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

Average Error: 38.3 → 26.9

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.29040423208750134 \cdot 10^{146}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 2.5655741827255853 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\frac{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt{\frac{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}{\sqrt[3]{3}}}\\ \mathbf{elif}\;x \le 9.27267059557563286 \cdot 10^{-80}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;x \le 1.86734213565976303 \cdot 10^{76}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + \left(x \cdot x + y \cdot y\right)}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	38.3
Target	25.7
Herbie	26.9

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

Derivation

1. Split input into 5 regimes

2. if x < -3.29040423208750134e146

   1. Initial program 62.0

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

   2. Taylor expanded around -inf 15.2

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)}\]

   3. Simplified 15.2

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

   if -3.29040423208750134e146 < x < 2.5655741827255853e-127

   1. Initial program 30.5

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

   3. Applied add-cube-cbrt 30.5

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

   4. Applied add-sqr-sqrt 30.5

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

   5. Applied times-frac 30.5

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

   6. Applied sqrt-prod 30.6

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

   7. Simplified 30.6

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

   8. Simplified 30.6

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

   if 2.5655741827255853e-127 < x < 9.27267059557563286e-80

   1. Initial program 28.7

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

   3. Applied add-sqr-sqrt 28.9

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

   4. Applied add-sqr-sqrt 28.9

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

   5. Applied times-frac 28.8

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

   6. Applied sqrt-prod 29.0

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

   7. Simplified 29.0

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

   8. Simplified 29.0

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

   9. Taylor expanded around 0 48.0

      \[\leadsto \color{blue}{\frac{z}{\sqrt{3}}}\]

   if 9.27267059557563286e-80 < x < 1.86734213565976303e76

   1. Initial program 27.8

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

   if 1.86734213565976303e76 < x

   1. Initial program 52.6

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

   2. Taylor expanded around inf 20.0

      \[\leadsto \color{blue}{x \cdot \sqrt{0.333333333333333315}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 26.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.29040423208750134 \cdot 10^{146}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 2.5655741827255853 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\frac{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt{\frac{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}{\sqrt[3]{3}}}\\ \mathbf{elif}\;x \le 9.27267059557563286 \cdot 10^{-80}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;x \le 1.86734213565976303 \cdot 10^{76}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + \left(x \cdot x + y \cdot y\right)}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))