Average Error: 33.4 → 9.9
Time: 20.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -1.0027271082217074 \cdot 10^{+110}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r719397 = b;
        double r719398 = -r719397;
        double r719399 = r719397 * r719397;
        double r719400 = 4.0;
        double r719401 = a;
        double r719402 = r719400 * r719401;
        double r719403 = c;
        double r719404 = r719402 * r719403;
        double r719405 = r719399 - r719404;
        double r719406 = sqrt(r719405);
        double r719407 = r719398 + r719406;
        double r719408 = 2.0;
        double r719409 = r719408 * r719401;
        double r719410 = r719407 / r719409;
        return r719410;
}
↓
double f(double a, double b, double c) {
        double r719411 = b;
        double r719412 = -1.0027271082217074e+110;
        bool r719413 = r719411 <= r719412;
        double r719414 = c;
        double r719415 = r719414 / r719411;
        double r719416 = a;
        double r719417 = r719411 / r719416;
        double r719418 = r719415 - r719417;
        double r719419 = 2.0;
        double r719420 = r719418 * r719419;
        double r719421 = r719420 / r719419;
        double r719422 = 2.326372645943808e-74;
        bool r719423 = r719411 <= r719422;
        double r719424 = -4.0;
        double r719425 = r719424 * r719416;
        double r719426 = r719411 * r719411;
        double r719427 = fma(r719414, r719425, r719426);
        double r719428 = sqrt(r719427);
        double r719429 = r719428 - r719411;
        double r719430 = r719429 / r719416;
        double r719431 = r719430 / r719419;
        double r719432 = -2.0;
        double r719433 = r719415 * r719432;
        double r719434 = r719433 / r719419;
        double r719435 = r719423 ? r719431 : r719434;
        double r719436 = r719413 ? r719421 : r719435;
        return r719436;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0027271082217074e+110

    1. Initial program 46.7

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

    2. Simplified46.7

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

    3. Taylor expanded around -inf 3.6

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified3.6

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

    if -1.0027271082217074e+110 < b < 2.326372645943808e-74

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around -inf 12.8

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

    4. Simplified12.8

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

    if 2.326372645943808e-74 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 8.8

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))