Average Error: 33.7 → 11.4
Time: 22.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.918306653269124147694529335594372917926 \cdot 10^{51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 0.03941192619295448562599659680927288718522:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.918306653269124147694529335594372917926 \cdot 10^{51}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\

\mathbf{elif}\;b_2 \le 0.03941192619295448562599659680927288718522:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{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 r2291721 = b_2;
        double r2291722 = -r2291721;
        double r2291723 = r2291721 * r2291721;
        double r2291724 = a;
        double r2291725 = c;
        double r2291726 = r2291724 * r2291725;
        double r2291727 = r2291723 - r2291726;
        double r2291728 = sqrt(r2291727);
        double r2291729 = r2291722 + r2291728;
        double r2291730 = r2291729 / r2291724;
        return r2291730;
}
↓
double f(double a, double b_2, double c) {
        double r2291731 = b_2;
        double r2291732 = -1.918306653269124e+51;
        bool r2291733 = r2291731 <= r2291732;
        double r2291734 = c;
        double r2291735 = r2291734 / r2291731;
        double r2291736 = 0.5;
        double r2291737 = -2.0;
        double r2291738 = a;
        double r2291739 = r2291731 / r2291738;
        double r2291740 = r2291737 * r2291739;
        double r2291741 = fma(r2291735, r2291736, r2291740);
        double r2291742 = 0.039411926192954486;
        bool r2291743 = r2291731 <= r2291742;
        double r2291744 = r2291731 * r2291731;
        double r2291745 = r2291738 * r2291734;
        double r2291746 = r2291744 - r2291745;
        double r2291747 = sqrt(r2291746);
        double r2291748 = r2291747 / r2291738;
        double r2291749 = r2291748 - r2291739;
        double r2291750 = -0.5;
        double r2291751 = r2291750 * r2291735;
        double r2291752 = r2291743 ? r2291749 : r2291751;
        double r2291753 = r2291733 ? r2291741 : r2291752;
        return r2291753;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.918306653269124e+51

    1. Initial program 38.3

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

    2. Simplified38.3

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

    4. Applied div-sub38.3

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

    5. Taylor expanded around -inf 5.7

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

    6. Simplified5.7

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

    if -1.918306653269124e+51 < b_2 < 0.039411926192954486

    1. Initial program 17.2

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

    2. Simplified17.2

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

    4. Applied div-sub17.2

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

    if 0.039411926192954486 < b_2

    1. Initial program 54.9

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

    2. Simplified54.9

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

    3. Taylor expanded around inf 6.4

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.918306653269124147694529335594372917926 \cdot 10^{51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 0.03941192619295448562599659680927288718522:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))