Average Error: 33.1 → 10.4
Time: 22.0s
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 r3056519 = b_2;
        double r3056520 = -r3056519;
        double r3056521 = r3056519 * r3056519;
        double r3056522 = a;
        double r3056523 = c;
        double r3056524 = r3056522 * r3056523;
        double r3056525 = r3056521 - r3056524;
        double r3056526 = sqrt(r3056525);
        double r3056527 = r3056520 - r3056526;
        double r3056528 = r3056527 / r3056522;
        return r3056528;
}
↓
double f(double a, double b_2, double c) {
        double r3056529 = b_2;
        double r3056530 = -1.1962309819144974e-65;
        bool r3056531 = r3056529 <= r3056530;
        double r3056532 = -0.5;
        double r3056533 = c;
        double r3056534 = r3056533 / r3056529;
        double r3056535 = r3056532 * r3056534;
        double r3056536 = 5.6488521390017767e+48;
        bool r3056537 = r3056529 <= r3056536;
        double r3056538 = 1.0;
        double r3056539 = a;
        double r3056540 = r3056538 / r3056539;
        double r3056541 = -r3056529;
        double r3056542 = r3056529 * r3056529;
        double r3056543 = r3056533 * r3056539;
        double r3056544 = r3056542 - r3056543;
        double r3056545 = sqrt(r3056544);
        double r3056546 = r3056541 - r3056545;
        double r3056547 = r3056540 * r3056546;
        double r3056548 = r3056529 / r3056539;
        double r3056549 = -2.0;
        double r3056550 = 2.0;
        double r3056551 = r3056534 / r3056550;
        double r3056552 = fma(r3056548, r3056549, r3056551);
        double r3056553 = r3056537 ? r3056547 : r3056552;
        double r3056554 = r3056531 ? r3056535 : r3056553;
        return r3056554;
}

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 "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))