Average Error: 33.6 → 10.4
Time: 16.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2137335 = b;
        double r2137336 = -r2137335;
        double r2137337 = r2137335 * r2137335;
        double r2137338 = 4.0;
        double r2137339 = a;
        double r2137340 = r2137338 * r2137339;
        double r2137341 = c;
        double r2137342 = r2137340 * r2137341;
        double r2137343 = r2137337 - r2137342;
        double r2137344 = sqrt(r2137343);
        double r2137345 = r2137336 + r2137344;
        double r2137346 = 2.0;
        double r2137347 = r2137346 * r2137339;
        double r2137348 = r2137345 / r2137347;
        return r2137348;
}
↓
double f(double a, double b, double c) {
        double r2137349 = b;
        double r2137350 = -2.1144981103869975e+131;
        bool r2137351 = r2137349 <= r2137350;
        double r2137352 = c;
        double r2137353 = r2137352 / r2137349;
        double r2137354 = a;
        double r2137355 = r2137349 / r2137354;
        double r2137356 = r2137353 - r2137355;
        double r2137357 = 4.5810084990875205e-68;
        bool r2137358 = r2137349 <= r2137357;
        double r2137359 = 1.0;
        double r2137360 = 2.0;
        double r2137361 = r2137354 * r2137360;
        double r2137362 = -4.0;
        double r2137363 = r2137352 * r2137362;
        double r2137364 = r2137363 * r2137354;
        double r2137365 = fma(r2137349, r2137349, r2137364);
        double r2137366 = sqrt(r2137365);
        double r2137367 = r2137366 - r2137349;
        double r2137368 = r2137361 / r2137367;
        double r2137369 = r2137359 / r2137368;
        double r2137370 = -r2137353;
        double r2137371 = r2137358 ? r2137369 : r2137370;
        double r2137372 = r2137351 ? r2137356 : r2137371;
        return r2137372;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.1144981103869975e+131

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 2.6

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

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

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

    3. Applied clear-num13.4

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

    4. Simplified13.4

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

    if 4.5810084990875205e-68 < b

    1. Initial program 52.0

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))