Average Error: 34.3 → 9.8
Time: 21.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.295171830459029919055067501609567990033 \cdot 10^{-256}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1920982614230223.5:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.295171830459029919055067501609567990033 \cdot 10^{-256}:\\
\;\;\;\;\frac{\frac{a \cdot c}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 1920982614230223.5:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r28245 = b_2;
        double r28246 = -r28245;
        double r28247 = r28245 * r28245;
        double r28248 = a;
        double r28249 = c;
        double r28250 = r28248 * r28249;
        double r28251 = r28247 - r28250;
        double r28252 = sqrt(r28251);
        double r28253 = r28246 - r28252;
        double r28254 = r28253 / r28248;
        return r28254;
}
↓
double f(double a, double b_2, double c) {
        double r28255 = b_2;
        double r28256 = -7.359940312872037e+54;
        bool r28257 = r28255 <= r28256;
        double r28258 = -0.5;
        double r28259 = c;
        double r28260 = r28259 / r28255;
        double r28261 = r28258 * r28260;
        double r28262 = -1.2951718304590299e-256;
        bool r28263 = r28255 <= r28262;
        double r28264 = a;
        double r28265 = r28264 * r28259;
        double r28266 = -r28259;
        double r28267 = r28255 * r28255;
        double r28268 = fma(r28266, r28264, r28267);
        double r28269 = sqrt(r28268);
        double r28270 = r28269 - r28255;
        double r28271 = r28265 / r28270;
        double r28272 = r28271 / r28264;
        double r28273 = 1920982614230223.5;
        bool r28274 = r28255 <= r28273;
        double r28275 = 1.0;
        double r28276 = -r28255;
        double r28277 = 2.0;
        double r28278 = pow(r28255, r28277);
        double r28279 = r28259 * r28264;
        double r28280 = r28278 - r28279;
        double r28281 = sqrt(r28280);
        double r28282 = r28276 - r28281;
        double r28283 = r28264 / r28282;
        double r28284 = r28275 / r28283;
        double r28285 = 0.5;
        double r28286 = r28255 / r28264;
        double r28287 = -2.0;
        double r28288 = r28286 * r28287;
        double r28289 = fma(r28260, r28285, r28288);
        double r28290 = r28274 ? r28284 : r28289;
        double r28291 = r28263 ? r28272 : r28290;
        double r28292 = r28257 ? r28261 : r28291;
        return r28292;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -7.359940312872037e+54

    1. Initial program 57.6

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

    2. Taylor expanded around -inf 3.8

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

    if -7.359940312872037e+54 < b_2 < -1.2951718304590299e-256

    1. Initial program 33.6

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

    3. Applied flip--33.6

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

    4. Simplified18.4

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified18.4

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}{a}\]

    if -1.2951718304590299e-256 < b_2 < 1920982614230223.5

    1. Initial program 10.8

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

    3. Applied clear-num10.9

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

    4. Simplified10.9

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

    if 1920982614230223.5 < b_2

    1. Initial program 34.0

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

    3. Applied div-inv34.1

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

    4. Taylor expanded around inf 6.8

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

    5. Simplified6.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.295171830459029919055067501609567990033 \cdot 10^{-256}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1920982614230223.5:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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))