Average Error: 34.4 → 10.4
Time: 5.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r16358 = b_2;
        double r16359 = -r16358;
        double r16360 = r16358 * r16358;
        double r16361 = a;
        double r16362 = c;
        double r16363 = r16361 * r16362;
        double r16364 = r16360 - r16363;
        double r16365 = sqrt(r16364);
        double r16366 = r16359 - r16365;
        double r16367 = r16366 / r16361;
        return r16367;
}
↓
double f(double a, double b_2, double c) {
        double r16368 = b_2;
        double r16369 = -1.0674124610604968e-82;
        bool r16370 = r16368 <= r16369;
        double r16371 = -0.5;
        double r16372 = c;
        double r16373 = r16372 / r16368;
        double r16374 = r16371 * r16373;
        double r16375 = 5.968766258400916e+107;
        bool r16376 = r16368 <= r16375;
        double r16377 = -r16368;
        double r16378 = cbrt(r16377);
        double r16379 = r16378 * r16378;
        double r16380 = r16368 * r16368;
        double r16381 = a;
        double r16382 = r16381 * r16372;
        double r16383 = r16380 - r16382;
        double r16384 = sqrt(r16383);
        double r16385 = -r16384;
        double r16386 = fma(r16379, r16378, r16385);
        double r16387 = r16386 / r16381;
        double r16388 = 0.5;
        double r16389 = r16388 * r16373;
        double r16390 = 2.0;
        double r16391 = r16368 / r16381;
        double r16392 = r16390 * r16391;
        double r16393 = r16389 - r16392;
        double r16394 = r16376 ? r16387 : r16393;
        double r16395 = r16370 ? r16374 : r16394;
        return r16395;
}

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 < -1.0674124610604968e-82

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 8.8

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

    if -1.0674124610604968e-82 < b_2 < 5.968766258400916e+107

    1. Initial program 13.7

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

    3. Applied add-cube-cbrt13.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \sqrt[3]{-b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    4. Applied fma-neg13.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]

    if 5.968766258400916e+107 < b_2

    1. Initial program 50.0

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

    2. Taylor expanded around inf 3.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))