Average Error: 33.1 → 10.3
Time: 21.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.7729369216517423 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 9.831724396970673 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.7729369216517423 \cdot 10^{+64}:\\
\;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 9.831724396970673 \cdot 10^{-110}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r521463 = b_2;
        double r521464 = -r521463;
        double r521465 = r521463 * r521463;
        double r521466 = a;
        double r521467 = c;
        double r521468 = r521466 * r521467;
        double r521469 = r521465 - r521468;
        double r521470 = sqrt(r521469);
        double r521471 = r521464 + r521470;
        double r521472 = r521471 / r521466;
        return r521472;
}
↓
double f(double a, double b_2, double c) {
        double r521473 = b_2;
        double r521474 = -1.7729369216517423e+64;
        bool r521475 = r521473 <= r521474;
        double r521476 = -2.0;
        double r521477 = a;
        double r521478 = r521473 / r521477;
        double r521479 = 0.5;
        double r521480 = c;
        double r521481 = r521480 / r521473;
        double r521482 = r521479 * r521481;
        double r521483 = fma(r521476, r521478, r521482);
        double r521484 = 9.831724396970673e-110;
        bool r521485 = r521473 <= r521484;
        double r521486 = r521473 * r521473;
        double r521487 = r521480 * r521477;
        double r521488 = r521486 - r521487;
        double r521489 = sqrt(r521488);
        double r521490 = r521489 - r521473;
        double r521491 = r521490 / r521477;
        double r521492 = -0.5;
        double r521493 = r521481 * r521492;
        double r521494 = r521485 ? r521491 : r521493;
        double r521495 = r521475 ? r521483 : r521494;
        return r521495;
}

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 < -1.7729369216517423e+64

    1. Initial program 37.7

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

    2. Simplified37.7

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

    3. Taylor expanded around -inf 5.2

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

    4. Simplified5.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)}\]

    if -1.7729369216517423e+64 < b_2 < 9.831724396970673e-110

    1. Initial program 12.0

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

    2. Simplified12.0

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

    3. Taylor expanded around inf 12.0

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

    4. Simplified12.0

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

    if 9.831724396970673e-110 < b_2

    1. Initial program 50.9

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

    2. Simplified50.9

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

    3. Taylor expanded around inf 10.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.7729369216517423 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 9.831724396970673 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))