Average Error: 33.5 → 10.4
Time: 1.7m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(b_2 \cdot -2\right)\right)}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r8025524 = b_2;
        double r8025525 = -r8025524;
        double r8025526 = r8025524 * r8025524;
        double r8025527 = a;
        double r8025528 = c;
        double r8025529 = r8025527 * r8025528;
        double r8025530 = r8025526 - r8025529;
        double r8025531 = sqrt(r8025530);
        double r8025532 = r8025525 - r8025531;
        double r8025533 = r8025532 / r8025527;
        return r8025533;
}
↓
double f(double a, double b_2, double c) {
        double r8025534 = b_2;
        double r8025535 = -5.691277786452672e-38;
        bool r8025536 = r8025534 <= r8025535;
        double r8025537 = -0.5;
        double r8025538 = c;
        double r8025539 = r8025538 / r8025534;
        double r8025540 = r8025537 * r8025539;
        double r8025541 = 1.502350718288979e+75;
        bool r8025542 = r8025534 <= r8025541;
        double r8025543 = -r8025534;
        double r8025544 = r8025534 * r8025534;
        double r8025545 = a;
        double r8025546 = r8025545 * r8025538;
        double r8025547 = r8025544 - r8025546;
        double r8025548 = sqrt(r8025547);
        double r8025549 = r8025543 - r8025548;
        double r8025550 = r8025549 / r8025545;
        double r8025551 = 0.5;
        double r8025552 = r8025534 / r8025538;
        double r8025553 = r8025545 / r8025552;
        double r8025554 = -2.0;
        double r8025555 = r8025534 * r8025554;
        double r8025556 = fma(r8025551, r8025553, r8025555);
        double r8025557 = r8025556 / r8025545;
        double r8025558 = r8025542 ? r8025550 : r8025557;
        double r8025559 = r8025536 ? r8025540 : r8025558;
        return r8025559;
}

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 < -5.691277786452672e-38

    1. Initial program 54.0

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

    2. Taylor expanded around -inf 54.0

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified54.0

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

    4. Taylor expanded around -inf 7.9

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

    if -5.691277786452672e-38 < b_2 < 1.502350718288979e+75

    1. Initial program 14.7

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

    2. Taylor expanded around -inf 14.7

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified14.7

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

    if 1.502350718288979e+75 < b_2

    1. Initial program 40.8

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

    2. Taylor expanded around -inf 40.8

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified40.8

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

    4. Taylor expanded around inf 9.8

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

    5. Simplified4.6

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(b_2 \cdot -2\right)\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))