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

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 12

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 99.9%

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

   1. div-sub 99.9%

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{x}{x + y} - \frac{y}{x + y}} \]
5. Add Preprocessing

Alternative 2: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-29} \lor \neg \left(x \leq 1.7 \cdot 10^{-10}\right):\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e-29) (not (<= x 1.7e-10)))
   (+ 1.0 (* -2.0 (/ y x)))
   (+ (* 2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e-29) || !(x <= 1.7e-10)) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = (2.0 * (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 ((x <= (-1d-29)) .or. (.not. (x <= 1.7d-10))) then
        tmp = 1.0d0 + ((-2.0d0) * (y / x))
    else
        tmp = (2.0d0 * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1e-29) || !(x <= 1.7e-10)) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = (2.0 * (x / y)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1e-29) or not (x <= 1.7e-10):
		tmp = 1.0 + (-2.0 * (y / x))
	else:
		tmp = (2.0 * (x / y)) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1e-29) || !(x <= 1.7e-10))
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(2.0 * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1e-29) || ~((x <= 1.7e-10)))
		tmp = 1.0 + (-2.0 * (y / x));
	else
		tmp = (2.0 * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1e-29], N[Not[LessEqual[x, 1.7e-10]], $MachinePrecision]], N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999943e-30 or 1.70000000000000007e-10 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 77.9%

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

   if -9.99999999999999943e-30 < x < 1.70000000000000007e-10

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 82.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-29} \lor \neg \left(x \leq 1.7 \cdot 10^{-10}\right):\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-30} \lor \neg \left(x \leq 4.1 \cdot 10^{-11}\right):\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6e-30) (not (<= x 4.1e-11)))
   (+ 1.0 (* -2.0 (/ y x)))
   (/ (- x y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6e-30) || !(x <= 4.1e-11)) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = (x - y) / 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 <= (-6d-30)) .or. (.not. (x <= 4.1d-11))) then
        tmp = 1.0d0 + ((-2.0d0) * (y / x))
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6e-30) || !(x <= 4.1e-11)) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6e-30) or not (x <= 4.1e-11):
		tmp = 1.0 + (-2.0 * (y / x))
	else:
		tmp = (x - y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6e-30) || !(x <= 4.1e-11))
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6e-30) || ~((x <= 4.1e-11)))
		tmp = 1.0 + (-2.0 * (y / x));
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6e-30], N[Not[LessEqual[x, 4.1e-11]], $MachinePrecision]], N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.9999999999999998e-30 or 4.1000000000000001e-11 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 77.9%

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

   if -5.9999999999999998e-30 < x < 4.1000000000000001e-11

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-30} \lor \neg \left(x \leq 4.1 \cdot 10^{-11}\right):\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-29} \lor \neg \left(x \leq 2.8 \cdot 10^{-11}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e-29) (not (<= x 2.8e-11))) (- 1.0 (/ y x)) (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e-29) || !(x <= 2.8e-11)) {
		tmp = 1.0 - (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 ((x <= (-1d-29)) .or. (.not. (x <= 2.8d-11))) then
        tmp = 1.0d0 - (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 ((x <= -1e-29) || !(x <= 2.8e-11)) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1e-29) or not (x <= 2.8e-11):
		tmp = 1.0 - (y / x)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1e-29) || !(x <= 2.8e-11))
		tmp = Float64(1.0 - 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 ((x <= -1e-29) || ~((x <= 2.8e-11)))
		tmp = 1.0 - (y / x);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1e-29], N[Not[LessEqual[x, 2.8e-11]], $MachinePrecision]], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999943e-30 or 2.8e-11 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 76.7%

