Average Error: 20.4 → 7.7
Time: 7.1s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3459740091915103 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le -1.078731019298903 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \left(e^{\log \left(\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.6883396060119998 \cdot 10^{69}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.3459740091915103 \cdot 10^{154}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le -1.078731019298903 \cdot 10^{-310}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \left(e^{\log \left(\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right)}\\

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

\end{array}\\

\mathbf{elif}\;b \le 3.6883396060119998 \cdot 10^{69}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right)\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r42395 = b;
        double r42396 = 0.0;
        bool r42397 = r42395 >= r42396;
        double r42398 = 2.0;
        double r42399 = c;
        double r42400 = r42398 * r42399;
        double r42401 = -r42395;
        double r42402 = r42395 * r42395;
        double r42403 = 4.0;
        double r42404 = a;
        double r42405 = r42403 * r42404;
        double r42406 = r42405 * r42399;
        double r42407 = r42402 - r42406;
        double r42408 = sqrt(r42407);
        double r42409 = r42401 - r42408;
        double r42410 = r42400 / r42409;
        double r42411 = r42401 + r42408;
        double r42412 = r42398 * r42404;
        double r42413 = r42411 / r42412;
        double r42414 = r42397 ? r42410 : r42413;
        return r42414;
}
↓
double f(double a, double b, double c) {
        double r42415 = b;
        double r42416 = -1.3459740091915103e+154;
        bool r42417 = r42415 <= r42416;
        double r42418 = 0.0;
        bool r42419 = r42415 >= r42418;
        double r42420 = 2.0;
        double r42421 = c;
        double r42422 = r42420 * r42421;
        double r42423 = -r42415;
        double r42424 = r42415 * r42415;
        double r42425 = 4.0;
        double r42426 = a;
        double r42427 = r42425 * r42426;
        double r42428 = r42427 * r42421;
        double r42429 = r42424 - r42428;
        double r42430 = sqrt(r42429);
        double r42431 = r42423 - r42430;
        double r42432 = r42422 / r42431;
        double r42433 = r42426 * r42421;
        double r42434 = r42433 / r42415;
        double r42435 = r42420 * r42434;
        double r42436 = r42435 - r42415;
        double r42437 = r42423 + r42436;
        double r42438 = r42420 * r42426;
        double r42439 = r42437 / r42438;
        double r42440 = r42419 ? r42432 : r42439;
        double r42441 = -1.0787310192989e-310;
        bool r42442 = r42415 <= r42441;
        double r42443 = sqrt(r42415);
        double r42444 = r42426 / r42443;
        double r42445 = 0.5;
        double r42446 = pow(r42415, r42445);
        double r42447 = r42421 / r42446;
        double r42448 = cbrt(r42447);
        double r42449 = log(r42448);
        double r42450 = exp(r42449);
        double r42451 = r42450 * r42448;
        double r42452 = r42444 * r42451;
        double r42453 = r42452 * r42448;
        double r42454 = r42420 * r42453;
        double r42455 = r42415 - r42454;
        double r42456 = r42423 - r42455;
        double r42457 = r42422 / r42456;
        double r42458 = r42423 + r42430;
        double r42459 = r42458 / r42438;
        double r42460 = r42419 ? r42457 : r42459;
        double r42461 = 3.688339606012e+69;
        bool r42462 = r42415 <= r42461;
        double r42463 = r42444 * r42447;
        double r42464 = r42420 * r42463;
        double r42465 = r42415 - r42464;
        double r42466 = r42423 - r42465;
        double r42467 = r42422 / r42466;
        double r42468 = -r42429;
        double r42469 = fma(r42415, r42415, r42468);
        double r42470 = r42469 / r42431;
        double r42471 = r42470 / r42438;
        double r42472 = r42419 ? r42467 : r42471;
        double r42473 = r42462 ? r42440 : r42472;
        double r42474 = r42442 ? r42460 : r42473;
        double r42475 = r42417 ? r42440 : r42474;
        return r42475;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3459740091915103e+154 or -1.0787310192989e-310 < b < 3.688339606012e+69

    1. Initial program 24.9

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

    2. Taylor expanded around -inf 9.8

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

    if -1.3459740091915103e+154 < b < -1.0787310192989e-310

    1. Initial program 8.6

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

    2. Taylor expanded around inf 8.6

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

    4. Applied add-sqr-sqrt8.6

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

    5. Applied times-frac8.6

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

    6. Simplified8.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \color{blue}{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt8.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    9. Applied associate-*r*8.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \color{blue}{\left(\left(\frac{a}{\sqrt{b}} \cdot \left(\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    10. Using strategy rm

    11. Applied add-exp-log8.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \left(\color{blue}{e^{\log \left(\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)}} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    if 3.688339606012e+69 < b

    1. Initial program 28.9

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

    2. Taylor expanded around inf 7.4

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

    4. Applied add-sqr-sqrt7.4

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

    5. Applied times-frac3.4

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

    6. Simplified3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \color{blue}{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    7. Using strategy rm

    8. Applied flip-+3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    9. Simplified3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3459740091915103 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le -1.078731019298903 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \left(e^{\log \left(\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.6883396060119998 \cdot 10^{69}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))