Average Error: 0.2 → 0.2
Time: 2.2s
Precision: binary64
\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]
↓
\[y \cdot \left(3 \cdot x\right) + y \cdot \left(-0.413793103448275856\right)\]

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[y \cdot \left(x \cdot 3 - 0.413793103448275856\right)\]

Derivation

  1. Initial program 0.2

    \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right) - 0.413793103448275856 \cdot y}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{y \cdot \left(3 \cdot x - 0.413793103448275856\right)}\]
  4. Using strategy rm

  5. Applied sub-neg0.2

    \[\leadsto y \cdot \color{blue}{\left(3 \cdot x + \left(-0.413793103448275856\right)\right)}\]

  6. Applied distribute-lft-in0.2

    \[\leadsto \color{blue}{y \cdot \left(3 \cdot x\right) + y \cdot \left(-0.413793103448275856\right)}\]

  7. Final simplification0.2

    \[\leadsto y \cdot \left(3 \cdot x\right) + y \cdot \left(-0.413793103448275856\right)\]

Reproduce

herbie shell --seed 2020147 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (* y (- (* x 3.0) 0.41379310344827586))

  (* (* (- x (/ 16.0 116.0)) 3.0) y))