Average Error: 33.6 → 10.8
Time: 19.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2 \cdot -2}{a}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.136683434005781 \cdot 10^{-32}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.0410715251838527 \cdot 10^{+49}:\\
\;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2 \cdot -2}{a}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r764024 = b_2;
        double r764025 = -r764024;
        double r764026 = r764024 * r764024;
        double r764027 = a;
        double r764028 = c;
        double r764029 = r764027 * r764028;
        double r764030 = r764026 - r764029;
        double r764031 = sqrt(r764030);
        double r764032 = r764025 - r764031;
        double r764033 = r764032 / r764027;
        return r764033;
}
↓
double f(double a, double b_2, double c) {
        double r764034 = b_2;
        double r764035 = -3.136683434005781e-32;
        bool r764036 = r764034 <= r764035;
        double r764037 = -0.5;
        double r764038 = c;
        double r764039 = r764038 / r764034;
        double r764040 = r764037 * r764039;
        double r764041 = 2.0410715251838527e+49;
        bool r764042 = r764034 <= r764041;
        double r764043 = r764034 * r764034;
        double r764044 = a;
        double r764045 = r764038 * r764044;
        double r764046 = r764043 - r764045;
        double r764047 = sqrt(r764046);
        double r764048 = r764047 + r764034;
        double r764049 = -r764048;
        double r764050 = r764049 / r764044;
        double r764051 = 0.5;
        double r764052 = -2.0;
        double r764053 = r764034 * r764052;
        double r764054 = r764053 / r764044;
        double r764055 = fma(r764051, r764039, r764054);
        double r764056 = r764042 ? r764050 : r764055;
        double r764057 = r764036 ? r764040 : r764056;
        return r764057;
}

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 < -3.136683434005781e-32

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 7.3

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

    if -3.136683434005781e-32 < b_2 < 2.0410715251838527e+49

    1. Initial program 15.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity15.8

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

    4. Applied associate-/r*15.8

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

    5. Simplified15.8

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

    if 2.0410715251838527e+49 < b_2

    1. Initial program 36.2

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

    2. Taylor expanded around inf 6.2

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

    3. Simplified6.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{-2 \cdot b_2}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2 \cdot -2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))