quad2p (problem 3.2.1, positive)

Average Error: 34.0 → 6.5

Time: 6.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.7431685240570133 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 9.658303386763521 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 1.0551401351209752 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r23499 = b_2;
        double r23500 = -r23499;
        double r23501 = r23499 * r23499;
        double r23502 = a;
        double r23503 = c;
        double r23504 = r23502 * r23503;
        double r23505 = r23501 - r23504;
        double r23506 = sqrt(r23505);
        double r23507 = r23500 + r23506;
        double r23508 = r23507 / r23502;
        return r23508;
}

↓

double f(double a, double b_2, double c) {
        double r23509 = b_2;
        double r23510 = -1.7431685240570133e+102;
        bool r23511 = r23509 <= r23510;
        double r23512 = 0.5;
        double r23513 = c;
        double r23514 = r23513 / r23509;
        double r23515 = r23512 * r23514;
        double r23516 = 2.0;
        double r23517 = a;
        double r23518 = r23509 / r23517;
        double r23519 = r23516 * r23518;
        double r23520 = r23515 - r23519;
        double r23521 = 9.658303386763521e-268;
        bool r23522 = r23509 <= r23521;
        double r23523 = -r23509;
        double r23524 = r23509 * r23509;
        double r23525 = r23517 * r23513;
        double r23526 = r23524 - r23525;
        double r23527 = sqrt(r23526);
        double r23528 = r23523 + r23527;
        double r23529 = r23528 / r23517;
        double r23530 = 1.0551401351209752e+102;
        bool r23531 = r23509 <= r23530;
        double r23532 = 1.0;
        double r23533 = r23523 - r23527;
        double r23534 = r23533 / r23513;
        double r23535 = r23532 / r23534;
        double r23536 = -0.5;
        double r23537 = r23536 * r23514;
        double r23538 = r23531 ? r23535 : r23537;
        double r23539 = r23522 ? r23529 : r23538;
        double r23540 = r23511 ? r23520 : r23539;
        return r23540;
}

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 < -1.7431685240570133e+102

   1. Initial program 47.5

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

   2. Taylor expanded around -inf 3.0

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

   if -1.7431685240570133e+102 < b_2 < 9.658303386763521e-268

   1. Initial program 9.4

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

   if 9.658303386763521e-268 < b_2 < 1.0551401351209752e+102

   1. Initial program 34.6

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

   3. Applied flip-+ 34.7

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

   4. Simplified 17.1

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

   6. Applied *-un-lft-identity 17.1

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

   7. Applied associate-/r* 17.1

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

   8. Simplified 14.7

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

   10. Applied *-un-lft-identity 14.7

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

   11. Applied *-un-lft-identity 14.7

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

   12. Applied times-frac 14.7

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

   13. Applied associate-/l* 14.7

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

   14. Simplified 8.5

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

   if 1.0551401351209752e+102 < b_2

   1. Initial program 59.8

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

   2. Taylor expanded around inf 2.3

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.7431685240570133 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 9.658303386763521 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 1.0551401351209752 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))