Average Error: 33.8 → 6.8
Time: 11.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 2.523104187564355660465456399839335705723 \cdot 10^{61}:\\ \;\;\;\;1 \cdot \frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 2.523104187564355660465456399839335705723 \cdot 10^{61}:\\
\;\;\;\;1 \cdot \frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r29248 = b_2;
        double r29249 = -r29248;
        double r29250 = r29248 * r29248;
        double r29251 = a;
        double r29252 = c;
        double r29253 = r29251 * r29252;
        double r29254 = r29250 - r29253;
        double r29255 = sqrt(r29254);
        double r29256 = r29249 + r29255;
        double r29257 = r29256 / r29251;
        return r29257;
}
↓
double f(double a, double b_2, double c) {
        double r29258 = b_2;
        double r29259 = -8.301687926884189e+98;
        bool r29260 = r29258 <= r29259;
        double r29261 = 0.5;
        double r29262 = c;
        double r29263 = r29262 / r29258;
        double r29264 = r29261 * r29263;
        double r29265 = 2.0;
        double r29266 = a;
        double r29267 = r29258 / r29266;
        double r29268 = r29265 * r29267;
        double r29269 = r29264 - r29268;
        double r29270 = -1077853067741081.0;
        double r29271 = 1.3656093558537942e+244;
        double r29272 = r29270 / r29271;
        bool r29273 = r29258 <= r29272;
        double r29274 = -r29258;
        double r29275 = r29258 * r29258;
        double r29276 = r29266 * r29262;
        double r29277 = r29275 - r29276;
        double r29278 = sqrt(r29277);
        double r29279 = r29274 + r29278;
        double r29280 = r29279 / r29266;
        double r29281 = 2.5231041875643557e+61;
        bool r29282 = r29258 <= r29281;
        double r29283 = 1.0;
        double r29284 = r29274 - r29278;
        double r29285 = r29262 / r29284;
        double r29286 = r29283 * r29285;
        double r29287 = -0.5;
        double r29288 = r29287 * r29263;
        double r29289 = r29282 ? r29286 : r29288;
        double r29290 = r29273 ? r29280 : r29289;
        double r29291 = r29260 ? r29269 : r29290;
        return r29291;
}

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 < -8.301687926884189e+98

    1. Initial program 46.2

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

    2. Taylor expanded around -inf 3.6

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

    if -8.301687926884189e+98 < b_2 < -7.892835993842436e-230

    1. Initial program 8.0

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

    if -7.892835993842436e-230 < b_2 < 2.5231041875643557e+61

    1. Initial program 27.6

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

    3. Applied flip-+27.7

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

    4. Simplified16.1

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

    6. Applied clear-num16.2

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a \cdot c}}}}{a}\]
    7. Using strategy rm

    8. Applied div-inv16.7

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

    9. Applied *-un-lft-identity16.7

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

    10. Applied times-frac16.6

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

    11. Applied associate-/l*16.4

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

    12. Simplified10.2

      \[\leadsto \frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{\frac{1}{c}}}\]
    13. Using strategy rm

    14. Applied *-un-lft-identity10.2

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

    15. Applied *-un-lft-identity10.2

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

    16. Applied times-frac10.2

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

    17. Applied *-un-lft-identity10.2

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

    18. Applied *-un-lft-identity10.2

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

    19. Applied times-frac10.2

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

    20. Applied times-frac10.2

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

    21. Simplified10.2

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

    22. Simplified10.0

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

    if 2.5231041875643557e+61 < b_2

    1. Initial program 57.4

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

    2. Taylor expanded around inf 3.9

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 2.523104187564355660465456399839335705723 \cdot 10^{61}:\\ \;\;\;\;1 \cdot \frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))