Average Error: 0.4 → 0.2
Time: 3.1s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(z, 6, -3\right), y \cdot 4\right) + z \cdot \left(y \cdot -6\right) \]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(z, 6, -3\right), y \cdot 4\right) + z \cdot \left(y \cdot -6\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 z 6.0 -3.0) (* y 4.0)) (* 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 fma(x, fma(z, 6.0, -3.0), (y * 4.0)) + (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. Applied fma-udef_binary640.2

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

  7. Applied associate-+r+_binary640.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2021313 
(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))))