Average Error: 33.1 → 10.4
Time: 23.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.1962309819144974 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.6488521390017767 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.1962309819144974 \cdot 10^{-65}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.6488521390017767 \cdot 10^{+48}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r899655 = b_2;
        double r899656 = -r899655;
        double r899657 = r899655 * r899655;
        double r899658 = a;
        double r899659 = c;
        double r899660 = r899658 * r899659;
        double r899661 = r899657 - r899660;
        double r899662 = sqrt(r899661);
        double r899663 = r899656 - r899662;
        double r899664 = r899663 / r899658;
        return r899664;
}
↓
double f(double a, double b_2, double c) {
        double r899665 = b_2;
        double r899666 = -1.1962309819144974e-65;
        bool r899667 = r899665 <= r899666;
        double r899668 = -0.5;
        double r899669 = c;
        double r899670 = r899669 / r899665;
        double r899671 = r899668 * r899670;
        double r899672 = 5.6488521390017767e+48;
        bool r899673 = r899665 <= r899672;
        double r899674 = 1.0;
        double r899675 = a;
        double r899676 = r899674 / r899675;
        double r899677 = -r899665;
        double r899678 = r899665 * r899665;
        double r899679 = r899669 * r899675;
        double r899680 = r899678 - r899679;
        double r899681 = sqrt(r899680);
        double r899682 = r899677 - r899681;
        double r899683 = r899676 * r899682;
        double r899684 = r899665 / r899675;
        double r899685 = -2.0;
        double r899686 = 2.0;
        double r899687 = r899670 / r899686;
        double r899688 = fma(r899684, r899685, r899687);
        double r899689 = r899673 ? r899683 : r899688;
        double r899690 = r899667 ? r899671 : r899689;
        return r899690;
}

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 < -1.1962309819144974e-65

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 8.8

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

    if -1.1962309819144974e-65 < b_2 < 5.6488521390017767e+48

    1. Initial program 14.1

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

    3. Applied clear-num14.2

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

    5. Applied div-inv14.3

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

    6. Applied associate-/r*14.3

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

    8. Applied div-inv14.3

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

    9. Simplified14.2

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

    if 5.6488521390017767e+48 < b_2

    1. Initial program 35.6

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

    2. Taylor expanded around inf 5.1

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

    3. Simplified5.1

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.1962309819144974 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.6488521390017767 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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))