quad2p (problem 3.2.1, positive)

Average Error: 33.3 → 10.3

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{c}{b_2}\right), \frac{1}{2}, \left(\frac{b_2}{a} \cdot -2\right)\right)\\ \mathbf{elif}\;b_2 \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r329305 = b_2;
        double r329306 = -r329305;
        double r329307 = r329305 * r329305;
        double r329308 = a;
        double r329309 = c;
        double r329310 = r329308 * r329309;
        double r329311 = r329307 - r329310;
        double r329312 = sqrt(r329311);
        double r329313 = r329306 + r329312;
        double r329314 = r329313 / r329308;
        return r329314;
}

↓

double f(double a, double b_2, double c) {
        double r329315 = b_2;
        double r329316 = -3.263941314600607e+152;
        bool r329317 = r329315 <= r329316;
        double r329318 = c;
        double r329319 = r329318 / r329315;
        double r329320 = 0.5;
        double r329321 = a;
        double r329322 = r329315 / r329321;
        double r329323 = -2.0;
        double r329324 = r329322 * r329323;
        double r329325 = fma(r329319, r329320, r329324);
        double r329326 = 1.8378252714625124e-19;
        bool r329327 = r329315 <= r329326;
        double r329328 = r329315 * r329315;
        double r329329 = r329318 * r329321;
        double r329330 = r329328 - r329329;
        double r329331 = sqrt(r329330);
        double r329332 = r329331 - r329315;
        double r329333 = r329332 / r329321;
        double r329334 = -0.5;
        double r329335 = r329319 * r329334;
        double r329336 = r329327 ? r329333 : r329335;
        double r329337 = r329317 ? r329325 : r329336;
        return r329337;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.263941314600607e+152

   1. Initial program 60.1

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

   2. Simplified 60.1

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

   4. Applied *-un-lft-identity 60.1

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

   5. Applied *-un-lft-identity 60.1

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

   6. Applied distribute-lft-out-- 60.1

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

   7. Applied associate-/l* 60.1

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

   9. Applied associate-/r/ 60.1

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\]
   10. Using strategy rm

   11. Applied associate-*l/ 60.1

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

   12. Simplified 60.1

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

   13. Taylor expanded around -inf 2.3

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

   14. Simplified 2.3

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{c}{b_2}\right), \frac{1}{2}, \left(\frac{b_2}{a} \cdot -2\right)\right)}\]

   if -3.263941314600607e+152 < b_2 < 1.8378252714625124e-19

   1. Initial program 14.1

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

   2. Simplified 14.1

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

   4. Applied *-un-lft-identity 14.1

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

   5. Applied *-un-lft-identity 14.1

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

   6. Applied distribute-lft-out-- 14.1

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

   7. Applied associate-/l* 14.3

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

   9. Applied associate-/r/ 14.3

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\]
   10. Using strategy rm

   11. Applied associate-*l/ 14.1

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

   12. Simplified 14.1

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

   if 1.8378252714625124e-19 < b_2

   1. Initial program 54.4

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

   2. Simplified 54.4

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

   3. Taylor expanded around inf 7.0

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{c}{b_2}\right), \frac{1}{2}, \left(\frac{b_2}{a} \cdot -2\right)\right)\\ \mathbf{elif}\;b_2 \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))