Average Error: 34.4 → 7.1
Time: 35.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -9.97798043992307827 \cdot 10^{-261}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.94192058439483012 \cdot 10^{74}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{\sqrt[3]{1}}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.30101840923646093 \cdot 10^{98}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -9.97798043992307827 \cdot 10^{-261}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 5.94192058439483012 \cdot 10^{74}:\\
\;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{\sqrt[3]{1}}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r152410 = b_2;
        double r152411 = -r152410;
        double r152412 = r152410 * r152410;
        double r152413 = a;
        double r152414 = c;
        double r152415 = r152413 * r152414;
        double r152416 = r152412 - r152415;
        double r152417 = sqrt(r152416);
        double r152418 = r152411 + r152417;
        double r152419 = r152418 / r152413;
        return r152419;
}
↓
double f(double a, double b_2, double c) {
        double r152420 = b_2;
        double r152421 = -4.301018409236461e+98;
        bool r152422 = r152420 <= r152421;
        double r152423 = 0.5;
        double r152424 = c;
        double r152425 = r152424 / r152420;
        double r152426 = r152423 * r152425;
        double r152427 = 2.0;
        double r152428 = a;
        double r152429 = r152420 / r152428;
        double r152430 = r152427 * r152429;
        double r152431 = r152426 - r152430;
        double r152432 = -9.977980439923078e-261;
        bool r152433 = r152420 <= r152432;
        double r152434 = -r152420;
        double r152435 = r152420 * r152420;
        double r152436 = r152428 * r152424;
        double r152437 = r152435 - r152436;
        double r152438 = sqrt(r152437);
        double r152439 = r152434 + r152438;
        double r152440 = 1.0;
        double r152441 = r152440 / r152428;
        double r152442 = r152439 * r152441;
        double r152443 = 5.94192058439483e+74;
        bool r152444 = r152420 <= r152443;
        double r152445 = cbrt(r152440);
        double r152446 = r152445 * r152445;
        double r152447 = cbrt(r152428);
        double r152448 = r152447 * r152447;
        double r152449 = r152440 / r152448;
        double r152450 = r152448 * r152449;
        double r152451 = r152446 / r152450;
        double r152452 = r152434 - r152438;
        double r152453 = r152452 / r152447;
        double r152454 = r152453 / r152424;
        double r152455 = r152445 / r152454;
        double r152456 = r152455 / r152447;
        double r152457 = r152451 * r152456;
        double r152458 = -0.5;
        double r152459 = r152458 * r152425;
        double r152460 = r152444 ? r152457 : r152459;
        double r152461 = r152433 ? r152442 : r152460;
        double r152462 = r152422 ? r152431 : r152461;
        return r152462;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -4.301018409236461e+98

    1. Initial program 47.2

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

    2. Taylor expanded around -inf 3.9

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

    if -4.301018409236461e+98 < b_2 < -9.977980439923078e-261

    1. Initial program 8.4

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

    3. Applied div-inv8.6

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

    if -9.977980439923078e-261 < b_2 < 5.94192058439483e+74

    1. Initial program 29.3

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

    3. Applied flip-+29.4

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

    4. Simplified16.1

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

    6. Applied clear-num16.3

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

    7. Simplified14.8

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

    9. Applied add-cube-cbrt15.6

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

    10. Applied *-un-lft-identity15.6

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

    11. Applied add-cube-cbrt14.9

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

    12. Applied *-un-lft-identity14.9

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

    13. Applied times-frac14.9

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

    14. Applied times-frac14.4

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

    15. Applied add-cube-cbrt14.4

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

    16. Applied times-frac14.0

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

    17. Applied times-frac10.7

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

    18. Simplified10.7

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

    if 5.94192058439483e+74 < b_2

    1. Initial program 59.0

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

    2. Taylor expanded around inf 3.3

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -9.97798043992307827 \cdot 10^{-261}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.94192058439483012 \cdot 10^{74}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{\sqrt[3]{1}}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))