Average Error: 43.9 → 11.3
Time: 7.9s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.114113106997123639:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 0.114113106997123639:\\
\;\;\;\;\frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r21866 = b;
        double r21867 = -r21866;
        double r21868 = r21866 * r21866;
        double r21869 = 4.0;
        double r21870 = a;
        double r21871 = r21869 * r21870;
        double r21872 = c;
        double r21873 = r21871 * r21872;
        double r21874 = r21868 - r21873;
        double r21875 = sqrt(r21874);
        double r21876 = r21867 + r21875;
        double r21877 = 2.0;
        double r21878 = r21877 * r21870;
        double r21879 = r21876 / r21878;
        return r21879;
}
↓
double f(double a, double b, double c) {
        double r21880 = b;
        double r21881 = 0.11411310699712364;
        bool r21882 = r21880 <= r21881;
        double r21883 = r21880 * r21880;
        double r21884 = 4.0;
        double r21885 = a;
        double r21886 = r21884 * r21885;
        double r21887 = c;
        double r21888 = r21886 * r21887;
        double r21889 = fma(r21880, r21880, r21888);
        double r21890 = r21883 - r21889;
        double r21891 = r21883 - r21888;
        double r21892 = sqrt(r21891);
        double r21893 = r21892 + r21880;
        double r21894 = r21890 / r21893;
        double r21895 = 2.0;
        double r21896 = r21894 / r21895;
        double r21897 = r21896 / r21885;
        double r21898 = -1.0;
        double r21899 = r21887 / r21880;
        double r21900 = r21898 * r21899;
        double r21901 = r21882 ? r21897 : r21900;
        return r21901;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.11411310699712364

    1. Initial program 23.2

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

    2. Simplified23.2

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

    4. Applied flip--23.3

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

    5. Simplified22.6

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

    if 0.11411310699712364 < b

    1. Initial program 46.9

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

    2. Simplified46.9

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

    3. Taylor expanded around inf 9.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.114113106997123639:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))