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

      1. div-sub 76.7%

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

      2. *-inverses 76.7%

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

      3. sub-neg 76.7%

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

   5. Applied egg-rr 76.7%

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

   6. Taylor expanded in y around 0 76.7%

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

      1. mul-1-neg 76.7%

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

      2. sub-neg 76.7%

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

   8. Simplified 76.7%

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

   if -9.99999999999999943e-30 < x < 2.8e-11

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.3%

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

      1. neg-mul-1 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in y around inf 81.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-29} \lor \neg \left(x \leq 2.8 \cdot 10^{-11}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26 \cdot 10^{-74} \lor \neg \left(x \leq 1.6 \cdot 10^{-10}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.26e-74) (not (<= x 1.6e-10))) (- 1.0 (/ y x)) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.26e-74) || !(x <= 1.6e-10)) {
		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 ((x <= (-1.26d-74)) .or. (.not. (x <= 1.6d-10))) 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 ((x <= -1.26e-74) || !(x <= 1.6e-10)) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.26e-74) or not (x <= 1.6e-10):
		tmp = 1.0 - (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.26e-74) || !(x <= 1.6e-10))
		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 ((x <= -1.26e-74) || ~((x <= 1.6e-10)))
		tmp = 1.0 - (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.26e-74], N[Not[LessEqual[x, 1.6e-10]], $MachinePrecision]], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.25999999999999997e-74 or 1.5999999999999999e-10 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 74.8%

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

      1. div-sub 74.8%

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

      2. *-inverses 74.8%

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

      3. sub-neg 74.8%

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

   5. Applied egg-rr 74.8%

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

   6. Taylor expanded in y around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. sub-neg 74.8%

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

   8. Simplified 74.8%

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

   if -1.25999999999999997e-74 < x < 1.5999999999999999e-10

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26 \cdot 10^{-74} \lor \neg \left(x \leq 1.6 \cdot 10^{-10}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-10}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.6e-30)
   (/ x (+ x y))
   (if (<= x 6e-10) (/ (- x y) y) (/ (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.6e-30) {
		tmp = x / (x + y);
	} else if (x <= 6e-10) {
		tmp = (x - y) / y;
	} else {
		tmp = (x - 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 (x <= (-9.6d-30)) then
        tmp = x / (x + y)
    else if (x <= 6d-10) then
        tmp = (x - y) / y
    else
        tmp = (x - y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.6e-30) {
		tmp = x / (x + y);
	} else if (x <= 6e-10) {
		tmp = (x - y) / y;
	} else {
		tmp = (x - y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.6e-30:
		tmp = x / (x + y)
	elif x <= 6e-10:
		tmp = (x - y) / y
	else:
		tmp = (x - y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.6e-30)
		tmp = Float64(x / Float64(x + y));
	elseif (x <= 6e-10)
		tmp = Float64(Float64(x - y) / y);
	else
		tmp = Float64(Float64(x - y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.6e-30)
		tmp = x / (x + y);
	elseif (x <= 6e-10)
		tmp = (x - y) / y;
	else
		tmp = (x - y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.6e-30], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-10], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5999999999999994e-30

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 83.9%

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

   4. Taylor expanded in y around 0 76.9%

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

   if -9.5999999999999994e-30 < x < 6e-10

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.6%

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

   if 6e-10 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 76.4%

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

Alternative 7: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-30)
   (/ x (+ x y))
   (if (<= x 2.2e-11) (+ (/ x y) -1.0) (/ (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-30) {
		tmp = x / (x + y);
	} else if (x <= 2.2e-11) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = (x - 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 (x <= (-4.8d-30)) then
        tmp = x / (x + y)
    else if (x <= 2.2d-11) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = (x - y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-30) {
		tmp = x / (x + y);
	} else if (x <= 2.2e-11) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = (x - y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e-30:
		tmp = x / (x + y)
	elif x <= 2.2e-11:
		tmp = (x / y) + -1.0
	else:
		tmp = (x - y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-30)
		tmp = Float64(x / Float64(x + y));
	elseif (x <= 2.2e-11)
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = Float64(Float64(x - y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e-30)
		tmp = x / (x + y);
	elseif (x <= 2.2e-11)
		tmp = (x / y) + -1.0;
	else
		tmp = (x - y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e-30], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-11], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999997e-30

