Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D

Percentage Accurate: 99.5% → 99.8%

Time: 12.3s

Alternatives: 11

Speedup: 1.9×

• 22.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-6, z, 4\right), y - x, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (fma -6.0 z 4.0) (- y x) x))

C

double code(double x, double y, double z) {
	return fma(fma(-6.0, z, 4.0), (y - x), x);
}

Julia

function code(x, y, z)
	return fma(fma(-6.0, z, 4.0), Float64(y - x), x)
end

Wolfram

code[x_, y_, z_] := N[(N[(-6.0 * z + 4.0), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \left(\frac{2}{3} - z\right) + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(y - x\right)} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(6 \cdot \left(\frac{2}{3} - z\right), y - x, x\right)} \]

   8. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}, y - x, x\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(6 \cdot \color{blue}{\left(\frac{2}{3} + \left(\mathsf{neg}\left(z\right)\right)\right)}, y - x, x\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(6 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + \frac{2}{3}\right)}, y - x, x\right) \]

   11. distribute-lft-in N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{6 \cdot \left(\mathsf{neg}\left(z\right)\right) + 6 \cdot \frac{2}{3}}, y - x, x\right) \]

   12. neg-mul-1 N/A

       \[\leadsto \mathsf{fma}\left(6 \cdot \color{blue}{\left(-1 \cdot z\right)} + 6 \cdot \frac{2}{3}, y - x, x\right) \]

   13. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(6 \cdot -1\right) \cdot z} + 6 \cdot \frac{2}{3}, y - x, x\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-6} \cdot z + 6 \cdot \frac{2}{3}, y - x, x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot z + 6 \cdot \frac{2}{3}, y - x, x\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(6\right), z, 6 \cdot \frac{2}{3}\right)}, y - x, x\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-6}, z, 6 \cdot \frac{2}{3}\right), y - x, x\right) \]

   18. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-6, z, 6 \cdot \color{blue}{\frac{2}{3}}\right), y - x, x\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-6, z, 6 \cdot \color{blue}{\frac{2}{3}}\right), y - x, x\right) \]

   20. metadata-eval 99.8

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-6, z, \color{blue}{4}\right), y - x, x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-6, z, 4\right), y - x, x\right)} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \left(6 \cdot z\right) \cdot \left(x - y\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (* 6.0 z) (- x y))))
   (if (<= t_0 -2.0) t_1 (if (<= t_0 1.0) (fma (- y x) 4.0 x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = (6.0 * z) * (x - y);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(Float64(6.0 * z) * Float64(x - y))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(6.0 * z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\left(-1 \cdot 6\right)} \cdot \left(z \cdot \left(y - x\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{-1 \cdot \left(6 \cdot \left(z \cdot \left(y - x\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto -1 \cdot \left(6 \cdot \color{blue}{\left(\left(y - x\right) \cdot z\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(6 \cdot \left(y - x\right)\right) \cdot z\right)} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \left(\color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot z\right) \]

      6. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(y - x\right) \cdot \left(6 \cdot z\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right) \cdot \left(6 \cdot z\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(y - x\right)\right) \cdot \left(\color{blue}{\left(-6 \cdot -1\right)} \cdot z\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(y - x\right)\right) \cdot \color{blue}{\left(-6 \cdot \left(-1 \cdot z\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right) \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right)} \]

      11. sub-neg N/A

          \[\leadsto \left(-1 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      12. neg-mul-1 N/A

          \[\leadsto \left(-1 \cdot \left(y + \color{blue}{-1 \cdot x}\right)\right) \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(-1 \cdot \color{blue}{\left(-1 \cdot x + y\right)}\right) \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot x\right) + -1 \cdot y\right)} \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot -1\right) \cdot x} + -1 \cdot y\right) \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(\color{blue}{1} \cdot x + -1 \cdot y\right) \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \left(\color{blue}{x} + -1 \cdot y\right) \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \left(x + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      19. sub-neg N/A

          \[\leadsto \color{blue}{\left(x - y\right)} \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      20. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(x - y\right)} \cdot \left(-6 \cdot \left(-1 \cdot z\right)\right) \]

      21. associate-*r* N/A

          \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(\left(-6 \cdot -1\right) \cdot z\right)} \]

      22. metadata-eval N/A

          \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{6} \cdot z\right) \]

      23. *-commutative N/A

          \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(z \cdot 6\right)} \]

      24. lower-*.f64 98.0

          \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(z \cdot 6\right)} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \left(z \cdot 6\right)} \]

   if -2 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -2:\\ \;\;\;\;\left(6 \cdot z\right) \cdot \left(x - y\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))