Average Error: 43.8 → 11.3
Time: 19.4s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 4.197456508925139 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 4.197456508925139 \cdot 10^{-05}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r970722 = b;
        double r970723 = -r970722;
        double r970724 = r970722 * r970722;
        double r970725 = 4.0;
        double r970726 = a;
        double r970727 = r970725 * r970726;
        double r970728 = c;
        double r970729 = r970727 * r970728;
        double r970730 = r970724 - r970729;
        double r970731 = sqrt(r970730);
        double r970732 = r970723 + r970731;
        double r970733 = 2.0;
        double r970734 = r970733 * r970726;
        double r970735 = r970732 / r970734;
        return r970735;
}
↓
double f(double a, double b, double c) {
        double r970736 = b;
        double r970737 = 4.197456508925139e-05;
        bool r970738 = r970736 <= r970737;
        double r970739 = a;
        double r970740 = c;
        double r970741 = r970739 * r970740;
        double r970742 = -4.0;
        double r970743 = r970736 * r970736;
        double r970744 = fma(r970741, r970742, r970743);
        double r970745 = sqrt(r970744);
        double r970746 = r970745 * r970744;
        double r970747 = r970743 * r970736;
        double r970748 = r970746 - r970747;
        double r970749 = r970743 + r970744;
        double r970750 = fma(r970736, r970745, r970749);
        double r970751 = r970748 / r970750;
        double r970752 = r970751 / r970739;
        double r970753 = 2.0;
        double r970754 = r970752 / r970753;
        double r970755 = -2.0;
        double r970756 = r970740 / r970736;
        double r970757 = r970755 * r970756;
        double r970758 = r970757 / r970753;
        double r970759 = r970738 ? r970754 : r970758;
        return r970759;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 4.197456508925139e-05

    1. Initial program 18.7

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

    2. Simplified18.7

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

    4. Applied flip3--18.8

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

    5. Simplified18.1

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

    6. Simplified18.1

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

    if 4.197456508925139e-05 < b

    1. Initial program 45.3

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

    2. Simplified45.3

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

    3. Taylor expanded around inf 10.9

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.197456508925139 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))