Average Error: 34.6 → 10.7
Time: 26.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{-2 \cdot b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{-2 \cdot b_2}{a}\right)\\

\mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\
\;\;\;\;\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r737817 = b_2;
        double r737818 = -r737817;
        double r737819 = r737817 * r737817;
        double r737820 = a;
        double r737821 = c;
        double r737822 = r737820 * r737821;
        double r737823 = r737819 - r737822;
        double r737824 = sqrt(r737823);
        double r737825 = r737818 + r737824;
        double r737826 = r737825 / r737820;
        return r737826;
}
↓
double f(double a, double b_2, double c) {
        double r737827 = b_2;
        double r737828 = -2.7668189408748547e+100;
        bool r737829 = r737827 <= r737828;
        double r737830 = 0.5;
        double r737831 = c;
        double r737832 = r737831 / r737827;
        double r737833 = -2.0;
        double r737834 = r737833 * r737827;
        double r737835 = a;
        double r737836 = r737834 / r737835;
        double r737837 = fma(r737830, r737832, r737836);
        double r737838 = 7.923524897992037e-153;
        bool r737839 = r737827 <= r737838;
        double r737840 = r737827 * r737827;
        double r737841 = r737835 * r737831;
        double r737842 = r737840 - r737841;
        double r737843 = sqrt(r737842);
        double r737844 = r737843 - r737827;
        double r737845 = 1.0;
        double r737846 = r737845 / r737835;
        double r737847 = r737844 * r737846;
        double r737848 = -0.5;
        double r737849 = r737832 * r737848;
        double r737850 = r737839 ? r737847 : r737849;
        double r737851 = r737829 ? r737837 : r737850;
        return r737851;
}

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 < -2.7668189408748547e+100

    1. Initial program 47.1

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

    2. Simplified47.1

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

    3. Taylor expanded around -inf 4.0

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

    4. Simplified4.1

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

    if -2.7668189408748547e+100 < b_2 < 7.923524897992037e-153

    1. Initial program 10.8

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

    2. Simplified10.8

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

    4. Applied div-inv10.9

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

    if 7.923524897992037e-153 < b_2

    1. Initial program 50.5

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

    2. Simplified50.5

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

    3. Taylor expanded around inf 12.7

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{-2 \cdot b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))