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

Average Error: 38.1 → 25.7

Time: 5.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.9353333948222178 \cdot 10^{136}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 5.0542352973125761 \cdot 10^{73}:\\ \;\;\;\;\sqrt{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r915293 = x;
        double r915294 = r915293 * r915293;
        double r915295 = y;
        double r915296 = r915295 * r915295;
        double r915297 = r915294 + r915296;
        double r915298 = z;
        double r915299 = r915298 * r915298;
        double r915300 = r915297 + r915299;
        double r915301 = 3.0;
        double r915302 = r915300 / r915301;
        double r915303 = sqrt(r915302);
        return r915303;
}

↓

double f(double x, double y, double z) {
        double r915304 = x;
        double r915305 = -1.9353333948222178e+136;
        bool r915306 = r915304 <= r915305;
        double r915307 = 1.0;
        double r915308 = 3.0;
        double r915309 = cbrt(r915308);
        double r915310 = r915309 * r915309;
        double r915311 = r915307 / r915310;
        double r915312 = sqrt(r915311);
        double r915313 = -1.0;
        double r915314 = r915307 / r915309;
        double r915315 = sqrt(r915314);
        double r915316 = r915315 * r915304;
        double r915317 = r915313 * r915316;
        double r915318 = r915312 * r915317;
        double r915319 = 5.054235297312576e+73;
        bool r915320 = r915304 <= r915319;
        double r915321 = 0.3333333333333333;
        double r915322 = r915304 * r915304;
        double r915323 = y;
        double r915324 = r915323 * r915323;
        double r915325 = r915322 + r915324;
        double r915326 = z;
        double r915327 = r915326 * r915326;
        double r915328 = r915325 + r915327;
        double r915329 = r915321 * r915328;
        double r915330 = sqrt(r915329);
        double r915331 = sqrt(r915321);
        double r915332 = r915304 * r915331;
        double r915333 = r915320 ? r915330 : r915332;
        double r915334 = r915306 ? r915318 : r915333;
        return r915334;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	38.1
Target	26.0
Herbie	25.7

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

Derivation

1. Split input into 3 regimes

2. if x < -1.9353333948222178e+136

   1. Initial program 59.9

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

   3. Applied add-cube-cbrt 59.9

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

   4. Applied *-un-lft-identity 59.9

      \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}\]

   5. Applied times-frac 59.9

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]

   6. Applied sqrt-prod 59.9

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]

   7. Taylor expanded around -inf 15.9

      \[\leadsto \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)}\]

   if -1.9353333948222178e+136 < x < 5.054235297312576e+73

   1. Initial program 29.4

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

   2. Taylor expanded around 0 29.4

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

   3. Simplified 29.4

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

   if 5.054235297312576e+73 < x

   1. Initial program 52.2

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

   2. Taylor expanded around inf 19.8

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

4. Final simplification 25.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.9353333948222178 \cdot 10^{136}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 5.0542352973125761 \cdot 10^{73}:\\ \;\;\;\;\sqrt{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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) (/ (- z) (sqrt 3)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3)))