Average Error: 33.5 → 10.0
Time: 18.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(\frac{c}{b_2}\right), \left(\frac{b_2 \cdot -2}{a}\right)\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.2092322739463293 \cdot 10^{-86}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.891777552454845 \cdot 10^{+74}:\\
\;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r584634 = b_2;
        double r584635 = -r584634;
        double r584636 = r584634 * r584634;
        double r584637 = a;
        double r584638 = c;
        double r584639 = r584637 * r584638;
        double r584640 = r584636 - r584639;
        double r584641 = sqrt(r584640);
        double r584642 = r584635 - r584641;
        double r584643 = r584642 / r584637;
        return r584643;
}
↓
double f(double a, double b_2, double c) {
        double r584644 = b_2;
        double r584645 = -3.2092322739463293e-86;
        bool r584646 = r584644 <= r584645;
        double r584647 = -0.5;
        double r584648 = c;
        double r584649 = r584648 / r584644;
        double r584650 = r584647 * r584649;
        double r584651 = 2.891777552454845e+74;
        bool r584652 = r584644 <= r584651;
        double r584653 = r584644 * r584644;
        double r584654 = a;
        double r584655 = r584654 * r584648;
        double r584656 = r584653 - r584655;
        double r584657 = sqrt(r584656);
        double r584658 = r584657 + r584644;
        double r584659 = r584658 / r584654;
        double r584660 = -r584659;
        double r584661 = 0.5;
        double r584662 = -2.0;
        double r584663 = r584644 * r584662;
        double r584664 = r584663 / r584654;
        double r584665 = fma(r584661, r584649, r584664);
        double r584666 = r584652 ? r584660 : r584665;
        double r584667 = r584646 ? r584650 : r584666;
        return r584667;
}

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 < -3.2092322739463293e-86

    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 9.4

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

    if -3.2092322739463293e-86 < b_2 < 2.891777552454845e+74

    1. Initial program 13.1

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

    3. Applied add-sqr-sqrt13.1

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

    4. Applied sqrt-prod13.3

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

    6. Applied neg-sub013.3

      \[\leadsto \frac{\color{blue}{\left(0 - b_2\right)} - \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    7. Applied associate--l-13.3

      \[\leadsto \frac{\color{blue}{0 - \left(b_2 + \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}}{a}\]

    8. Simplified13.1

      \[\leadsto \frac{0 - \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}}{a}\]

    if 2.891777552454845e+74 < b_2

    1. Initial program 38.9

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

    2. Taylor expanded around inf 4.2

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

    3. Simplified4.2

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(\frac{c}{b_2}\right), \left(\frac{b_2 \cdot -2}{a}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +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))