Average Error: 19.4 → 13.4
Time: 21.6s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le 2.0410715251838527 \cdot 10^{+49}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r976901 = b;
        double r976902 = 0.0;
        bool r976903 = r976901 >= r976902;
        double r976904 = 2.0;
        double r976905 = c;
        double r976906 = r976904 * r976905;
        double r976907 = -r976901;
        double r976908 = r976901 * r976901;
        double r976909 = 4.0;
        double r976910 = a;
        double r976911 = r976909 * r976910;
        double r976912 = r976911 * r976905;
        double r976913 = r976908 - r976912;
        double r976914 = sqrt(r976913);
        double r976915 = r976907 - r976914;
        double r976916 = r976906 / r976915;
        double r976917 = r976907 + r976914;
        double r976918 = r976904 * r976910;
        double r976919 = r976917 / r976918;
        double r976920 = r976903 ? r976916 : r976919;
        return r976920;
}
↓
double f(double a, double b, double c) {
        double r976921 = b;
        double r976922 = 2.0410715251838527e+49;
        bool r976923 = r976921 <= r976922;
        double r976924 = 0.0;
        bool r976925 = r976921 >= r976924;
        double r976926 = 2.0;
        double r976927 = c;
        double r976928 = r976926 * r976927;
        double r976929 = -r976921;
        double r976930 = -4.0;
        double r976931 = a;
        double r976932 = r976931 * r976927;
        double r976933 = r976921 * r976921;
        double r976934 = fma(r976930, r976932, r976933);
        double r976935 = sqrt(r976934);
        double r976936 = sqrt(r976935);
        double r976937 = r976936 * r976936;
        double r976938 = r976929 - r976937;
        double r976939 = r976928 / r976938;
        double r976940 = r976935 - r976921;
        double r976941 = r976940 / r976926;
        double r976942 = r976941 / r976931;
        double r976943 = r976925 ? r976939 : r976942;
        double r976944 = r976931 / r976921;
        double r976945 = r976944 * r976927;
        double r976946 = r976945 - r976921;
        double r976947 = r976926 * r976946;
        double r976948 = r976928 / r976947;
        double r976949 = r976925 ? r976948 : r976942;
        double r976950 = r976923 ? r976943 : r976949;
        return r976950;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2.0410715251838527e+49

    1. Initial program 16.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified16.8

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt16.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    5. Applied sqrt-prod16.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    if 2.0410715251838527e+49 < b

    1. Initial program 25.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified25.8

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt25.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    5. Applied sqrt-prod25.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    6. Taylor expanded around inf 7.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    7. Simplified4.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))