Average Error: 33.6 → 10.8
Time: 18.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2 \cdot -2}{a}\right)\\ \end{array}\]
\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{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{-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 r2767497 = b_2;
        double r2767498 = -r2767497;
        double r2767499 = r2767497 * r2767497;
        double r2767500 = a;
        double r2767501 = c;
        double r2767502 = r2767500 * r2767501;
        double r2767503 = r2767499 - r2767502;
        double r2767504 = sqrt(r2767503);
        double r2767505 = r2767498 - r2767504;
        double r2767506 = r2767505 / r2767500;
        return r2767506;
}
↓
double f(double a, double b_2, double c) {
        double r2767507 = b_2;
        double r2767508 = -3.136683434005781e-32;
        bool r2767509 = r2767507 <= r2767508;
        double r2767510 = -0.5;
        double r2767511 = c;
        double r2767512 = r2767511 / r2767507;
        double r2767513 = r2767510 * r2767512;
        double r2767514 = 2.0410715251838527e+49;
        bool r2767515 = r2767507 <= r2767514;
        double r2767516 = r2767507 * r2767507;
        double r2767517 = a;
        double r2767518 = r2767511 * r2767517;
        double r2767519 = r2767516 - r2767518;
        double r2767520 = sqrt(r2767519);
        double r2767521 = r2767520 + r2767507;
        double r2767522 = -r2767517;
        double r2767523 = r2767521 / r2767522;
        double r2767524 = 0.5;
        double r2767525 = -2.0;
        double r2767526 = r2767507 * r2767525;
        double r2767527 = r2767526 / r2767517;
        double r2767528 = fma(r2767524, r2767512, r2767527);
        double r2767529 = r2767515 ? r2767523 : r2767528;
        double r2767530 = r2767509 ? r2767513 : r2767529;
        return r2767530;
}

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 frac-2neg15.8

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

    4. Simplified15.8

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

    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

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

Reproduce

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