Average Error: 33.4 → 10.6
Time: 27.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -227369802444031.66:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 2.0617732603635578 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -227369802444031.66:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 2.0617732603635578 \cdot 10^{-61}:\\
\;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r630412 = b_2;
        double r630413 = -r630412;
        double r630414 = r630412 * r630412;
        double r630415 = a;
        double r630416 = c;
        double r630417 = r630415 * r630416;
        double r630418 = r630414 - r630417;
        double r630419 = sqrt(r630418);
        double r630420 = r630413 + r630419;
        double r630421 = r630420 / r630415;
        return r630421;
}
↓
double f(double a, double b_2, double c) {
        double r630422 = b_2;
        double r630423 = -227369802444031.66;
        bool r630424 = r630422 <= r630423;
        double r630425 = 0.5;
        double r630426 = c;
        double r630427 = r630426 / r630422;
        double r630428 = r630425 * r630427;
        double r630429 = a;
        double r630430 = r630422 / r630429;
        double r630431 = 2.0;
        double r630432 = r630430 * r630431;
        double r630433 = r630428 - r630432;
        double r630434 = 2.0617732603635578e-61;
        bool r630435 = r630422 <= r630434;
        double r630436 = 1.0;
        double r630437 = r630436 / r630429;
        double r630438 = r630422 * r630422;
        double r630439 = r630426 * r630429;
        double r630440 = r630438 - r630439;
        double r630441 = sqrt(r630440);
        double r630442 = r630441 - r630422;
        double r630443 = r630436 / r630442;
        double r630444 = r630437 / r630443;
        double r630445 = -0.5;
        double r630446 = r630445 * r630427;
        double r630447 = r630435 ? r630444 : r630446;
        double r630448 = r630424 ? r630433 : r630447;
        return r630448;
}

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 < -227369802444031.66

    1. Initial program 32.9

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

    2. Simplified32.9

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

    3. Taylor expanded around -inf 6.7

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

    if -227369802444031.66 < b_2 < 2.0617732603635578e-61

    1. Initial program 15.0

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

    2. Simplified15.0

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

    4. Applied clear-num15.0

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

    6. Applied div-inv15.1

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

    7. Applied associate-/r*15.1

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

    if 2.0617732603635578e-61 < b_2

    1. Initial program 52.8

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

    2. Simplified52.8

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

    3. Taylor expanded around inf 8.2

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -227369802444031.66:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 2.0617732603635578 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))