Average Error: 33.1 → 8.0
Time: 36.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\
\;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\

\mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r5257460 = b_2;
        double r5257461 = -r5257460;
        double r5257462 = r5257460 * r5257460;
        double r5257463 = a;
        double r5257464 = c;
        double r5257465 = r5257463 * r5257464;
        double r5257466 = r5257462 - r5257465;
        double r5257467 = sqrt(r5257466);
        double r5257468 = r5257461 - r5257467;
        double r5257469 = r5257468 / r5257463;
        return r5257469;
}
↓
double f(double a, double b_2, double c) {
        double r5257470 = b_2;
        double r5257471 = -6.473972066548491e+100;
        bool r5257472 = r5257470 <= r5257471;
        double r5257473 = -0.5;
        double r5257474 = c;
        double r5257475 = r5257474 / r5257470;
        double r5257476 = r5257473 * r5257475;
        double r5257477 = -3.554031892664371e-133;
        bool r5257478 = r5257470 <= r5257477;
        double r5257479 = a;
        double r5257480 = cbrt(r5257479);
        double r5257481 = r5257474 / r5257480;
        double r5257482 = r5257480 * r5257480;
        double r5257483 = r5257479 / r5257482;
        double r5257484 = r5257481 * r5257483;
        double r5257485 = r5257470 * r5257470;
        double r5257486 = r5257474 * r5257479;
        double r5257487 = r5257485 - r5257486;
        double r5257488 = sqrt(r5257487);
        double r5257489 = -r5257470;
        double r5257490 = r5257488 + r5257489;
        double r5257491 = r5257484 / r5257490;
        double r5257492 = 1.983916337927056e+89;
        bool r5257493 = r5257470 <= r5257492;
        double r5257494 = r5257470 / r5257479;
        double r5257495 = -r5257494;
        double r5257496 = r5257488 / r5257479;
        double r5257497 = r5257495 - r5257496;
        double r5257498 = 0.5;
        double r5257499 = r5257498 * r5257475;
        double r5257500 = 2.0;
        double r5257501 = r5257494 * r5257500;
        double r5257502 = r5257499 - r5257501;
        double r5257503 = r5257493 ? r5257497 : r5257502;
        double r5257504 = r5257478 ? r5257491 : r5257503;
        double r5257505 = r5257472 ? r5257476 : r5257504;
        return r5257505;
}

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 < -6.473972066548491e+100

    1. Initial program 58.8

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

    2. Taylor expanded around -inf 2.3

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

    if -6.473972066548491e+100 < b_2 < -3.554031892664371e-133

    1. Initial program 39.0

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

    3. Applied div-inv39.1

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

    5. Applied flip--39.2

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

    6. Applied associate-*l/39.2

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

    7. Simplified13.1

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

    9. Applied add-cube-cbrt14.0

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

    10. Applied times-frac10.6

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

    if -3.554031892664371e-133 < b_2 < 1.983916337927056e+89

    1. Initial program 11.5

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

    3. Applied div-sub11.5

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

    if 1.983916337927056e+89 < b_2

    1. Initial program 42.0

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))