Average Error: 32.9 → 10.4
Time: 1.1m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.805535571809126 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.4120493174601926 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{a \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{c}{b_2}\right), \frac{1}{2}, \left(b_2 \cdot \frac{-2}{a}\right)\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.805535571809126 \cdot 10^{-39}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.4120493174601926 \cdot 10^{+147}:\\
\;\;\;\;\frac{1}{a \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r6538459 = b_2;
        double r6538460 = -r6538459;
        double r6538461 = r6538459 * r6538459;
        double r6538462 = a;
        double r6538463 = c;
        double r6538464 = r6538462 * r6538463;
        double r6538465 = r6538461 - r6538464;
        double r6538466 = sqrt(r6538465);
        double r6538467 = r6538460 - r6538466;
        double r6538468 = r6538467 / r6538462;
        return r6538468;
}
↓
double f(double a, double b_2, double c) {
        double r6538469 = b_2;
        double r6538470 = -3.805535571809126e-39;
        bool r6538471 = r6538469 <= r6538470;
        double r6538472 = -0.5;
        double r6538473 = c;
        double r6538474 = r6538473 / r6538469;
        double r6538475 = r6538472 * r6538474;
        double r6538476 = 3.4120493174601926e+147;
        bool r6538477 = r6538469 <= r6538476;
        double r6538478 = 1.0;
        double r6538479 = a;
        double r6538480 = -r6538469;
        double r6538481 = r6538469 * r6538469;
        double r6538482 = r6538473 * r6538479;
        double r6538483 = r6538481 - r6538482;
        double r6538484 = sqrt(r6538483);
        double r6538485 = r6538480 - r6538484;
        double r6538486 = r6538478 / r6538485;
        double r6538487 = r6538479 * r6538486;
        double r6538488 = r6538478 / r6538487;
        double r6538489 = 0.5;
        double r6538490 = -2.0;
        double r6538491 = r6538490 / r6538479;
        double r6538492 = r6538469 * r6538491;
        double r6538493 = fma(r6538474, r6538489, r6538492);
        double r6538494 = r6538477 ? r6538488 : r6538493;
        double r6538495 = r6538471 ? r6538475 : r6538494;
        return r6538495;
}

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 < -3.805535571809126e-39

    1. Initial program 53.6

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

    2. Taylor expanded around -inf 7.7

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

    if -3.805535571809126e-39 < b_2 < 3.4120493174601926e+147

    1. Initial program 13.5

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

    3. Applied *-un-lft-identity13.5

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

    4. Applied *-un-lft-identity13.5

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

    5. Applied distribute-lft-out--13.5

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

    6. Applied associate-/l*13.6

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

    8. Applied div-inv13.7

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

    if 3.4120493174601926e+147 < b_2

    1. Initial program 58.6

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

    2. Taylor expanded around inf 3.3

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

    3. Simplified3.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.805535571809126 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.4120493174601926 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{a \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{c}{b_2}\right), \frac{1}{2}, \left(b_2 \cdot \frac{-2}{a}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))