Average Error: 34.2 → 9.5
Time: 19.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{c}{b} \cdot 2\right)}{2}\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{c}{b} \cdot 2\right)}{2}\\

\mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a} - \frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1273610 = b;
        double r1273611 = -r1273610;
        double r1273612 = r1273610 * r1273610;
        double r1273613 = 4.0;
        double r1273614 = a;
        double r1273615 = r1273613 * r1273614;
        double r1273616 = c;
        double r1273617 = r1273615 * r1273616;
        double r1273618 = r1273612 - r1273617;
        double r1273619 = sqrt(r1273618);
        double r1273620 = r1273611 + r1273619;
        double r1273621 = 2.0;
        double r1273622 = r1273621 * r1273614;
        double r1273623 = r1273620 / r1273622;
        return r1273623;
}
↓
double f(double a, double b, double c) {
        double r1273624 = b;
        double r1273625 = -3.7108875578650606e+138;
        bool r1273626 = r1273624 <= r1273625;
        double r1273627 = -2.0;
        double r1273628 = a;
        double r1273629 = r1273624 / r1273628;
        double r1273630 = c;
        double r1273631 = r1273630 / r1273624;
        double r1273632 = 2.0;
        double r1273633 = r1273631 * r1273632;
        double r1273634 = fma(r1273627, r1273629, r1273633);
        double r1273635 = r1273634 / r1273632;
        double r1273636 = 4.626043257219638e-62;
        bool r1273637 = r1273624 <= r1273636;
        double r1273638 = r1273624 * r1273624;
        double r1273639 = 4.0;
        double r1273640 = r1273639 * r1273630;
        double r1273641 = r1273640 * r1273628;
        double r1273642 = r1273638 - r1273641;
        double r1273643 = sqrt(r1273642);
        double r1273644 = r1273643 / r1273628;
        double r1273645 = r1273644 - r1273629;
        double r1273646 = r1273645 / r1273632;
        double r1273647 = -2.0;
        double r1273648 = r1273647 * r1273631;
        double r1273649 = r1273648 / r1273632;
        double r1273650 = r1273637 ? r1273646 : r1273649;
        double r1273651 = r1273626 ? r1273635 : r1273650;
        return r1273651;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -3.7108875578650606e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded around -inf 2.0

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

    4. Simplified2.0

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

    if -3.7108875578650606e+138 < b < 4.626043257219638e-62

    1. Initial program 12.3

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

    2. Simplified12.3

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

    4. Applied div-sub12.3

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

    if 4.626043257219638e-62 < b

    1. Initial program 53.7

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

    2. Simplified53.7

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

    3. Taylor expanded around inf 8.5

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{c}{b} \cdot 2\right)}{2}\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))