Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I

Average Error: 3.4 → 0.2

Time: 2.9s

Precision: binary64

Math

\[x \cdot \left(1 - y \cdot z\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \le -1.1617888851699199 \cdot 10^{191} \lor \neg \left(y \cdot z \le 1.6714011123855759 \cdot 10^{211}\right):\\ \;\;\;\;x \cdot 1 + z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 + \left(y \cdot z\right) \cdot \left(-x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Split input into 2 regimes

2. if (* y z) < -1.1617888851699199e191 or 1.6714011123855759e211 < (* y z)

   1. Initial program 27.3

      \[x \cdot \left(1 - y \cdot z\right)\]
   2. Using strategy rm

   3. Applied sub-neg 27.3

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(-y \cdot z\right)\right)}\]

   4. Applied distribute-lft-in 27.3

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-y \cdot z\right)}\]

   5. Simplified 27.3

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(y \cdot \left(-z\right)\right)}\]
   6. Using strategy rm

   7. Applied associate-*r* 0.8

      \[\leadsto x \cdot 1 + \color{blue}{\left(x \cdot y\right) \cdot \left(-z\right)}\]

   if -1.1617888851699199e191 < (* y z) < 1.6714011123855759e211

   1. Initial program 0.1

      \[x \cdot \left(1 - y \cdot z\right)\]
   2. Using strategy rm

   3. Applied sub-neg 0.1

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(-y \cdot z\right)\right)}\]

   4. Applied distribute-lft-in 0.1

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-y \cdot z\right)}\]

   5. Simplified 0.1

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(y \cdot \left(-z\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \le -1.1617888851699199 \cdot 10^{191} \lor \neg \left(y \cdot z \le 1.6714011123855759 \cdot 10^{211}\right):\\ \;\;\;\;x \cdot 1 + z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 + \left(y \cdot z\right) \cdot \left(-x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))