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

Percentage Accurate: 100.0% → 100.0%

Time: 12.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{x + y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{x + y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + y} - \frac{y}{x + y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. div-sub 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-8} \lor \neg \left(y \leq 4.6 \cdot 10^{-15}\right):\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.8e-8) (not (<= y 4.6e-15)))
   (+ (* 2.0 (/ x y)) -1.0)
   (+ 1.0 (* -2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.8e-8) || !(y <= 4.6e-15)) {
		tmp = (2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (-2.0 * (y / 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 <= (-2.8d-8)) .or. (.not. (y <= 4.6d-15))) then
        tmp = (2.0d0 * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0 + ((-2.0d0) * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.8e-8) || !(y <= 4.6e-15)) {
		tmp = (2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (-2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.8e-8) or not (y <= 4.6e-15):
		tmp = (2.0 * (x / y)) + -1.0
	else:
		tmp = 1.0 + (-2.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.8e-8) || !(y <= 4.6e-15))
		tmp = Float64(Float64(2.0 * Float64(x / y)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.8e-8) || ~((y <= 4.6e-15)))
		tmp = (2.0 * (x / y)) + -1.0;
	else
		tmp = 1.0 + (-2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.8e-8], N[Not[LessEqual[y, 4.6e-15]], $MachinePrecision]], N[(N[(2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7999999999999999e-8 or 4.59999999999999981e-15 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 80.7%

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

   if -2.7999999999999999e-8 < y < 4.59999999999999981e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-8} \lor \neg \left(y \leq 4.6 \cdot 10^{-15}\right):\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-y}{x + y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-27}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.5e-11)
   (/ (- y) (+ x y))
   (if (<= y 1.9e-27) (+ 1.0 (* -2.0 (/ y x))) (+ (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.5e-11) {
		tmp = -y / (x + y);
	} else if (y <= 1.9e-27) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = (x / y) + -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 <= (-3.5d-11)) then
        tmp = -y / (x + y)
    else if (y <= 1.9d-27) then
        tmp = 1.0d0 + ((-2.0d0) * (y / x))
    else
        tmp = (x / y) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.5e-11) {
		tmp = -y / (x + y);
	} else if (y <= 1.9e-27) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.5e-11:
		tmp = -y / (x + y)
	elif y <= 1.9e-27:
		tmp = 1.0 + (-2.0 * (y / x))
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.5e-11)
		tmp = Float64(Float64(-y) / Float64(x + y));
	elseif (y <= 1.9e-27)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.5e-11)
		tmp = -y / (x + y);
	elseif (y <= 1.9e-27)
		tmp = 1.0 + (-2.0 * (y / x));
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.5e-11], N[((-y) / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e-27], N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.50000000000000019e-11

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 48.5%

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

      2. associate-/r* 49.7%

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

   5. Applied egg-rr 49.7%

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

   6. Taylor expanded in x around 0 83.5%

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

      1. neg-mul-1 83.5%

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

   8. Simplified 83.5%

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

   if -3.50000000000000019e-11 < y < 1.9e-27

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.8%

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

   if 1.9e-27 < y

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 48.0%

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

      2. associate-/r* 49.2%

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

   5. Applied egg-rr 49.2%

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

   6. Taylor expanded in x around 0 77.1%

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

      1. neg-mul-1 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in y around inf 77.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-y}{x + y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-27}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 4: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-10} \lor \neg \left(y \leq 4.5 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.05e-10) (not (<= y 4.5e-26)))
   (+ (/ x y) -1.0)
   (- 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.05e-10) || !(y <= 4.5e-26)) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 1.0 - (y / 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 <= (-2.05d-10)) .or. (.not. (y <= 4.5d-26))) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = 1.0d0 - (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.05e-10) || !(y <= 4.5e-26)) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 1.0 - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.05e-10) or not (y <= 4.5e-26):
		tmp = (x / y) + -1.0
	else:
		tmp = 1.0 - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.05e-10) || !(y <= 4.5e-26))
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = Float64(1.0 - Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.05e-10) || ~((y <= 4.5e-26)))
		tmp = (x / y) + -1.0;
	else
		tmp = 1.0 - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.05e-10], N[Not[LessEqual[y, 4.5e-26]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0499999999999999e-10 or 4.4999999999999999e-26 < y

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 48.2%

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

      2. associate-/r* 49.4%

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

   5. Applied egg-rr 49.4%

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

   6. Taylor expanded in x around 0 80.1%

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

      1. neg-mul-1 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in y around inf 80.0%

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

   if -2.0499999999999999e-10 < y < 4.4999999999999999e-26

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 63.9%

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

      2. associate-/r* 64.9%

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

   5. Applied egg-rr 64.9%

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

   6. Taylor expanded in x around inf 77.4%

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

   7. Taylor expanded in x around inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. unsub-neg 77.4%

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

   9. Simplified 77.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-10} \lor \neg \left(y \leq 4.5 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \]

