Average Error: 33.2 → 10.4
Time: 25.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.571206846913461 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{c}{b} - \frac{b}{a}\right) - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -5.571206846913461 \cdot 10^{+106}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{c}{b} - \frac{b}{a}\right) - \frac{b}{a}}{2}\\

\mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{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 r1248701 = b;
        double r1248702 = -r1248701;
        double r1248703 = r1248701 * r1248701;
        double r1248704 = 4.0;
        double r1248705 = a;
        double r1248706 = r1248704 * r1248705;
        double r1248707 = c;
        double r1248708 = r1248706 * r1248707;
        double r1248709 = r1248703 - r1248708;
        double r1248710 = sqrt(r1248709);
        double r1248711 = r1248702 + r1248710;
        double r1248712 = 2.0;
        double r1248713 = r1248712 * r1248705;
        double r1248714 = r1248711 / r1248713;
        return r1248714;
}
↓
double f(double a, double b, double c) {
        double r1248715 = b;
        double r1248716 = -5.571206846913461e+106;
        bool r1248717 = r1248715 <= r1248716;
        double r1248718 = 2.0;
        double r1248719 = c;
        double r1248720 = r1248719 / r1248715;
        double r1248721 = r1248718 * r1248720;
        double r1248722 = a;
        double r1248723 = r1248715 / r1248722;
        double r1248724 = r1248721 - r1248723;
        double r1248725 = r1248724 - r1248723;
        double r1248726 = r1248725 / r1248718;
        double r1248727 = 3.821014310434392e-21;
        bool r1248728 = r1248715 <= r1248727;
        double r1248729 = -4.0;
        double r1248730 = r1248729 * r1248722;
        double r1248731 = r1248719 * r1248730;
        double r1248732 = fma(r1248715, r1248715, r1248731);
        double r1248733 = sqrt(r1248732);
        double r1248734 = r1248733 / r1248722;
        double r1248735 = r1248734 - r1248723;
        double r1248736 = r1248735 / r1248718;
        double r1248737 = -2.0;
        double r1248738 = r1248737 * r1248720;
        double r1248739 = r1248738 / r1248718;
        double r1248740 = r1248728 ? r1248736 : r1248739;
        double r1248741 = r1248717 ? r1248726 : r1248740;
        return r1248741;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -5.571206846913461e+106

    1. Initial program 46.5

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

    2. Simplified46.5

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

    4. Applied div-sub46.5

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

    5. Taylor expanded around -inf 3.5

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

    if -5.571206846913461e+106 < b < 3.821014310434392e-21

    1. Initial program 14.8

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

    2. Simplified14.8

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

    4. Applied div-sub14.8

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

    if 3.821014310434392e-21 < b

    1. Initial program 54.7

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

    2. Simplified54.7

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

    4. Applied div-sub55.4

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

    5. Taylor expanded around inf 6.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.571206846913461 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{c}{b} - \frac{b}{a}\right) - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))