Average Error: 32.9 → 9.5
Time: 17.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.376414644198452 \cdot 10^{-55}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.3362227636224895 \cdot 10^{+83}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.376414644198452 \cdot 10^{-55}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.3362227636224895 \cdot 10^{+83}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r582320 = b_2;
        double r582321 = -r582320;
        double r582322 = r582320 * r582320;
        double r582323 = a;
        double r582324 = c;
        double r582325 = r582323 * r582324;
        double r582326 = r582322 - r582325;
        double r582327 = sqrt(r582326);
        double r582328 = r582321 - r582327;
        double r582329 = r582328 / r582323;
        return r582329;
}
↓
double f(double a, double b_2, double c) {
        double r582330 = b_2;
        double r582331 = -1.376414644198452e-55;
        bool r582332 = r582330 <= r582331;
        double r582333 = -0.5;
        double r582334 = c;
        double r582335 = r582334 / r582330;
        double r582336 = r582333 * r582335;
        double r582337 = 2.3362227636224895e+83;
        bool r582338 = r582330 <= r582337;
        double r582339 = -r582330;
        double r582340 = a;
        double r582341 = r582339 / r582340;
        double r582342 = r582330 * r582330;
        double r582343 = r582334 * r582340;
        double r582344 = r582342 - r582343;
        double r582345 = sqrt(r582344);
        double r582346 = r582345 / r582340;
        double r582347 = r582341 - r582346;
        double r582348 = 0.5;
        double r582349 = r582335 * r582348;
        double r582350 = 2.0;
        double r582351 = r582330 / r582340;
        double r582352 = r582350 * r582351;
        double r582353 = r582349 - r582352;
        double r582354 = r582338 ? r582347 : r582353;
        double r582355 = r582332 ? r582336 : r582354;
        return r582355;
}

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 < -1.376414644198452e-55

    1. Initial program 53.3

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

    3. Applied div-sub54.0

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

    4. Taylor expanded around -inf 8.1

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

    if -1.376414644198452e-55 < b_2 < 2.3362227636224895e+83

    1. Initial program 12.9

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

    3. Applied div-sub12.9

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

    if 2.3362227636224895e+83 < b_2

    1. Initial program 41.5

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

    3. Applied div-sub41.5

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt41.5

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

    6. Applied sqrt-prod41.5

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

    7. Applied associate-/l*41.5

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

    8. 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 simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.376414644198452 \cdot 10^{-55}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.3362227636224895 \cdot 10^{+83}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))