Average Error: 34.4 → 8.9
Time: 7.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.889080831912834239838349081155498349678 \cdot 10^{153}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.666823646884851555969061278738005639466 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 9.549485962776621401293196082424409645337 \cdot 10^{89}:\\ \;\;\;\;\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.889080831912834239838349081155498349678 \cdot 10^{153}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 7.666823646884851555969061278738005639466 \cdot 10^{-125}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 9.549485962776621401293196082424409645337 \cdot 10^{89}:\\
\;\;\;\;\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \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 r23294 = b_2;
        double r23295 = -r23294;
        double r23296 = r23294 * r23294;
        double r23297 = a;
        double r23298 = c;
        double r23299 = r23297 * r23298;
        double r23300 = r23296 - r23299;
        double r23301 = sqrt(r23300);
        double r23302 = r23295 + r23301;
        double r23303 = r23302 / r23297;
        return r23303;
}
↓
double f(double a, double b_2, double c) {
        double r23304 = b_2;
        double r23305 = -8.889080831912834e+153;
        bool r23306 = r23304 <= r23305;
        double r23307 = 0.5;
        double r23308 = c;
        double r23309 = r23308 / r23304;
        double r23310 = r23307 * r23309;
        double r23311 = 2.0;
        double r23312 = a;
        double r23313 = r23304 / r23312;
        double r23314 = r23311 * r23313;
        double r23315 = r23310 - r23314;
        double r23316 = 7.666823646884852e-125;
        bool r23317 = r23304 <= r23316;
        double r23318 = -r23304;
        double r23319 = r23304 * r23304;
        double r23320 = r23312 * r23308;
        double r23321 = r23319 - r23320;
        double r23322 = sqrt(r23321);
        double r23323 = r23318 + r23322;
        double r23324 = r23323 / r23312;
        double r23325 = 9.549485962776621e+89;
        bool r23326 = r23304 <= r23325;
        double r23327 = 0.0;
        double r23328 = r23327 + r23320;
        double r23329 = r23318 - r23322;
        double r23330 = r23328 / r23329;
        double r23331 = 1.0;
        double r23332 = r23331 / r23312;
        double r23333 = r23330 * r23332;
        double r23334 = -0.5;
        double r23335 = r23334 * r23309;
        double r23336 = r23326 ? r23333 : r23335;
        double r23337 = r23317 ? r23324 : r23336;
        double r23338 = r23306 ? r23315 : r23337;
        return r23338;
}

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 4 regimes

  2. if b_2 < -8.889080831912834e+153

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 2.8

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

    if -8.889080831912834e+153 < b_2 < 7.666823646884852e-125

    1. Initial program 11.0

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

    if 7.666823646884852e-125 < b_2 < 9.549485962776621e+89

    1. Initial program 40.0

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

    3. Applied flip-+40.1

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

    4. Simplified15.8

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

    6. Applied div-inv15.9

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}}\]

    if 9.549485962776621e+89 < b_2

    1. Initial program 59.5

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

    2. Taylor expanded around inf 2.5

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.889080831912834239838349081155498349678 \cdot 10^{153}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.666823646884851555969061278738005639466 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 9.549485962776621401293196082424409645337 \cdot 10^{89}:\\ \;\;\;\;\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))