Average Error: 20.0 → 7.9
Time: 12.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.50519849908317111 \cdot 10^{136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}}}\right) - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.50519849908317111 \cdot 10^{136}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}}}\right) - b\right) - b}\\

\end{array}\\

\mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\

\end{array}
double f(double a, double b, double c) {
        double r46331 = b;
        double r46332 = 0.0;
        bool r46333 = r46331 >= r46332;
        double r46334 = -r46331;
        double r46335 = r46331 * r46331;
        double r46336 = 4.0;
        double r46337 = a;
        double r46338 = r46336 * r46337;
        double r46339 = c;
        double r46340 = r46338 * r46339;
        double r46341 = r46335 - r46340;
        double r46342 = sqrt(r46341);
        double r46343 = r46334 - r46342;
        double r46344 = 2.0;
        double r46345 = r46344 * r46337;
        double r46346 = r46343 / r46345;
        double r46347 = r46344 * r46339;
        double r46348 = r46334 + r46342;
        double r46349 = r46347 / r46348;
        double r46350 = r46333 ? r46346 : r46349;
        return r46350;
}
↓
double f(double a, double b, double c) {
        double r46351 = b;
        double r46352 = -3.505198499083171e+136;
        bool r46353 = r46351 <= r46352;
        double r46354 = 0.0;
        bool r46355 = r46351 >= r46354;
        double r46356 = -r46351;
        double r46357 = r46351 * r46351;
        double r46358 = 4.0;
        double r46359 = a;
        double r46360 = r46358 * r46359;
        double r46361 = c;
        double r46362 = r46360 * r46361;
        double r46363 = r46357 - r46362;
        double r46364 = sqrt(r46363);
        double r46365 = cbrt(r46364);
        double r46366 = r46365 * r46365;
        double r46367 = r46366 * r46365;
        double r46368 = r46356 - r46367;
        double r46369 = 2.0;
        double r46370 = r46369 * r46359;
        double r46371 = r46368 / r46370;
        double r46372 = r46369 * r46361;
        double r46373 = cbrt(r46351);
        double r46374 = r46373 * r46373;
        double r46375 = r46359 / r46374;
        double r46376 = cbrt(r46361);
        double r46377 = r46376 * r46376;
        double r46378 = cbrt(r46373);
        double r46379 = r46378 * r46378;
        double r46380 = r46377 / r46379;
        double r46381 = r46375 * r46380;
        double r46382 = r46376 / r46378;
        double r46383 = r46381 * r46382;
        double r46384 = r46369 * r46383;
        double r46385 = r46384 - r46351;
        double r46386 = r46385 - r46351;
        double r46387 = r46372 / r46386;
        double r46388 = r46355 ? r46371 : r46387;
        double r46389 = 4.1199128263687574e+46;
        bool r46390 = r46351 <= r46389;
        double r46391 = r46356 - r46364;
        double r46392 = r46391 / r46370;
        double r46393 = sqrt(r46364);
        double r46394 = r46393 * r46393;
        double r46395 = r46394 - r46351;
        double r46396 = r46372 / r46395;
        double r46397 = r46355 ? r46392 : r46396;
        double r46398 = r46359 * r46361;
        double r46399 = r46398 / r46351;
        double r46400 = r46369 * r46399;
        double r46401 = r46351 - r46400;
        double r46402 = r46356 - r46401;
        double r46403 = r46402 / r46370;
        double r46404 = r46364 - r46351;
        double r46405 = r46372 / r46404;
        double r46406 = r46355 ? r46403 : r46405;
        double r46407 = r46390 ? r46397 : r46406;
        double r46408 = r46353 ? r46388 : r46407;
        return r46408;
}

Error

Bits error versus a

Bits error versus b

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 < -3.505198499083171e+136

    1. Initial program 34.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified34.4

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]

    3. Taylor expanded around -inf 5.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}\\ \end{array}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt5.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \frac{a \cdot c}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}} - b\right) - b}\\ \end{array}\]

    6. Applied times-frac1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - b\right) - b}\\ \end{array}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\left(\sqrt[3]{\sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}\right) \cdot \sqrt[3]{\sqrt[3]{b}}}\right) - b\right) - b}\\ \end{array}\]

    9. Applied add-cube-cbrt1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}}{\left(\sqrt[3]{\sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}\right) \cdot \sqrt[3]{\sqrt[3]{b}}}\right) - b\right) - b}\\ \end{array}\]

    10. Applied times-frac1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \left(\frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}} \cdot \frac{\sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}}}\right)\right) - b\right) - b}\\ \end{array}\]

    11. Applied associate-*r*1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}}}\right) - b\right) - b}\\ \end{array}\]
    12. Using strategy rm

    13. Applied add-cube-cbrt1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}}}\right) - b\right) - b}\\ \end{array}\]

    if -3.505198499083171e+136 < b < 4.1199128263687574e+46

    1. Initial program 9.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified9.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt9.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2} \cdot c}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\]

    5. Applied sqrt-prod9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\]

    if 4.1199128263687574e+46 < b

    1. Initial program 36.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified36.8

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]

    3. Taylor expanded around inf 10.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.50519849908317111 \cdot 10^{136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt[3]{\sqrt[3]{b}}}\right) - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))