Average Error: 34.6 → 9.9
Time: 27.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.6581383089037873 \cdot 10^{81}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 2.45811587950602871 \cdot 10^{-136}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 4.40565710546396028 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.1310446734884525 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.75875355528826554 \cdot 10^{78}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.6581383089037873 \cdot 10^{81}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 2.45811587950602871 \cdot 10^{-136}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 4.40565710546396028 \cdot 10^{-70}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.1310446734884525 \cdot 10^{-47}:\\
\;\;\;\;\frac{\frac{a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\

\mathbf{elif}\;b_2 \le 4.75875355528826554 \cdot 10^{78}:\\
\;\;\;\;\frac{a \cdot c}{a \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r26270 = b_2;
        double r26271 = -r26270;
        double r26272 = r26270 * r26270;
        double r26273 = a;
        double r26274 = c;
        double r26275 = r26273 * r26274;
        double r26276 = r26272 - r26275;
        double r26277 = sqrt(r26276);
        double r26278 = r26271 + r26277;
        double r26279 = r26278 / r26273;
        return r26279;
}
↓
double f(double a, double b_2, double c) {
        double r26280 = b_2;
        double r26281 = -1.6581383089037873e+81;
        bool r26282 = r26280 <= r26281;
        double r26283 = 0.5;
        double r26284 = c;
        double r26285 = r26284 / r26280;
        double r26286 = r26283 * r26285;
        double r26287 = 2.0;
        double r26288 = a;
        double r26289 = r26280 / r26288;
        double r26290 = r26287 * r26289;
        double r26291 = r26286 - r26290;
        double r26292 = 2.4581158795060287e-136;
        bool r26293 = r26280 <= r26292;
        double r26294 = -r26280;
        double r26295 = r26280 * r26280;
        double r26296 = r26288 * r26284;
        double r26297 = r26295 - r26296;
        double r26298 = sqrt(r26297);
        double r26299 = r26294 + r26298;
        double r26300 = 1.0;
        double r26301 = r26300 / r26288;
        double r26302 = r26299 * r26301;
        double r26303 = 4.40565710546396e-70;
        bool r26304 = r26280 <= r26303;
        double r26305 = -0.5;
        double r26306 = r26305 * r26285;
        double r26307 = 1.1310446734884525e-47;
        bool r26308 = r26280 <= r26307;
        double r26309 = r26294 - r26298;
        double r26310 = r26296 / r26309;
        double r26311 = r26310 / r26288;
        double r26312 = 4.7587535552882655e+78;
        bool r26313 = r26280 <= r26312;
        double r26314 = r26288 * r26309;
        double r26315 = r26296 / r26314;
        double r26316 = r26313 ? r26315 : r26306;
        double r26317 = r26308 ? r26311 : r26316;
        double r26318 = r26304 ? r26306 : r26317;
        double r26319 = r26293 ? r26302 : r26318;
        double r26320 = r26282 ? r26291 : r26319;
        return r26320;
}

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 5 regimes

  2. if b_2 < -1.6581383089037873e+81

    1. Initial program 43.9

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

    2. Taylor expanded around -inf 3.7

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

    if -1.6581383089037873e+81 < b_2 < 2.4581158795060287e-136

    1. Initial program 11.6

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

    3. Applied div-inv11.7

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

    if 2.4581158795060287e-136 < b_2 < 4.40565710546396e-70 or 4.7587535552882655e+78 < b_2

    1. Initial program 53.8

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

    2. Taylor expanded around inf 9.1

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

    if 4.40565710546396e-70 < b_2 < 1.1310446734884525e-47

    1. Initial program 31.8

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

    3. Applied flip-+31.9

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

    4. Simplified13.4

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

    if 1.1310446734884525e-47 < b_2 < 4.7587535552882655e+78

    1. Initial program 45.7

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

    3. Applied flip-+45.7

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

    4. Simplified14.4

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

    6. Applied div-inv14.5

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

    7. Applied associate-/l*15.2

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

    8. Simplified15.1

      \[\leadsto \frac{0 + a \cdot c}{\color{blue}{a \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.6581383089037873 \cdot 10^{81}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 2.45811587950602871 \cdot 10^{-136}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 4.40565710546396028 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.1310446734884525 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.75875355528826554 \cdot 10^{78}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))