Average Error: 33.7 → 10.9
Time: 5.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.98276540088900058 \cdot 10^{134}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r18306 = b_2;
        double r18307 = -r18306;
        double r18308 = r18306 * r18306;
        double r18309 = a;
        double r18310 = c;
        double r18311 = r18309 * r18310;
        double r18312 = r18308 - r18311;
        double r18313 = sqrt(r18312);
        double r18314 = r18307 + r18313;
        double r18315 = r18314 / r18309;
        return r18315;
}
↓
double f(double a, double b_2, double c) {
        double r18316 = b_2;
        double r18317 = -1.9827654008890006e+134;
        bool r18318 = r18316 <= r18317;
        double r18319 = 0.5;
        double r18320 = c;
        double r18321 = r18320 / r18316;
        double r18322 = r18319 * r18321;
        double r18323 = 2.0;
        double r18324 = a;
        double r18325 = r18316 / r18324;
        double r18326 = r18323 * r18325;
        double r18327 = r18322 - r18326;
        double r18328 = 1.1860189201379418e-161;
        bool r18329 = r18316 <= r18328;
        double r18330 = -r18316;
        double r18331 = r18316 * r18316;
        double r18332 = r18324 * r18320;
        double r18333 = r18331 - r18332;
        double r18334 = sqrt(r18333);
        double r18335 = r18330 + r18334;
        double r18336 = 1.0;
        double r18337 = r18336 / r18324;
        double r18338 = r18335 * r18337;
        double r18339 = -0.5;
        double r18340 = r18339 * r18321;
        double r18341 = r18329 ? r18338 : r18340;
        double r18342 = r18318 ? r18327 : r18341;
        return r18342;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.9827654008890006e+134

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 3.1

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

    if -1.9827654008890006e+134 < b_2 < 1.1860189201379418e-161

    1. Initial program 10.3

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

    3. Applied div-inv10.4

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

    if 1.1860189201379418e-161 < b_2

    1. Initial program 49.6

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

    2. Taylor expanded around inf 13.6

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))