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

Percentage Accurate: 100.0% → 99.9%

Time: 6.5s

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: 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: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-22}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(-y\right) - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e-79)
   (+ (/ x y) -1.0)
   (if (<= y 2.8e-22) (+ 1.0 (* -2.0 (/ y x))) (/ y (- (- y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e-79) {
		tmp = (x / y) + -1.0;
	} else if (y <= 2.8e-22) {
		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 (y <= (-2.6d-79)) then
        tmp = (x / y) + (-1.0d0)
    else if (y <= 2.8d-22) 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 (y <= -2.6e-79) {
		tmp = (x / y) + -1.0;
	} else if (y <= 2.8e-22) {
		tmp = 1.0 + (-2.0 * (y / x));
	} else {
		tmp = y / (-y - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.6e-79:
		tmp = (x / y) + -1.0
	elif y <= 2.8e-22:
		tmp = 1.0 + (-2.0 * (y / x))
	else:
		tmp = y / (-y - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e-79)
		tmp = Float64(Float64(x / y) + -1.0);
	elseif (y <= 2.8e-22)
		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 (y <= -2.6e-79)
		tmp = (x / y) + -1.0;
	elseif (y <= 2.8e-22)
		tmp = 1.0 + (-2.0 * (y / x));
	else
		tmp = y / (-y - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e-79], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, 2.8e-22], N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[((-y) - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.59999999999999994e-79

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.8%

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

      1. neg-mul-1 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in y around inf 75.8%

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

   if -2.59999999999999994e-79 < y < 2.79999999999999995e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 81.1%

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

   if 2.79999999999999995e-22 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.6%

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

      1. neg-mul-1 75.6%

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

   5. Simplified 75.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-22}:\\ \;\;\;\;1 + -2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(-y\right) - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(-y\right) - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.2e-79)
   (+ (/ x y) -1.0)
   (if (<= y 2.9e-22) (/ (- x y) x) (/ y (- (- y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.2e-79) {
		tmp = (x / y) + -1.0;
	} else if (y <= 2.9e-22) {
		tmp = (x - 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 (y <= (-8.2d-79)) then
        tmp = (x / y) + (-1.0d0)
    else if (y <= 2.9d-22) then
        tmp = (x - 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 (y <= -8.2e-79) {
		tmp = (x / y) + -1.0;
	} else if (y <= 2.9e-22) {
		tmp = (x - y) / x;
	} else {
		tmp = y / (-y - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.2e-79:
		tmp = (x / y) + -1.0
	elif y <= 2.9e-22:
		tmp = (x - y) / x
	else:
		tmp = y / (-y - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.2e-79)
		tmp = Float64(Float64(x / y) + -1.0);
	elseif (y <= 2.9e-22)
		tmp = Float64(Float64(x - 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 (y <= -8.2e-79)
		tmp = (x / y) + -1.0;
	elseif (y <= 2.9e-22)
		tmp = (x - y) / x;
	else
		tmp = y / (-y - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.2e-79], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, 2.9e-22], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], N[(y / N[((-y) - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.19999999999999987e-79

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.8%

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

      1. neg-mul-1 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in y around inf 75.8%

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

   if -8.19999999999999987e-79 < y < 2.9000000000000002e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.9%

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

   4. Taylor expanded in x around inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

   6. Simplified 80.3%

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

   7. Taylor expanded in x around 0 80.3%

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

   if 2.9000000000000002e-22 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.6%

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

      1. neg-mul-1 75.6%

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

   5. Simplified 75.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(-y\right) - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{-79} \lor \neg \left(y \leq 2.8 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.4e-79) (not (<= y 2.8e-22))) (+ (/ x y) -1.0) (/ (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.4e-79) || !(y <= 2.8e-22)) {
		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 ((y <= (-9.4d-79)) .or. (.not. (y <= 2.8d-22))) 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 ((y <= -9.4e-79) || !(y <= 2.8e-22)) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = (x - y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.4e-79) or not (y <= 2.8e-22):
		tmp = (x / y) + -1.0
	else:
		tmp = (x - y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.4e-79) || !(y <= 2.8e-22))
		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 ((y <= -9.4e-79) || ~((y <= 2.8e-22)))
		tmp = (x / y) + -1.0;
	else
		tmp = (x - y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.4e-79], N[Not[LessEqual[y, 2.8e-22]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.4000000000000003e-79 or 2.79999999999999995e-22 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.7%

