Average Error: 0.4 → 0.1
Time: 4.1s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(6, z, -3\right), y \cdot \left(4 - 6 \cdot z\right)\right) \]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(6, z, -3\right), y \cdot \left(4 - 6 \cdot z\right)\right)

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

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. Simplified0.1

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

  5. Taylor expanded in z around 0 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022125 
(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))))