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

Percentage Accurate: 99.7% → 99.7%

Time: 9.3s

Alternatives: 11

Speedup: 1.1×

• 21.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * 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) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * 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) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

   \[\leadsto z \cdot \left(6 \cdot \left(y - x\right)\right) + x \]
4. Add Preprocessing

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-12}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot \left(x - y\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(6 \cdot \left(y - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.8e-12)
   (* (* -6.0 z) (- x y))
   (if (<= z 2.9e-6) (fma (* z y) 6.0 x) (* z (* 6.0 (- y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.8e-12) {
		tmp = (-6.0 * z) * (x - y);
	} else if (z <= 2.9e-6) {
		tmp = fma((z * y), 6.0, x);
	} else {
		tmp = z * (6.0 * (y - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.8e-12)
		tmp = Float64(Float64(-6.0 * z) * Float64(x - y));
	elseif (z <= 2.9e-6)
		tmp = fma(Float64(z * y), 6.0, x);
	else
		tmp = Float64(z * Float64(6.0 * Float64(y - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.8e-12], N[(N[(-6.0 * z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e-6], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(z * N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.8000000000000001e-12

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. distribute-lft-neg-in N/A

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

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(y - x\right)\right)} \cdot \left(-6 \cdot z\right) \]

      8. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(0 - y\right) + x\right)} \cdot \left(-6 \cdot z\right) \]

      9. neg-sub0 N/A

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. *-inverses N/A

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

      13. associate-*l/ N/A

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

      14. associate-*r/ N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-in N/A

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

      18. +-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 97.3%

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

   if -6.8000000000000001e-12 < z < 2.9000000000000002e-6

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 2.9000000000000002e-6 < z

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-neg-in N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-in N/A

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

      18. +-commutative N/A

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

      19. mul-1-neg N/A

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

      20. sub-neg N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower--.f64 98.2

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

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-12}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot \left(x - y\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(6 \cdot \left(y - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- x (* (* 6.0 z) (- x y))))

C

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

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 - ((6.0d0 * z) * (x - y))
end function

Java

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

Python

def code(x, y, z):
	return x - ((6.0 * z) * (x - y))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = x - ((6.0 * z) * (x - y));
end

Wolfram

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (* 6 z) (- x y))))

  (+ x (* (* (- y x) 6.0) z)))