tmp = -1.0d0
    else if (x <= 1.35d+73) 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 <= -1e+23) {
		tmp = -1.0;
	} else if (x <= 1.35e+73) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e+23:
		tmp = -1.0
	elif x <= 1.35e+73:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e+23)
		tmp = -1.0;
	elseif (x <= 1.35e+73)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+23)
		tmp = -1.0;
	elseif (x <= 1.35e+73)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e+23], -1.0, If[LessEqual[x, 1.35e+73], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999992e22 or 1.35e73 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.1%

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

   if -9.9999999999999992e22 < x < 1.35e73

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 53.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+73}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9: 38.7% 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. Taylor expanded in x around inf 36.7%

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

3. Final simplification 36.7%

   \[\leadsto -1 \]

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 2023221 
(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))))