Average Error: 34.0 → 10.3
Time: 19.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.177289780863109 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 7.296044290893796 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.177289780863109 \cdot 10^{+94}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 7.296044290893796 \cdot 10^{-114}:\\
\;\;\;\;\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}
double f(double a, double b_2, double c) {
        double r441407 = b_2;
        double r441408 = -r441407;
        double r441409 = r441407 * r441407;
        double r441410 = a;
        double r441411 = c;
        double r441412 = r441410 * r441411;
        double r441413 = r441409 - r441412;
        double r441414 = sqrt(r441413);
        double r441415 = r441408 + r441414;
        double r441416 = r441415 / r441410;
        return r441416;
}
↓
double f(double a, double b_2, double c) {
        double r441417 = b_2;
        double r441418 = -3.177289780863109e+94;
        bool r441419 = r441417 <= r441418;
        double r441420 = 0.5;
        double r441421 = c;
        double r441422 = r441421 / r441417;
        double r441423 = r441420 * r441422;
        double r441424 = a;
        double r441425 = r441417 / r441424;
        double r441426 = 2.0;
        double r441427 = r441425 * r441426;
        double r441428 = r441423 - r441427;
        double r441429 = 7.296044290893796e-114;
        bool r441430 = r441417 <= r441429;
        double r441431 = r441417 * r441417;
        double r441432 = r441421 * r441424;
        double r441433 = r441431 - r441432;
        double r441434 = sqrt(r441433);
        double r441435 = r441434 - r441417;
        double r441436 = r441435 / r441424;
        double r441437 = -0.5;
        double r441438 = r441422 * r441437;
        double r441439 = r441430 ? r441436 : r441438;
        double r441440 = r441419 ? r441428 : r441439;
        return r441440;
}

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 < -3.177289780863109e+94

    1. Initial program 43.5

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

    2. Simplified43.5

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

    3. Taylor expanded around -inf 3.7

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

    if -3.177289780863109e+94 < b_2 < 7.296044290893796e-114

    1. Initial program 12.3

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

    2. Simplified12.3

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

    3. Taylor expanded around inf 12.3

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

    4. Simplified12.3

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

    if 7.296044290893796e-114 < b_2

    1. Initial program 51.6

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

    2. Simplified51.6

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

    3. Taylor expanded around inf 10.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.177289780863109 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 7.296044290893796 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))