Average Error: 18.9 → 12.6
Time: 22.9s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 3.8085667890961203 \cdot 10^{+87}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le 3.8085667890961203 \cdot 10^{+87}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\

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

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r699948 = b;
        double r699949 = 0.0;
        bool r699950 = r699948 >= r699949;
        double r699951 = -r699948;
        double r699952 = r699948 * r699948;
        double r699953 = 4.0;
        double r699954 = a;
        double r699955 = r699953 * r699954;
        double r699956 = c;
        double r699957 = r699955 * r699956;
        double r699958 = r699952 - r699957;
        double r699959 = sqrt(r699958);
        double r699960 = r699951 - r699959;
        double r699961 = 2.0;
        double r699962 = r699961 * r699954;
        double r699963 = r699960 / r699962;
        double r699964 = r699961 * r699956;
        double r699965 = r699951 + r699959;
        double r699966 = r699964 / r699965;
        double r699967 = r699950 ? r699963 : r699966;
        return r699967;
}
↓
double f(double a, double b, double c) {
        double r699968 = b;
        double r699969 = 3.8085667890961203e+87;
        bool r699970 = r699968 <= r699969;
        double r699971 = 0.0;
        bool r699972 = r699968 >= r699971;
        double r699973 = -r699968;
        double r699974 = a;
        double r699975 = -4.0;
        double r699976 = r699974 * r699975;
        double r699977 = c;
        double r699978 = r699968 * r699968;
        double r699979 = fma(r699976, r699977, r699978);
        double r699980 = sqrt(r699979);
        double r699981 = r699973 - r699980;
        double r699982 = 2.0;
        double r699983 = r699982 * r699974;
        double r699984 = r699981 / r699983;
        double r699985 = r699982 * r699977;
        double r699986 = r699980 - r699968;
        double r699987 = sqrt(r699986);
        double r699988 = r699987 * r699987;
        double r699989 = r699985 / r699988;
        double r699990 = r699972 ? r699984 : r699989;
        double r699991 = r699973 - r699968;
        double r699992 = r699991 / r699983;
        double r699993 = r699985 / r699986;
        double r699994 = r699972 ? r699992 : r699993;
        double r699995 = r699970 ? r699990 : r699994;
        return r699995;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 3.8085667890961203e+87

    1. Initial program 14.4

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

    2. Simplified14.4

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

    4. Applied add-sqr-sqrt14.5

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

    if 3.8085667890961203e+87 < b

    1. Initial program 41.3

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

    2. Simplified41.2

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

    3. Taylor expanded around 0 3.3

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

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.8085667890961203 \cdot 10^{+87}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))