Average Error: 34.4 → 10.0
Time: 19.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.090731821246914194343512212169473389565 \cdot 10^{152}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.090731821246914194343512212169473389565 \cdot 10^{152}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\

\mathbf{elif}\;b_2 \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - 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 r23934 = b_2;
        double r23935 = -r23934;
        double r23936 = r23934 * r23934;
        double r23937 = a;
        double r23938 = c;
        double r23939 = r23937 * r23938;
        double r23940 = r23936 - r23939;
        double r23941 = sqrt(r23940);
        double r23942 = r23935 + r23941;
        double r23943 = r23942 / r23937;
        return r23943;
}
↓
double f(double a, double b_2, double c) {
        double r23944 = b_2;
        double r23945 = -2.0907318212469142e+152;
        bool r23946 = r23944 <= r23945;
        double r23947 = c;
        double r23948 = r23947 / r23944;
        double r23949 = 0.5;
        double r23950 = a;
        double r23951 = r23944 / r23950;
        double r23952 = -2.0;
        double r23953 = r23951 * r23952;
        double r23954 = fma(r23948, r23949, r23953);
        double r23955 = 8.70366778308292e-52;
        bool r23956 = r23944 <= r23955;
        double r23957 = r23944 * r23944;
        double r23958 = r23947 * r23950;
        double r23959 = r23957 - r23958;
        double r23960 = sqrt(r23959);
        double r23961 = r23960 - r23944;
        double r23962 = r23961 / r23950;
        double r23963 = -0.5;
        double r23964 = r23963 * r23948;
        double r23965 = r23956 ? r23962 : r23964;
        double r23966 = r23946 ? r23954 : r23965;
        return r23966;
}

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 < -2.0907318212469142e+152

    1. Initial program 63.3

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

    2. Simplified63.3

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

    3. Taylor expanded around -inf 2.2

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

    4. Simplified2.2

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

    if -2.0907318212469142e+152 < b_2 < 8.70366778308292e-52

    1. Initial program 13.2

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

    2. Simplified13.2

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

    3. Taylor expanded around 0 13.2

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

    4. Simplified13.2

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

    if 8.70366778308292e-52 < b_2

    1. Initial program 54.1

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

    2. Simplified54.1

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

    3. Taylor expanded around inf 7.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.090731821246914194343512212169473389565 \cdot 10^{152}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))