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

Percentage Accurate: 100.0% → 99.9%

Time: 7.4s

Alternatives: 13

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: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. prod-diff 99.9%

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

   4. *-commutative 99.9%

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

   5. *-un-lft-identity 99.9%

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

   6. fma-neg 99.9%

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

   7. *-un-lft-identity 99.9%

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

   8. *-commutative 99.9%

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

   9. *-un-lft-identity 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. *-rgt-identity 99.9%

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

   3. associate-+r+ 99.9%

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

   4. +-commutative 99.9%

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

   5. sub-neg 99.9%

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

   6. associate-+l+ 100.0%

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

   7. neg-mul-1 100.0%

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

   8. neg-mul-1 100.0%

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

   9. distribute-rgt-out 100.0%

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

   10. metadata-eval 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \frac{y + \left(x + y \cdot -2\right)}{y + x} \]
8. Add Preprocessing

Alternative 2: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85} \lor \neg \left(x \leq 3.9 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{x + y \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5e-85) (not (<= x 3.9e+121)))
   (/ (+ x (* y -2.0)) x)
   (+ (* 2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5e-85) || !(x <= 3.9e+121)) {
		tmp = (x + (y * -2.0)) / 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 <= (-5d-85)) .or. (.not. (x <= 3.9d+121))) then
        tmp = (x + (y * (-2.0d0))) / 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 <= -5e-85) || !(x <= 3.9e+121)) {
		tmp = (x + (y * -2.0)) / x;
	} else {
		tmp = (2.0 * (x / y)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5e-85) or not (x <= 3.9e+121):
		tmp = (x + (y * -2.0)) / x
	else:
		tmp = (2.0 * (x / y)) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5e-85) || !(x <= 3.9e+121))
		tmp = Float64(Float64(x + Float64(y * -2.0)) / 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 <= -5e-85) || ~((x <= 3.9e+121)))
		tmp = (x + (y * -2.0)) / x;
	else
		tmp = (2.0 * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5e-85], N[Not[LessEqual[x, 3.9e+121]], $MachinePrecision]], N[(N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-85 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.2%

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

   4. Taylor expanded in x around 0 80.2%

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

   if -5.0000000000000002e-85 < x < 3.89999999999999984e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 76.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85} \lor \neg \left(x \leq 3.9 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{x + y \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-85} \lor \neg \left(x \leq 4 \cdot 10^{+121}\right):\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.65e-85) (not (<= x 4e+121)))
   (+ 1.0 (* -2.0 (/ y x)))
   (+ (* 2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.65e-85) || !(x <= 4e+121)) {
		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 <= (-1.65d-85)) .or. (.not. (x <= 4d+121))) 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 <= -1.65e-85) || !(x <= 4e+121)) {
		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 <= -1.65e-85) or not (x <= 4e+121):
		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 <= -1.65e-85) || !(x <= 4e+121))
		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 <= -1.65e-85) || ~((x <= 4e+121)))
		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, -1.65e-85], N[Not[LessEqual[x, 4e+121]], $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 < -1.64999999999999986e-85 or 4.00000000000000015e121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.2%

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

   if -1.64999999999999986e-85 < x < 4.00000000000000015e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 76.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-85} \lor \neg \left(x \leq 4 \cdot 10^{+121}\right):\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-37} \lor \neg \left(x \leq 3.9 \cdot 10^{+121}\right):\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(-y\right) - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.5e-37) (not (<= x 3.9e+121)))
   (+ 1.0 (* -2.0 (/ y x)))
   (/ y (- (- y) x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6.5e-37) || !(x <= 3.9e+121)) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = y / (-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 <= (-6.5d-37)) .or. (.not. (x <= 3.9d+121))) then
        tmp = 1.0d0 + ((-2.0d0) * (y / x))
    else
        tmp = y / (-y - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6.5e-37) || !(x <= 3.9e+121)) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = y / (-y - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6.5e-37) or not (x <= 3.9e+121):
		tmp = 1.0 + (-2.0 * (y / x))
	else:
		tmp = y / (-y - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6.5e-37) || !(x <= 3.9e+121))
		tmp = Float64(1.0 + Float64(-2.0 * Float64(y / x)));
	else
		tmp = Float64(y / Float64(Float64(-y) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6.5e-37) || ~((x <= 3.9e+121)))
		tmp = 1.0 + (-2.0 * (y / x));
	else
		tmp = y / (-y - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6.5e-37], N[Not[LessEqual[x, 3.9e+121]], $MachinePrecision]], N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[((-y) - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5000000000000001e-37 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 82.4%

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

   if -6.5000000000000001e-37 < x < 3.89999999999999984e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 74.6%

