Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1

Average Error: 38.6 → 26.3

Time: 4.0s

Precision: binary64

Math

\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.4940319537607468 \cdot 10^{66}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 3.88863330867275493 \cdot 10^{130}:\\ \;\;\;\;\sqrt{\left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right) \cdot 0.333333333333333315}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	38.6
Target	25.5
Herbie	26.3

\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -4.4940319537607468e66

   1. Initial program 51.1

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
   2. Using strategy rm

   3. Applied sqrt-div 51.1

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

   4. Simplified 51.1

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

   5. Taylor expanded around -inf 20.8

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\sqrt{3}}\]

   6. Simplified 20.8

      \[\leadsto \frac{\color{blue}{-x}}{\sqrt{3}}\]

   if -4.4940319537607468e66 < x < 3.88863330867275493e130

   1. Initial program 30.2

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]

   2. Taylor expanded around 0 30.3

      \[\leadsto \sqrt{\color{blue}{0.333333333333333315 \cdot {z}^{2} + \left(0.333333333333333315 \cdot {x}^{2} + 0.333333333333333315 \cdot {y}^{2}\right)}}\]

   3. Simplified 30.3

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right) \cdot 0.333333333333333315}}\]

   if 3.88863330867275493e130 < x

   1. Initial program 59.7

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]

   2. Taylor expanded around inf 16.0

      \[\leadsto \color{blue}{x \cdot \sqrt{0.333333333333333315}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 26.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4940319537607468 \cdot 10^{66}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 3.88863330867275493 \cdot 10^{130}:\\ \;\;\;\;\sqrt{\left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right) \cdot 0.333333333333333315}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (/ (neg z) (sqrt 3.0)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))