Average Error: 0.4 → 0.2
Time: 3.7s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
↓
\[\left(4 \cdot y + 6 \cdot \left(z \cdot x\right)\right) + \mathsf{fma}\left(x, -3, z \cdot \left(y \cdot -6\right)\right) \]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\left(4 \cdot y + 6 \cdot \left(z \cdot x\right)\right) + \mathsf{fma}\left(x, -3, z \cdot \left(y \cdot -6\right)\right)

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))
↓
(FPCore (x y z)
 :precision binary64
 (+ (+ (* 4.0 y) (* 6.0 (* z x))) (fma x -3.0 (* z (* y -6.0)))))
double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}
↓
double code(double x, double y, double z) {
	return ((4.0 * y) + (6.0 * (z * x))) + fma(x, -3.0, (z * (y * -6.0)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.4

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

  2. Simplified0.2

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

  3. Taylor expanded in y around 0 0.2

    \[\leadsto \color{blue}{\left(4 \cdot y + 6 \cdot \left(z \cdot x\right)\right) - \left(6 \cdot \left(y \cdot z\right) + 3 \cdot x\right)} \]

  4. Applied sub-neg_binary640.2

    \[\leadsto \color{blue}{\left(4 \cdot y + 6 \cdot \left(z \cdot x\right)\right) + \left(-\left(6 \cdot \left(y \cdot z\right) + 3 \cdot x\right)\right)} \]

  5. Simplified0.2

    \[\leadsto \left(4 \cdot y + 6 \cdot \left(z \cdot x\right)\right) + \color{blue}{\mathsf{fma}\left(x, -3, z \cdot \left(y \cdot -6\right)\right)} \]

  6. Final simplification0.2

    \[\leadsto \left(4 \cdot y + 6 \cdot \left(z \cdot x\right)\right) + \mathsf{fma}\left(x, -3, z \cdot \left(y \cdot -6\right)\right) \]

Reproduce

herbie shell --seed 2022088 
(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))))