Average Error: 34.4 → 9.1
Time: 5.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0461303908572575 \cdot 10^{65}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.47093887587755467 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 1026034526.7184467:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.0461303908572575 \cdot 10^{65}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.47093887587755467 \cdot 10^{-146}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\mathbf{elif}\;b \le 1026034526.7184467:\\
\;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r55450 = b;
        double r55451 = -r55450;
        double r55452 = r55450 * r55450;
        double r55453 = 4.0;
        double r55454 = a;
        double r55455 = r55453 * r55454;
        double r55456 = c;
        double r55457 = r55455 * r55456;
        double r55458 = r55452 - r55457;
        double r55459 = sqrt(r55458);
        double r55460 = r55451 + r55459;
        double r55461 = 2.0;
        double r55462 = r55461 * r55454;
        double r55463 = r55460 / r55462;
        return r55463;
}
↓
double f(double a, double b, double c) {
        double r55464 = b;
        double r55465 = -1.0461303908572575e+65;
        bool r55466 = r55464 <= r55465;
        double r55467 = 1.0;
        double r55468 = c;
        double r55469 = r55468 / r55464;
        double r55470 = a;
        double r55471 = r55464 / r55470;
        double r55472 = r55469 - r55471;
        double r55473 = r55467 * r55472;
        double r55474 = 3.4709388758775547e-146;
        bool r55475 = r55464 <= r55474;
        double r55476 = 1.0;
        double r55477 = 2.0;
        double r55478 = r55477 * r55470;
        double r55479 = -r55464;
        double r55480 = r55464 * r55464;
        double r55481 = 4.0;
        double r55482 = r55481 * r55470;
        double r55483 = r55482 * r55468;
        double r55484 = r55480 - r55483;
        double r55485 = sqrt(r55484);
        double r55486 = r55479 + r55485;
        double r55487 = r55478 / r55486;
        double r55488 = r55476 / r55487;
        double r55489 = 1026034526.7184467;
        bool r55490 = r55464 <= r55489;
        double r55491 = 0.0;
        double r55492 = r55470 * r55468;
        double r55493 = r55481 * r55492;
        double r55494 = r55491 + r55493;
        double r55495 = r55479 - r55485;
        double r55496 = r55494 / r55495;
        double r55497 = r55496 / r55478;
        double r55498 = -1.0;
        double r55499 = r55498 * r55469;
        double r55500 = r55490 ? r55497 : r55499;
        double r55501 = r55475 ? r55488 : r55500;
        double r55502 = r55466 ? r55473 : r55501;
        return r55502;
}

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 4 regimes

  2. if b < -1.0461303908572575e+65

    1. Initial program 41.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around -inf 4.6

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

    3. Simplified4.6

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

    if -1.0461303908572575e+65 < b < 3.4709388758775547e-146

    1. Initial program 12.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied clear-num12.3

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\]

    if 3.4709388758775547e-146 < b < 1026034526.7184467

    1. Initial program 33.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+33.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    4. Simplified17.5

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]

    if 1026034526.7184467 < b

    1. Initial program 56.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around inf 4.8

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0461303908572575 \cdot 10^{65}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.47093887587755467 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 1026034526.7184467:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))