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

↓

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

\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 r60342 = b_2;
        double r60343 = -r60342;
        double r60344 = r60342 * r60342;
        double r60345 = a;
        double r60346 = c;
        double r60347 = r60345 * r60346;
        double r60348 = r60344 - r60347;
        double r60349 = sqrt(r60348);
        double r60350 = r60343 - r60349;
        double r60351 = r60350 / r60345;
        return r60351;
}

↓

double f(double a, double b_2, double c) {
        double r60352 = b_2;
        double r60353 = -7.359940312872037e+54;
        bool r60354 = r60352 <= r60353;
        double r60355 = -0.5;
        double r60356 = c;
        double r60357 = r60356 / r60352;
        double r60358 = r60355 * r60357;
        double r60359 = -1.2951718304590299e-256;
        bool r60360 = r60352 <= r60359;
        double r60361 = a;
        double r60362 = r60361 * r60356;
        double r60363 = -r60356;
        double r60364 = r60352 * r60352;
        double r60365 = fma(r60363, r60361, r60364);
        double r60366 = sqrt(r60365);
        double r60367 = r60366 - r60352;
        double r60368 = r60362 / r60367;
        double r60369 = r60368 / r60361;
        double r60370 = 1920982614230223.5;
        bool r60371 = r60352 <= r60370;
        double r60372 = 1.0;
        double r60373 = -r60352;
        double r60374 = 2.0;
        double r60375 = pow(r60352, r60374);
        double r60376 = r60356 * r60361;
        double r60377 = r60375 - r60376;
        double r60378 = sqrt(r60377);
        double r60379 = r60373 - r60378;
        double r60380 = r60361 / r60379;
        double r60381 = r60372 / r60380;
        double r60382 = 0.5;
        double r60383 = r60352 / r60361;
        double r60384 = -2.0;
        double r60385 = r60383 * r60384;
        double r60386 = fma(r60357, r60382, r60385);
        double r60387 = r60371 ? r60381 : r60386;
        double r60388 = r60360 ? r60369 : r60387;
        double r60389 = r60354 ? r60358 : r60388;
        return r60389;
}

Derivation

Split input into 4 regimes
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
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)}\]
Recombined 4 regimes into one program.
Final simplification9.8
\[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.295171830459029919055067501609567990033 \cdot 10^{-256}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1920982614230223.5:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Error

Derivation

`if b_2 < -7.359940312872037e+54`

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

`if -1.2951718304590299e-256 < b_2 < 1920982614230223.5`

`if 1920982614230223.5 < b_2`

Reproduce