Average Error: 34.5 → 9.6
Time: 43.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.585466010660367 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.585466010660367 \cdot 10^{+150}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.3635892865650846 \cdot 10^{-93}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r911360 = b_2;
        double r911361 = -r911360;
        double r911362 = r911360 * r911360;
        double r911363 = a;
        double r911364 = c;
        double r911365 = r911363 * r911364;
        double r911366 = r911362 - r911365;
        double r911367 = sqrt(r911366);
        double r911368 = r911361 + r911367;
        double r911369 = r911368 / r911363;
        return r911369;
}
↓
double f(double a, double b_2, double c) {
        double r911370 = b_2;
        double r911371 = -1.585466010660367e+150;
        bool r911372 = r911370 <= r911371;
        double r911373 = 0.5;
        double r911374 = c;
        double r911375 = r911374 / r911370;
        double r911376 = r911373 * r911375;
        double r911377 = a;
        double r911378 = r911370 / r911377;
        double r911379 = 2.0;
        double r911380 = r911378 * r911379;
        double r911381 = r911376 - r911380;
        double r911382 = 1.3635892865650846e-93;
        bool r911383 = r911370 <= r911382;
        double r911384 = r911370 * r911370;
        double r911385 = r911374 * r911377;
        double r911386 = r911384 - r911385;
        double r911387 = sqrt(r911386);
        double r911388 = r911387 - r911370;
        double r911389 = r911388 / r911377;
        double r911390 = -0.5;
        double r911391 = r911390 * r911375;
        double r911392 = r911383 ? r911389 : r911391;
        double r911393 = r911372 ? r911381 : r911392;
        return r911393;
}

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 3 regimes

  2. if b_2 < -1.585466010660367e+150

    1. Initial program 62.8

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

    2. Simplified62.8

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

    3. Taylor expanded around -inf 2.8

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

    if -1.585466010660367e+150 < b_2 < 1.3635892865650846e-93

    1. Initial program 11.6

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

    2. Simplified11.6

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

    4. Applied clear-num11.7

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

    6. Applied *-un-lft-identity11.7

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

    7. Applied *-un-lft-identity11.7

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

    8. Applied times-frac11.7

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

    9. Applied add-cube-cbrt11.7

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

    10. Applied times-frac11.7

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

    11. Simplified11.7

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

    12. Simplified11.6

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

    if 1.3635892865650846e-93 < b_2

    1. Initial program 53.0

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

    2. Simplified53.0

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

    3. Taylor expanded around inf 9.1

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.585466010660367 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))