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

Average Error: 38.1 → 16.5

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.11966401356535409 \cdot 10^{154}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 2.5319878970433065 \cdot 10^{144}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r960207 = x;
        double r960208 = r960207 * r960207;
        double r960209 = y;
        double r960210 = r960209 * r960209;
        double r960211 = r960208 + r960210;
        double r960212 = z;
        double r960213 = r960212 * r960212;
        double r960214 = r960211 + r960213;
        double r960215 = 3.0;
        double r960216 = r960214 / r960215;
        double r960217 = sqrt(r960216);
        return r960217;
}

↓

double f(double x, double y, double z) {
        double r960218 = x;
        double r960219 = -1.1196640135653541e+154;
        bool r960220 = r960218 <= r960219;
        double r960221 = 1.0;
        double r960222 = 3.0;
        double r960223 = cbrt(r960222);
        double r960224 = r960223 * r960223;
        double r960225 = r960221 / r960224;
        double r960226 = sqrt(r960225);
        double r960227 = -1.0;
        double r960228 = r960221 / r960223;
        double r960229 = sqrt(r960228);
        double r960230 = r960229 * r960218;
        double r960231 = r960227 * r960230;
        double r960232 = r960226 * r960231;
        double r960233 = 2.5319878970433065e+144;
        bool r960234 = r960218 <= r960233;
        double r960235 = r960218 * r960218;
        double r960236 = y;
        double r960237 = r960236 * r960236;
        double r960238 = r960235 + r960237;
        double r960239 = sqrt(r960238);
        double r960240 = z;
        double r960241 = hypot(r960239, r960240);
        double r960242 = sqrt(r960222);
        double r960243 = r960241 / r960242;
        double r960244 = 0.3333333333333333;
        double r960245 = sqrt(r960244);
        double r960246 = r960218 * r960245;
        double r960247 = r960234 ? r960243 : r960246;
        double r960248 = r960220 ? r960232 : r960247;
        return r960248;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	38.1
Target	26.0
Herbie	16.5

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

Derivation

1. Split input into 3 regimes

2. if x < -1.1196640135653541e+154

   1. Initial program 63.9

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

   3. Applied add-cube-cbrt 63.9

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

   4. Applied *-un-lft-identity 63.9

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

   5. Applied times-frac 63.9

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

   6. Applied sqrt-prod 63.9

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

   7. Taylor expanded around -inf 14.2

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

   if -1.1196640135653541e+154 < x < 2.5319878970433065e+144

   1. Initial program 29.3

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

   3. Applied sqrt-div 29.4

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

   5. Applied add-sqr-sqrt 29.4

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

   6. Applied hypot-def 17.2

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

   if 2.5319878970433065e+144 < x

   1. Initial program 62.1

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

   2. Taylor expanded around inf 14.9

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

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.11966401356535409 \cdot 10^{154}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 2.5319878970433065 \cdot 10^{144}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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)))