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

      1. neg-mul-1 74.6%

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

   5. Simplified 74.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-37} \lor \neg \left(x \leq 3.9 \cdot 10^{+121}\right):\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(-y\right) - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+123}:\\ \;\;\;\;\frac{y}{\left(-y\right) - x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e-36)
   (/ x (+ y x))
   (if (<= x 1.26e+123) (/ y (- (- y) x)) (- 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-36) {
		tmp = x / (y + x);
	} else if (x <= 1.26e+123) {
		tmp = y / (-y - x);
	} 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 <= (-2.25d-36)) then
        tmp = x / (y + x)
    else if (x <= 1.26d+123) then
        tmp = y / (-y - x)
    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 <= -2.25e-36) {
		tmp = x / (y + x);
	} else if (x <= 1.26e+123) {
		tmp = y / (-y - x);
	} else {
		tmp = 1.0 - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.25e-36:
		tmp = x / (y + x)
	elif x <= 1.26e+123:
		tmp = y / (-y - x)
	else:
		tmp = 1.0 - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.25e-36)
		tmp = Float64(x / Float64(y + x));
	elseif (x <= 1.26e+123)
		tmp = Float64(y / Float64(Float64(-y) - x));
	else
		tmp = Float64(1.0 - Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.25e-36)
		tmp = x / (y + x);
	elseif (x <= 1.26e+123)
		tmp = y / (-y - x);
	else
		tmp = 1.0 - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.25e-36], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.26e+123], N[(y / N[((-y) - x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.25000000000000012e-36

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.1%

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

   if -2.25000000000000012e-36 < x < 1.26e123

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 74.6%

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

      1. neg-mul-1 74.6%

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

   5. Simplified 74.6%

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

   if 1.26e123 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 83.1%

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

   4. Taylor expanded in x around inf 83.8%

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

      1. mul-1-neg 83.8%

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

      2. unsub-neg 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+123}:\\ \;\;\;\;\frac{y}{\left(-y\right) - x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85} \lor \neg \left(x \leq 3.9 \cdot 10^{+121}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5e-85) (not (<= x 3.9e+121)))
   (- 1.0 (/ y x))
   (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5e-85) || !(x <= 3.9e+121)) {
		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 <= (-5d-85)) .or. (.not. (x <= 3.9d+121))) 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 <= -5e-85) || !(x <= 3.9e+121)) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5e-85) or not (x <= 3.9e+121):
		tmp = 1.0 - (y / x)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5e-85) || !(x <= 3.9e+121))
		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 <= -5e-85) || ~((x <= 3.9e+121)))
		tmp = 1.0 - (y / x);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5e-85], N[Not[LessEqual[x, 3.9e+121]], $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 < -5.0000000000000002e-85 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 79.6%

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

   4. Taylor expanded in x around inf 79.7%

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

      1. mul-1-neg 79.7%

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

      2. unsub-neg 79.7%

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

   6. Simplified 79.7%

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

   if -5.0000000000000002e-85 < x < 3.89999999999999984e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 76.3%

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

   4. Taylor expanded in x around 0 76.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85} \lor \neg \left(x \leq 3.9 \cdot 10^{+121}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85} \lor \neg \left(x \leq 3.9 \cdot 10^{+121}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5e-85) (not (<= x 3.9e+121))) (- 1.0 (/ y x)) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5e-85) || !(x <= 3.9e+121)) {
		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 <= (-5d-85)) .or. (.not. (x <= 3.9d+121))) 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 <= -5e-85) || !(x <= 3.9e+121)) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5e-85) or not (x <= 3.9e+121):
		tmp = 1.0 - (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5e-85) || !(x <= 3.9e+121))
		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 <= -5e-85) || ~((x <= 3.9e+121)))
		tmp = 1.0 - (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5e-85], N[Not[LessEqual[x, 3.9e+121]], $MachinePrecision]], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-85 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 79.6%

