Average Error: 33.6 → 10.8
Time: 18.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -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(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r2633498 = b_2;
        double r2633499 = -r2633498;
        double r2633500 = r2633498 * r2633498;
        double r2633501 = a;
        double r2633502 = c;
        double r2633503 = r2633501 * r2633502;
        double r2633504 = r2633500 - r2633503;
        double r2633505 = sqrt(r2633504);
        double r2633506 = r2633499 - r2633505;
        double r2633507 = r2633506 / r2633501;
        return r2633507;
}
↓
double f(double a, double b_2, double c) {
        double r2633508 = b_2;
        double r2633509 = -3.136683434005781e-32;
        bool r2633510 = r2633508 <= r2633509;
        double r2633511 = -0.5;
        double r2633512 = c;
        double r2633513 = r2633512 / r2633508;
        double r2633514 = r2633511 * r2633513;
        double r2633515 = 2.0410715251838527e+49;
        bool r2633516 = r2633508 <= r2633515;
        double r2633517 = r2633508 * r2633508;
        double r2633518 = a;
        double r2633519 = r2633512 * r2633518;
        double r2633520 = r2633517 - r2633519;
        double r2633521 = sqrt(r2633520);
        double r2633522 = r2633521 + r2633508;
        double r2633523 = -r2633518;
        double r2633524 = r2633522 / r2633523;
        double r2633525 = -2.0;
        double r2633526 = r2633508 / r2633518;
        double r2633527 = 0.5;
        double r2633528 = r2633513 * r2633527;
        double r2633529 = fma(r2633525, r2633526, r2633528);
        double r2633530 = r2633516 ? r2633524 : r2633529;
        double r2633531 = r2633510 ? r2633514 : r2633530;
        return r2633531;
}

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. Using strategy rm

    3. Applied frac-2neg36.2

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

    4. Simplified36.2

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

    5. Taylor expanded around inf 6.2

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

    6. Simplified6.2

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

  4. Final simplification10.8

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

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))