Average Error: 34.0 → 10.5
Time: 11.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.913471920057083513053706311365148123626 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 106106913250787377152:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.913471920057083513053706311365148123626 \cdot 10^{-110}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 106106913250787377152:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r81422 = b_2;
        double r81423 = -r81422;
        double r81424 = r81422 * r81422;
        double r81425 = a;
        double r81426 = c;
        double r81427 = r81425 * r81426;
        double r81428 = r81424 - r81427;
        double r81429 = sqrt(r81428);
        double r81430 = r81423 - r81429;
        double r81431 = r81430 / r81425;
        return r81431;
}
↓
double f(double a, double b_2, double c) {
        double r81432 = b_2;
        double r81433 = -1.9134719200570835e-110;
        bool r81434 = r81432 <= r81433;
        double r81435 = -0.5;
        double r81436 = c;
        double r81437 = r81436 / r81432;
        double r81438 = r81435 * r81437;
        double r81439 = 1.0610691325078738e+20;
        bool r81440 = r81432 <= r81439;
        double r81441 = -r81432;
        double r81442 = r81432 * r81432;
        double r81443 = a;
        double r81444 = r81443 * r81436;
        double r81445 = r81442 - r81444;
        double r81446 = sqrt(r81445);
        double r81447 = r81441 - r81446;
        double r81448 = r81447 / r81443;
        double r81449 = 0.5;
        double r81450 = r81432 / r81443;
        double r81451 = -2.0;
        double r81452 = r81450 * r81451;
        double r81453 = fma(r81449, r81437, r81452);
        double r81454 = r81440 ? r81448 : r81453;
        double r81455 = r81434 ? r81438 : r81454;
        return r81455;
}

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 < -1.9134719200570835e-110

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 10.7

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

    if -1.9134719200570835e-110 < b_2 < 1.0610691325078738e+20

    1. Initial program 12.7

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

    3. Applied *-un-lft-identity12.7

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

    4. Applied *-un-lft-identity12.7

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

    5. Applied times-frac12.7

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

    6. Simplified12.7

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

    if 1.0610691325078738e+20 < b_2

    1. Initial program 34.5

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

    2. Taylor expanded around inf 6.8

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

    3. Simplified6.8

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.913471920057083513053706311365148123626 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 106106913250787377152:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))