Average Error: 33.8 → 10.4
Time: 19.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.412776568687283300932456834981587297891 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 4.603517726908400645968266248286182254745 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.412776568687283300932456834981587297891 \cdot 10^{126}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 4.603517726908400645968266248286182254745 \cdot 10^{-74}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r29245 = b_2;
        double r29246 = -r29245;
        double r29247 = r29245 * r29245;
        double r29248 = a;
        double r29249 = c;
        double r29250 = r29248 * r29249;
        double r29251 = r29247 - r29250;
        double r29252 = sqrt(r29251);
        double r29253 = r29246 + r29252;
        double r29254 = r29253 / r29248;
        return r29254;
}
↓
double f(double a, double b_2, double c) {
        double r29255 = b_2;
        double r29256 = -3.4127765686872833e+126;
        bool r29257 = r29255 <= r29256;
        double r29258 = 0.5;
        double r29259 = c;
        double r29260 = r29259 / r29255;
        double r29261 = r29258 * r29260;
        double r29262 = 2.0;
        double r29263 = a;
        double r29264 = r29255 / r29263;
        double r29265 = r29262 * r29264;
        double r29266 = r29261 - r29265;
        double r29267 = 4.603517726908401e-74;
        bool r29268 = r29255 <= r29267;
        double r29269 = 1.0;
        double r29270 = r29255 * r29255;
        double r29271 = r29263 * r29259;
        double r29272 = r29270 - r29271;
        double r29273 = sqrt(r29272);
        double r29274 = r29273 - r29255;
        double r29275 = r29263 / r29274;
        double r29276 = r29269 / r29275;
        double r29277 = -0.5;
        double r29278 = r29277 * r29260;
        double r29279 = r29268 ? r29276 : r29278;
        double r29280 = r29257 ? r29266 : r29279;
        return r29280;
}

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 < -3.4127765686872833e+126

    1. Initial program 53.6

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

    2. Taylor expanded around -inf 3.2

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

    if -3.4127765686872833e+126 < b_2 < 4.603517726908401e-74

    1. Initial program 13.2

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

    3. Applied clear-num13.3

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

    4. Simplified13.3

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

    if 4.603517726908401e-74 < b_2

    1. Initial program 53.1

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

    2. Taylor expanded around inf 9.1

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.412776568687283300932456834981587297891 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 4.603517726908400645968266248286182254745 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))