Average Error: 0.4 → 0.2
Time: 3.5s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\left(y \cdot 4 - x \cdot 3\right) + \left(y - x\right) \cdot \left(6 \cdot \left(-z\right)\right)\]

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

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

  4. Applied sub-neg0.2

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

  5. Applied distribute-lft-in0.2

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

  6. Applied distribute-lft-in0.2

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

  7. Applied associate-+r+0.2

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

  8. Taylor expanded around 0 0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020179 
(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))))