NMSE problem 3.2.1

Average Error: 33.8 → 7.3

Time: 22.7s

Precision: 64

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.755118612359990099835323540998142276883 \cdot 10^{47}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.415217774929248159542813179693576626283 \cdot 10^{-267}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}}}{\sqrt[3]{1}} \cdot \frac{\frac{\sqrt[3]{\sqrt[3]{a}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c}}}}{\sqrt[3]{a}}\right)\\ \mathbf{elif}\;b_2 \le 3.561111989718156042582267854543789265968 \cdot 10^{98}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r83356 = b_2;
        double r83357 = -r83356;
        double r83358 = r83356 * r83356;
        double r83359 = a;
        double r83360 = c;
        double r83361 = r83359 * r83360;
        double r83362 = r83358 - r83361;
        double r83363 = sqrt(r83362);
        double r83364 = r83357 - r83363;
        double r83365 = r83364 / r83359;
        return r83365;
}

↓

double f(double a, double b_2, double c) {
        double r83366 = b_2;
        double r83367 = -3.75511861235999e+47;
        bool r83368 = r83366 <= r83367;
        double r83369 = -0.5;
        double r83370 = c;
        double r83371 = r83370 / r83366;
        double r83372 = r83369 * r83371;
        double r83373 = -1.4152177749292482e-267;
        bool r83374 = r83366 <= r83373;
        double r83375 = 1.0;
        double r83376 = a;
        double r83377 = cbrt(r83376);
        double r83378 = cbrt(r83377);
        double r83379 = r83378 * r83378;
        double r83380 = r83366 * r83366;
        double r83381 = r83376 * r83370;
        double r83382 = r83380 - r83381;
        double r83383 = sqrt(r83382);
        double r83384 = r83383 - r83366;
        double r83385 = sqrt(r83384);
        double r83386 = cbrt(r83370);
        double r83387 = r83386 * r83386;
        double r83388 = r83385 / r83387;
        double r83389 = r83379 / r83388;
        double r83390 = cbrt(r83375);
        double r83391 = r83389 / r83390;
        double r83392 = r83385 / r83386;
        double r83393 = r83378 / r83392;
        double r83394 = r83393 / r83377;
        double r83395 = r83391 * r83394;
        double r83396 = r83375 * r83395;
        double r83397 = 3.561111989718156e+98;
        bool r83398 = r83366 <= r83397;
        double r83399 = -r83366;
        double r83400 = r83399 - r83383;
        double r83401 = r83400 / r83376;
        double r83402 = 0.5;
        double r83403 = r83402 * r83371;
        double r83404 = 2.0;
        double r83405 = r83366 / r83376;
        double r83406 = r83404 * r83405;
        double r83407 = r83403 - r83406;
        double r83408 = r83398 ? r83401 : r83407;
        double r83409 = r83374 ? r83396 : r83408;
        double r83410 = r83368 ? r83372 : r83409;
        return r83410;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.75511861235999e+47

   1. Initial program 57.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around -inf 4.0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

   if -3.75511861235999e+47 < b_2 < -1.4152177749292482e-267

   1. Initial program 30.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-- 30.3

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 16.7

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 16.7

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 16.7

      \[\leadsto \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

   8. Applied associate-/r* 16.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{1}}{a}}\]

   9. Simplified 14.0

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{a}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 14.7

       \[\leadsto \frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   12. Applied *-un-lft-identity 14.7

       \[\leadsto \frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{\color{blue}{1 \cdot c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   13. Applied *-un-lft-identity 14.7

       \[\leadsto \frac{\frac{a}{\frac{\color{blue}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}{1 \cdot c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   14. Applied times-frac 14.7

       \[\leadsto \frac{\frac{a}{\color{blue}{\frac{1}{1} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   15. Applied add-cube-cbrt 14.0

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\frac{1}{1} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   16. Applied times-frac 14.0

       \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   17. Applied times-frac 10.2

       \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\frac{1}{1}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt[3]{a}}}\]

   18. Simplified 10.2

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt[3]{a}}\]
   19. Using strategy rm

   20. Applied *-un-lft-identity 10.2

       \[\leadsto 1 \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt[3]{\color{blue}{1 \cdot a}}}\]

   21. Applied cbrt-prod 10.2

       \[\leadsto 1 \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{a}}}\]

   22. Applied add-cube-cbrt 11.0

       \[\leadsto 1 \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{\color{blue}{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}}}}}{\sqrt[3]{1} \cdot \sqrt[3]{a}}\]

   23. Applied add-sqr-sqrt 11.0

       \[\leadsto 1 \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}}}}{\sqrt[3]{1} \cdot \sqrt[3]{a}}\]

   24. Applied times-frac 11.1

       \[\leadsto 1 \cdot \frac{\frac{\sqrt[3]{a}}{\color{blue}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c}}}}}{\sqrt[3]{1} \cdot \sqrt[3]{a}}\]

   25. Applied add-cube-cbrt 11.3

       \[\leadsto 1 \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}\right) \cdot \sqrt[3]{\sqrt[3]{a}}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c}}}}{\sqrt[3]{1} \cdot \sqrt[3]{a}}\]

   26. Applied times-frac 11.2

       \[\leadsto 1 \cdot \frac{\color{blue}{\frac{\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}} \cdot \frac{\sqrt[3]{\sqrt[3]{a}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c}}}}}{\sqrt[3]{1} \cdot \sqrt[3]{a}}\]

   27. Applied times-frac 10.3

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}}}{\sqrt[3]{1}} \cdot \frac{\frac{\sqrt[3]{\sqrt[3]{a}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c}}}}{\sqrt[3]{a}}\right)}\]

   if -1.4152177749292482e-267 < b_2 < 3.561111989718156e+98

   1. Initial program 9.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   if 3.561111989718156e+98 < b_2

   1. Initial program 44.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 3.6

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.755118612359990099835323540998142276883 \cdot 10^{47}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.415217774929248159542813179693576626283 \cdot 10^{-267}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}}}{\sqrt[3]{1}} \cdot \frac{\frac{\sqrt[3]{\sqrt[3]{a}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{c}}}}{\sqrt[3]{a}}\right)\\ \mathbf{elif}\;b_2 \le 3.561111989718156042582267854543789265968 \cdot 10^{98}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))