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

      1. neg-mul-1 75.7%

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

   5. Simplified 75.7%

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

   6. Taylor expanded in y around inf 75.7%

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

   if -9.4000000000000003e-79 < y < 2.79999999999999995e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.9%

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

   4. Taylor expanded in x around inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

   6. Simplified 80.3%

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

   7. Taylor expanded in x around 0 80.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{-79} \lor \neg \left(y \leq 2.8 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-79} \lor \neg \left(y \leq 3 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.2e-79) (not (<= y 3e-22))) (+ (/ x y) -1.0) (- 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.2e-79) || !(y <= 3e-22)) {
		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 <= (-5.2d-79)) .or. (.not. (y <= 3d-22))) 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 <= -5.2e-79) || !(y <= 3e-22)) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 1.0 - (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.2e-79) or not (y <= 3e-22):
		tmp = (x / y) + -1.0
	else:
		tmp = 1.0 - (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.2e-79) || !(y <= 3e-22))
		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 <= -5.2e-79) || ~((y <= 3e-22)))
		tmp = (x / y) + -1.0;
	else
		tmp = 1.0 - (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.2e-79], N[Not[LessEqual[y, 3e-22]], $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 < -5.19999999999999987e-79 or 2.9999999999999999e-22 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.7%

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

      1. neg-mul-1 75.7%

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

   5. Simplified 75.7%

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

   6. Taylor expanded in y around inf 75.7%

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

   if -5.19999999999999987e-79 < y < 2.9999999999999999e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.9%

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

   4. Taylor expanded in x around inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

   6. Simplified 80.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-79} \lor \neg \left(y \leq 3 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{-79}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-22}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.4e-79) -1.0 (if (<= y 2.85e-22) (- 1.0 (/ y x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.4e-79) {
		tmp = -1.0;
	} else if (y <= 2.85e-22) {
		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 <= (-9.4d-79)) then
        tmp = -1.0d0
    else if (y <= 2.85d-22) 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 <= -9.4e-79) {
		tmp = -1.0;
	} else if (y <= 2.85e-22) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.4e-79:
		tmp = -1.0
	elif y <= 2.85e-22:
		tmp = 1.0 - (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.4e-79)
		tmp = -1.0;
	elseif (y <= 2.85e-22)
		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 <= -9.4e-79)
		tmp = -1.0;
	elseif (y <= 2.85e-22)
		tmp = 1.0 - (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.4e-79], -1.0, If[LessEqual[y, 2.85e-22], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -9.4000000000000003e-79 or 2.8499999999999998e-22 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.0%

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

   if -9.4000000000000003e-79 < y < 2.8499999999999998e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.9%

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

   4. Taylor expanded in x around inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

   6. Simplified 80.3%

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

Alternative 7: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-79}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.2e-79) -1.0 (if (<= y 3e-22) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.2e-79) {
		tmp = -1.0;
	} else if (y <= 3e-22) {
		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 <= (-5.2d-79)) then
        tmp = -1.0d0
    else if (y <= 3d-22) 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 <= -5.2e-79) {
		tmp = -1.0;
	} else if (y <= 3e-22) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.2e-79:
		tmp = -1.0
	elif y <= 3e-22:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.2e-79)
		tmp = -1.0;
	elseif (y <= 3e-22)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.2e-79)
		tmp = -1.0;
	elseif (y <= 3e-22)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.2e-79], -1.0, If[LessEqual[y, 3e-22], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999987e-79 or 2.9999999999999999e-22 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.0%

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

   if -5.19999999999999987e-79 < y < 2.9999999999999999e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.5%

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

Alternative 8: 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. +-commutative 100.0%

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

   3. fma-undefine 100.0%

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

   4. associate-+r+ 99.9%

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

   5. *-commutative 99.9%

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

   6. distribute-rgt1-in 99.9%

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

   7. metadata-eval 99.9%

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

   8. +-commutative 99.9%

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

   9. mul-1-neg 99.9%

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

   10. unsub-neg 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

    \[\leadsto \frac{1}{\frac{y + x}{x - y}} \]
12. Add Preprocessing

Alternative 9: 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 10: 49.9% 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 51.1%

   \[\leadsto \color{blue}{-1} \]
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 2024129 
(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)))