Average Error: 34.2 → 9.4
Time: 13.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{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 r518468 = b_2;
        double r518469 = -r518468;
        double r518470 = r518468 * r518468;
        double r518471 = a;
        double r518472 = c;
        double r518473 = r518471 * r518472;
        double r518474 = r518470 - r518473;
        double r518475 = sqrt(r518474);
        double r518476 = r518469 + r518475;
        double r518477 = r518476 / r518471;
        return r518477;
}
↓
double f(double a, double b_2, double c) {
        double r518478 = b_2;
        double r518479 = -3.7108875578650606e+138;
        bool r518480 = r518478 <= r518479;
        double r518481 = 0.5;
        double r518482 = c;
        double r518483 = r518482 / r518478;
        double r518484 = r518481 * r518483;
        double r518485 = a;
        double r518486 = r518478 / r518485;
        double r518487 = r518484 - r518486;
        double r518488 = r518487 - r518486;
        double r518489 = 4.626043257219638e-62;
        bool r518490 = r518478 <= r518489;
        double r518491 = r518478 * r518478;
        double r518492 = r518485 * r518482;
        double r518493 = r518491 - r518492;
        double r518494 = sqrt(r518493);
        double r518495 = r518494 / r518485;
        double r518496 = r518495 - r518486;
        double r518497 = -0.5;
        double r518498 = r518483 * r518497;
        double r518499 = r518490 ? r518496 : r518498;
        double r518500 = r518480 ? r518488 : r518499;
        return r518500;
}

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 < -3.7108875578650606e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    4. Applied div-sub58.5

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

    5. Taylor expanded around -inf 2.0

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

    if -3.7108875578650606e+138 < b_2 < 4.626043257219638e-62

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied div-sub12.1

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

    if 4.626043257219638e-62 < b_2

    1. Initial program 53.7

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

    2. Simplified53.7

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

    3. Taylor expanded around inf 8.4

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))