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 83.9%

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

   4. Taylor expanded in y around 0 76.9%

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

   if -4.7999999999999997e-30 < x < 2.2000000000000002e-11

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.3%

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

      1. neg-mul-1 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in y around inf 81.6%

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

   if 2.2000000000000002e-11 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 76.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e-30)
   (/ x (+ x y))
   (if (<= x 5.4e-11) (+ (/ x y) -1.0) (- 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e-30) {
		tmp = x / (x + y);
	} else if (x <= 5.4e-11) {
		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 (x <= (-9d-30)) then
        tmp = x / (x + y)
    else if (x <= 5.4d-11) 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 (x <= -9e-30) {
		tmp = x / (x + y);
	} else if (x <= 5.4e-11) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 1.0 - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e-30:
		tmp = x / (x + y)
	elif x <= 5.4e-11:
		tmp = (x / y) + -1.0
	else:
		tmp = 1.0 - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e-30)
		tmp = Float64(x / Float64(x + y));
	elseif (x <= 5.4e-11)
		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 (x <= -9e-30)
		tmp = x / (x + y);
	elseif (x <= 5.4e-11)
		tmp = (x / y) + -1.0;
	else
		tmp = 1.0 - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e-30], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e-11], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999935e-30

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 83.9%

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

   4. Taylor expanded in y around 0 76.9%

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

   if -8.99999999999999935e-30 < x < 5.40000000000000009e-11

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.3%

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

      1. neg-mul-1 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in y around inf 81.6%

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

   if 5.40000000000000009e-11 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 76.4%

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

      1. div-sub 76.4%

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

      2. *-inverses 76.4%

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

      3. sub-neg 76.4%

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

   5. Applied egg-rr 76.4%

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

   6. Taylor expanded in y around 0 76.4%

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

      1. mul-1-neg 76.4%

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

      2. sub-neg 76.4%

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

   8. Simplified 76.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-10}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-31) 1.0 (if (<= x 3e-10) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-31) {
		tmp = 1.0;
	} else if (x <= 3e-10) {
		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 <= (-5d-31)) then
        tmp = 1.0d0
    else if (x <= 3d-10) 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 <= -5e-31) {
		tmp = 1.0;
	} else if (x <= 3e-10) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-31:
		tmp = 1.0
	elif x <= 3e-10:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-31)
		tmp = 1.0;
	elseif (x <= 3e-10)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-31)
		tmp = 1.0;
	elseif (x <= 3e-10)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-31], 1.0, If[LessEqual[x, 3e-10], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -5e-31 or 3e-10 < x

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 75.5%

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

   if -5e-31 < x < 3e-10

   1. Initial program 99.9%

      \[\frac{x - y}{x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.4%

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

Alternative 10: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x + y}{x - y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (/ (+ x y) (- x y))))

C

double code(double x, double y) {
	return 1.0 / ((x + y) / (x - y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / ((x + y) / (x - y))
end function

Java

public static double code(double x, double y) {
	return 1.0 / ((x + y) / (x - y));
}

Python

def code(x, y):
	return 1.0 / ((x + y) / (x - y))

Julia

function code(x, y)
	return Float64(1.0 / Float64(Float64(x + y) / Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / ((x + y) / (x - y));
end

Wolfram

code[x_, y_] := N[(1.0 / N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. associate-/r/ 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. clear-num 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{x - y}}} \]
7. Add Preprocessing

Alternative 11: 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 99.9%

   \[\frac{x - y}{x + y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 12: 49.5% 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 99.9%

   \[\frac{x - y}{x + y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 50.6%

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

Developer Target 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]

Reproduce

herbie shell --seed 2024118 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ x (+ x y)) (/ y (+ x y))))

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