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

   4. Taylor expanded in x around inf 79.7%

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

      1. mul-1-neg 79.7%

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

      2. unsub-neg 79.7%

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

   6. Simplified 79.7%

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

   if -5.0000000000000002e-85 < x < 3.89999999999999984e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85} \lor \neg \left(x \leq 3.9 \cdot 10^{+121}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e-87)
   (/ x (+ y x))
   (if (<= x 3.9e+121) (/ (- x y) y) (- 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-87) {
		tmp = x / (y + x);
	} else if (x <= 3.9e+121) {
		tmp = (x - y) / y;
	} 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 <= (-1.5d-87)) then
        tmp = x / (y + x)
    else if (x <= 3.9d+121) then
        tmp = (x - y) / y
    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 <= -1.5e-87) {
		tmp = x / (y + x);
	} else if (x <= 3.9e+121) {
		tmp = (x - y) / y;
	} else {
		tmp = 1.0 - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e-87:
		tmp = x / (y + x)
	elif x <= 3.9e+121:
		tmp = (x - y) / y
	else:
		tmp = 1.0 - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e-87)
		tmp = Float64(x / Float64(y + x));
	elseif (x <= 3.9e+121)
		tmp = Float64(Float64(x - y) / y);
	else
		tmp = Float64(1.0 - Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.5e-87)
		tmp = x / (y + x);
	elseif (x <= 3.9e+121)
		tmp = (x - y) / y;
	else
		tmp = 1.0 - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e-87], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e+121], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.50000000000000008e-87

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 77.9%

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

   if -1.50000000000000008e-87 < x < 3.89999999999999984e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 76.3%

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

   if 3.89999999999999984e121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 83.1%

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

   4. Taylor expanded in x around inf 83.8%

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

      1. mul-1-neg 83.8%

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

      2. unsub-neg 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-85)
   (/ x (+ y x))
   (if (<= x 3.9e+121) (+ (/ x y) -1.0) (- 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-85) {
		tmp = x / (y + x);
	} else if (x <= 3.9e+121) {
		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 <= (-5d-85)) then
        tmp = x / (y + x)
    else if (x <= 3.9d+121) 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 <= -5e-85) {
		tmp = x / (y + x);
	} else if (x <= 3.9e+121) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 1.0 - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-85:
		tmp = x / (y + x)
	elif x <= 3.9e+121:
		tmp = (x / y) + -1.0
	else:
		tmp = 1.0 - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-85)
		tmp = Float64(x / Float64(y + x));
	elseif (x <= 3.9e+121)
		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 <= -5e-85)
		tmp = x / (y + x);
	elseif (x <= 3.9e+121)
		tmp = (x / y) + -1.0;
	else
		tmp = 1.0 - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-85], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e+121], 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 < -5.0000000000000002e-85

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 77.9%

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

   if -5.0000000000000002e-85 < x < 3.89999999999999984e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 76.3%

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

   4. Taylor expanded in x around 0 76.3%

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

   if 3.89999999999999984e121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 83.1%

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

   4. Taylor expanded in x around inf 83.8%

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

      1. mul-1-neg 83.8%

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

      2. unsub-neg 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e-38) 1.0 (if (<= x 3.9e+121) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e-38) {
		tmp = 1.0;
	} else if (x <= 3.9e+121) {
		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 <= (-8.5d-38)) then
        tmp = 1.0d0
    else if (x <= 3.9d+121) 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 <= -8.5e-38) {
		tmp = 1.0;
	} else if (x <= 3.9e+121) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.5e-38:
		tmp = 1.0
	elif x <= 3.9e+121:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e-38)
		tmp = 1.0;
	elseif (x <= 3.9e+121)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.5e-38)
		tmp = 1.0;
	elseif (x <= 3.9e+121)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e-38], 1.0, If[LessEqual[x, 3.9e+121], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000046e-38 or 3.89999999999999984e121 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 81.4%

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

   if -8.50000000000000046e-38 < x < 3.89999999999999984e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.8%

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

Alternative 11: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / N[(N[(y + x), $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. *-un-lft-identity 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. prod-diff 99.9%

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

   4. *-commutative 99.9%

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

   5. *-un-lft-identity 99.9%

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

   6. fma-neg 99.9%

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

   7. *-un-lft-identity 99.9%

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

   8. *-commutative 99.9%

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

   9. *-un-lft-identity 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. *-rgt-identity 99.9%

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

   3. associate-+r+ 99.9%

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

   4. +-commutative 99.9%

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

   5. sub-neg 99.9%

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

   6. associate-+l+ 100.0%

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

   7. neg-mul-1 100.0%

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

   8. neg-mul-1 100.0%

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

   9. distribute-rgt-out 100.0%

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

   10. metadata-eval 100.0%

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

6. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-define 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\frac{y + x}{y + \mathsf{fma}\left(y, -2, x\right)}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

   4. *-commutative 100.0%

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

   5. distribute-rgt1-in 100.0%

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

   6. metadata-eval 100.0%

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

   7. neg-mul-1 100.0%

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

   8. +-commutative 100.0%

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

   9. sub-neg 100.0%

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

10. Simplified 100.0%

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

Alternative 12: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 13: 51.0% 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 99.9%

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

3. Taylor expanded in x around 0 46.2%

   \[\leadsto \color{blue}{-1} \]
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 2024145 
(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)))