Average Error: 19.5 → 8.6
Time: 39.7s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.5754430728629 \cdot 10^{+59}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 4.1432508585428765 \cdot 10^{+166}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot c}{b} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 -4.5754430728629 \cdot 10^{+59}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 4.1432508585428765 \cdot 10^{+166}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} + \left(-b\right)}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot c}{b} \cdot 2\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1892386 = b;
        double r1892387 = 0.0;
        bool r1892388 = r1892386 >= r1892387;
        double r1892389 = 2.0;
        double r1892390 = c;
        double r1892391 = r1892389 * r1892390;
        double r1892392 = -r1892386;
        double r1892393 = r1892386 * r1892386;
        double r1892394 = 4.0;
        double r1892395 = a;
        double r1892396 = r1892394 * r1892395;
        double r1892397 = r1892396 * r1892390;
        double r1892398 = r1892393 - r1892397;
        double r1892399 = sqrt(r1892398);
        double r1892400 = r1892392 - r1892399;
        double r1892401 = r1892391 / r1892400;
        double r1892402 = r1892392 + r1892399;
        double r1892403 = r1892389 * r1892395;
        double r1892404 = r1892402 / r1892403;
        double r1892405 = r1892388 ? r1892401 : r1892404;
        return r1892405;
}
↓
double f(double a, double b, double c) {
        double r1892406 = b;
        double r1892407 = -4.5754430728629e+59;
        bool r1892408 = r1892406 <= r1892407;
        double r1892409 = 0.0;
        bool r1892410 = r1892406 >= r1892409;
        double r1892411 = 2.0;
        double r1892412 = c;
        double r1892413 = r1892411 * r1892412;
        double r1892414 = -r1892406;
        double r1892415 = r1892406 * r1892406;
        double r1892416 = 4.0;
        double r1892417 = a;
        double r1892418 = r1892416 * r1892417;
        double r1892419 = r1892418 * r1892412;
        double r1892420 = r1892415 - r1892419;
        double r1892421 = sqrt(r1892420);
        double r1892422 = r1892414 - r1892421;
        double r1892423 = r1892413 / r1892422;
        double r1892424 = r1892412 / r1892406;
        double r1892425 = r1892417 * r1892424;
        double r1892426 = r1892425 - r1892406;
        double r1892427 = r1892411 * r1892426;
        double r1892428 = r1892411 * r1892417;
        double r1892429 = r1892427 / r1892428;
        double r1892430 = r1892410 ? r1892423 : r1892429;
        double r1892431 = 4.1432508585428765e+166;
        bool r1892432 = r1892406 <= r1892431;
        double r1892433 = cbrt(r1892420);
        double r1892434 = r1892433 * r1892433;
        double r1892435 = r1892433 * r1892434;
        double r1892436 = sqrt(r1892435);
        double r1892437 = r1892436 + r1892414;
        double r1892438 = r1892437 / r1892428;
        double r1892439 = r1892410 ? r1892423 : r1892438;
        double r1892440 = r1892417 * r1892412;
        double r1892441 = r1892440 / r1892406;
        double r1892442 = r1892441 * r1892411;
        double r1892443 = r1892406 - r1892442;
        double r1892444 = r1892414 - r1892443;
        double r1892445 = r1892413 / r1892444;
        double r1892446 = r1892421 + r1892414;
        double r1892447 = r1892446 / r1892428;
        double r1892448 = r1892410 ? r1892445 : r1892447;
        double r1892449 = r1892432 ? r1892439 : r1892448;
        double r1892450 = r1892408 ? r1892430 : r1892449;
        return r1892450;
}

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 < -4.5754430728629e+59

    1. Initial program 38.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt38.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Applied sqrt-prod38.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    5. Taylor expanded around -inf 11.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    6. Simplified5.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \frac{c}{b} - b\right) \cdot 2}{2 \cdot a}\\ \end{array}\]

    if -4.5754430728629e+59 < b < 4.1432508585428765e+166

    1. Initial program 9.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt9.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    if 4.1432508585428765e+166 < b

    1. Initial program 37.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 7.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.5754430728629 \cdot 10^{+59}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 4.1432508585428765 \cdot 10^{+166}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot c}{b} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))