Average Error: 34.0 → 10.0
Time: 19.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.2705286994550075 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2} \cdot c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 1.744031351412433 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.2705286994550075 \cdot 10^{+152}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2} \cdot c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 1.744031351412433 \cdot 10^{-142}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - 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 r774856 = b_2;
        double r774857 = -r774856;
        double r774858 = r774856 * r774856;
        double r774859 = a;
        double r774860 = c;
        double r774861 = r774859 * r774860;
        double r774862 = r774858 - r774861;
        double r774863 = sqrt(r774862);
        double r774864 = r774857 + r774863;
        double r774865 = r774864 / r774859;
        return r774865;
}
↓
double f(double a, double b_2, double c) {
        double r774866 = b_2;
        double r774867 = -1.2705286994550075e+152;
        bool r774868 = r774866 <= r774867;
        double r774869 = a;
        double r774870 = r774866 / r774869;
        double r774871 = -2.0;
        double r774872 = 0.5;
        double r774873 = c;
        double r774874 = r774872 * r774873;
        double r774875 = r774874 / r774866;
        double r774876 = fma(r774870, r774871, r774875);
        double r774877 = 1.744031351412433e-142;
        bool r774878 = r774866 <= r774877;
        double r774879 = r774866 * r774866;
        double r774880 = r774869 * r774873;
        double r774881 = r774879 - r774880;
        double r774882 = sqrt(r774881);
        double r774883 = r774882 - r774866;
        double r774884 = r774883 / r774869;
        double r774885 = -0.5;
        double r774886 = r774873 / r774866;
        double r774887 = r774885 * r774886;
        double r774888 = r774878 ? r774884 : r774887;
        double r774889 = r774868 ? r774876 : r774888;
        return r774889;
}

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 < -1.2705286994550075e+152

    1. Initial program 60.0

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

    2. Simplified60.0

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

    4. Applied div-inv60.0

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

    5. Taylor expanded around -inf 1.6

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

    6. Simplified1.6

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

    if -1.2705286994550075e+152 < b_2 < 1.744031351412433e-142

    1. Initial program 10.2

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

    2. Simplified10.2

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

    if 1.744031351412433e-142 < b_2

    1. Initial program 50.1

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

    2. Simplified50.1

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

    3. Taylor expanded around inf 12.0

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.2705286994550075 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2} \cdot c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 1.744031351412433 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))