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

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 6

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, 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 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{y + x} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{y}, x, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{x}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (+ y x)) -0.5)
   (fma (/ 2.0 y) x -1.0)
   (fma (/ -2.0 x) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (y + x)) <= -0.5) {
		tmp = fma((2.0 / y), x, -1.0);
	} else {
		tmp = fma((-2.0 / x), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(y + x)) <= -0.5)
		tmp = fma(Float64(2.0 / y), x, -1.0);
	else
		tmp = fma(Float64(-2.0 / x), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(2.0 / y), $MachinePrecision] * x + -1.0), $MachinePrecision], N[(N[(-2.0 / x), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (+.f64 x y)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -0.5 < (/.f64 (-.f64 x y) (+.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. remove-double-neg N/A

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

      14. distribute-neg-out N/A

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

      15. mul-1-neg N/A

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

      16. sub-neg N/A

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

      17. mul-1-neg N/A

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

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-2}{x}, y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{y + x} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{y}, x, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{x}, y, 1\right)\\ \end{array} \]
5. 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 2024230 
(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)))