Alternative 5: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-9} \lor \neg \left(y \leq 4.2 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.32e-9) (not (<= y 4.2e-24))) (+ (/ x y) -1.0) (/ x (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.32e-9) || !(y <= 4.2e-24)) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 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 <= (-1.32d-9)) .or. (.not. (y <= 4.2d-24))) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = x / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.32e-9) || !(y <= 4.2e-24)) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = x / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.32e-9) or not (y <= 4.2e-24):
		tmp = (x / y) + -1.0
	else:
		tmp = x / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.32e-9) || !(y <= 4.2e-24))
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = Float64(x / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.32e-9) || ~((y <= 4.2e-24)))
		tmp = (x / y) + -1.0;
	else
		tmp = x / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.32e-9], N[Not[LessEqual[y, 4.2e-24]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32e-9 or 4.1999999999999999e-24 < y

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 48.2%

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

      2. associate-/r* 49.4%

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

   5. Applied egg-rr 49.4%

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

   6. Taylor expanded in x around 0 80.1%

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

      1. neg-mul-1 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in y around inf 80.0%

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

   if -1.32e-9 < y < 4.1999999999999999e-24

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 63.9%

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

      2. associate-/r* 64.9%

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

   5. Applied egg-rr 64.9%

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

   6. Taylor expanded in x around inf 77.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-9} \lor \neg \left(y \leq 4.2 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y}\\ \end{array} \]

Alternative 6: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.85 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-20}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.85e-14) -1.0 (if (<= y 1.3e-20) (- 1.0 (/ y x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.85e-14) {
		tmp = -1.0;
	} else if (y <= 1.3e-20) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.85d-14)) then
        tmp = -1.0d0
    else if (y <= 1.3d-20) then
        tmp = 1.0d0 - (y / x)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.85e-14) {
		tmp = -1.0;
	} else if (y <= 1.3e-20) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.85e-14:
		tmp = -1.0
	elif y <= 1.3e-20:
		tmp = 1.0 - (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.85e-14)
		tmp = -1.0;
	elseif (y <= 1.3e-20)
		tmp = Float64(1.0 - Float64(y / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.85e-14)
		tmp = -1.0;
	elseif (y <= 1.3e-20)
		tmp = 1.0 - (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.85e-14], -1.0, If[LessEqual[y, 1.3e-20], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.85000000000000016e-14 or 1.29999999999999997e-20 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 79.4%

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

   if -4.85000000000000016e-14 < y < 1.29999999999999997e-20

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 63.9%

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

      2. associate-/r* 64.9%

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

   5. Applied egg-rr 64.9%

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

   6. Taylor expanded in x around inf 77.4%

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

   7. Taylor expanded in x around inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. unsub-neg 77.4%

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

   9. Simplified 77.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.85 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-20}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{-y}{x + y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e-20)
   (/ (- y) (+ x y))
   (if (<= y 1.6e-14) (/ x (+ x y)) (+ (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e-20) {
		tmp = -y / (x + y);
	} else if (y <= 1.6e-14) {
		tmp = x / (x + y);
	} else {
		tmp = (x / y) + -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 <= (-2.6d-20)) then
        tmp = -y / (x + y)
    else if (y <= 1.6d-14) then
        tmp = x / (x + y)
    else
        tmp = (x / y) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.6e-20) {
		tmp = -y / (x + y);
	} else if (y <= 1.6e-14) {
		tmp = x / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.6e-20:
		tmp = -y / (x + y)
	elif y <= 1.6e-14:
		tmp = x / (x + y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e-20)
		tmp = Float64(Float64(-y) / Float64(x + y));
	elseif (y <= 1.6e-14)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.6e-20)
		tmp = -y / (x + y);
	elseif (y <= 1.6e-14)
		tmp = x / (x + y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e-20], N[((-y) / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-14], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.59999999999999995e-20

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 48.5%

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

      2. associate-/r* 49.7%

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

   5. Applied egg-rr 49.7%

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

   6. Taylor expanded in x around 0 82.6%

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

      1. neg-mul-1 82.6%

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

   8. Simplified 82.6%

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

   if -2.59999999999999995e-20 < y < 1.6000000000000001e-14

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 64.2%

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

      2. associate-/r* 65.1%

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

   5. Applied egg-rr 65.1%

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

   6. Taylor expanded in x around inf 77.9%

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

   if 1.6000000000000001e-14 < y

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-sub 48.0%

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

      2. associate-/r* 49.2%

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

   5. Applied egg-rr 49.2%

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

   6. Taylor expanded in x around 0 77.1%

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

      1. neg-mul-1 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in y around inf 77.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{-y}{x + y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 8: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{x + y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 9: 73.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e-7) -1.0 (if (<= y 2e-21) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e-7) {
		tmp = -1.0;
	} else if (y <= 2e-21) {
		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 <= (-4.5d-7)) then
        tmp = -1.0d0
    else if (y <= 2d-21) 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 <= -4.5e-7) {
		tmp = -1.0;
	} else if (y <= 2e-21) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e-7:
		tmp = -1.0
	elif y <= 2e-21:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e-7)
		tmp = -1.0;
	elseif (y <= 2e-21)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e-7)
		tmp = -1.0;
	elseif (y <= 2e-21)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e-7], -1.0, If[LessEqual[y, 2e-21], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4999999999999998e-7 or 1.99999999999999982e-21 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 79.4%

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

   if -4.4999999999999998e-7 < y < 1.99999999999999982e-21

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 76.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 50.0% accurate, 7.0× 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%

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

2. Taylor expanded in x around 0 54.6%

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

3. Final simplification 54.6%

   \[\leadsto -1 \]

Developer target: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + y} - \frac{y}{x + y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"
  :precision binary64

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

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