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

Average Error: 3.4 → 0.2

Time: 7.7s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \le -1.00202048192943851 \cdot 10^{223}:\\ \;\;\;\;1 \cdot x + \left(-x \cdot y\right) \cdot z\\ \mathbf{elif}\;y \cdot z \le 8.1219315469979067 \cdot 10^{206}:\\ \;\;\;\;1 \cdot x + x \cdot \left(-y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x + \left(-y \cdot \left(x \cdot z\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Split input into 3 regimes

2. if (* y z) < -1.00202048192943851e223

   1. Initial program 30.6

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

   3. Applied sub-neg 30.6

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

   4. Applied distribute-lft-in 30.6

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

   5. Simplified 30.6

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

   7. Applied distribute-lft-neg-in 30.6

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

   8. Applied associate-*r* 0.5

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

   9. Simplified 0.5

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

   if -1.00202048192943851e223 < (* y z) < 8.1219315469979067e206

   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

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

   if 8.1219315469979067e206 < (* y z)

   1. Initial program 26.3

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

   3. Applied sub-neg 26.3

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

   4. Applied distribute-lft-in 26.3

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

   5. Simplified 26.3

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

   7. Applied distribute-rgt-neg-out 26.3

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

   8. Simplified 1.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \le -1.00202048192943851 \cdot 10^{223}:\\ \;\;\;\;1 \cdot x + \left(-x \cdot y\right) \cdot z\\ \mathbf{elif}\;y \cdot z \le 8.1219315469979067 \cdot 10^{206}:\\ \;\;\;\;1 \cdot x + x \cdot \left(-y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x + \left(-y \cdot \left(x \cdot z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))