Average Error: 19.7 → 6.4
Time: 47.5s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(2.0 \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2.0 \cdot \frac{a}{b}, c, b \cdot -2\right)}{a \cdot 2.0}\\ \end{array}\\ \mathbf{elif}\;b \le 8.119950078133889 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)} + \left(-b\right)}{a \cdot 2.0}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(2.0 \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2.0 \cdot \frac{a \cdot c}{b} - b\right) + \left(-b\right)}{a \cdot 2.0}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(2.0 \cdot c\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2.0 \cdot \frac{a}{b}, c, b \cdot -2\right)}{a \cdot 2.0}\\

\end{array}\\

\mathbf{elif}\;b \le 8.119950078133889 \cdot 10^{+99}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)} + \left(-b\right)}{a \cdot 2.0}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(2.0 \cdot c\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(2.0 \cdot \frac{a \cdot c}{b} - b\right) + \left(-b\right)}{a \cdot 2.0}\\

\end{array}
double f(double a, double b, double c) {
        double r1100943 = b;
        double r1100944 = 0.0;
        bool r1100945 = r1100943 >= r1100944;
        double r1100946 = 2.0;
        double r1100947 = c;
        double r1100948 = r1100946 * r1100947;
        double r1100949 = -r1100943;
        double r1100950 = r1100943 * r1100943;
        double r1100951 = 4.0;
        double r1100952 = a;
        double r1100953 = r1100951 * r1100952;
        double r1100954 = r1100953 * r1100947;
        double r1100955 = r1100950 - r1100954;
        double r1100956 = sqrt(r1100955);
        double r1100957 = r1100949 - r1100956;
        double r1100958 = r1100948 / r1100957;
        double r1100959 = r1100949 + r1100956;
        double r1100960 = r1100946 * r1100952;
        double r1100961 = r1100959 / r1100960;
        double r1100962 = r1100945 ? r1100958 : r1100961;
        return r1100962;
}
↓
double f(double a, double b, double c) {
        double r1100963 = b;
        double r1100964 = -2.0519886074223697e+155;
        bool r1100965 = r1100963 <= r1100964;
        double r1100966 = 0.0;
        bool r1100967 = r1100963 >= r1100966;
        double r1100968 = 2.0;
        double r1100969 = c;
        double r1100970 = r1100968 * r1100969;
        double r1100971 = -2.0;
        double r1100972 = a;
        double r1100973 = r1100972 / r1100963;
        double r1100974 = r1100973 * r1100970;
        double r1100975 = fma(r1100963, r1100971, r1100974);
        double r1100976 = r1100970 / r1100975;
        double r1100977 = r1100968 * r1100973;
        double r1100978 = r1100963 * r1100971;
        double r1100979 = fma(r1100977, r1100969, r1100978);
        double r1100980 = r1100972 * r1100968;
        double r1100981 = r1100979 / r1100980;
        double r1100982 = r1100967 ? r1100976 : r1100981;
        double r1100983 = 8.119950078133889e+99;
        bool r1100984 = r1100963 <= r1100983;
        double r1100985 = -r1100963;
        double r1100986 = r1100963 * r1100963;
        double r1100987 = 4.0;
        double r1100988 = r1100987 * r1100972;
        double r1100989 = r1100969 * r1100988;
        double r1100990 = r1100986 - r1100989;
        double r1100991 = sqrt(r1100990);
        double r1100992 = sqrt(r1100991);
        double r1100993 = r1100992 * r1100992;
        double r1100994 = r1100985 - r1100993;
        double r1100995 = r1100970 / r1100994;
        double r1100996 = r1100991 + r1100985;
        double r1100997 = r1100996 / r1100980;
        double r1100998 = r1100967 ? r1100995 : r1100997;
        double r1100999 = r1100972 * r1100969;
        double r1101000 = r1100999 / r1100963;
        double r1101001 = r1100968 * r1101000;
        double r1101002 = r1101001 - r1100963;
        double r1101003 = r1101002 + r1100985;
        double r1101004 = r1101003 / r1100980;
        double r1101005 = r1100967 ? r1100976 : r1101004;
        double r1101006 = r1100984 ? r1100998 : r1101005;
        double r1101007 = r1100965 ? r1100982 : r1101006;
        return r1101007;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.0519886074223697e+155

    1. Initial program 64.0

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 64.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\color{blue}{2.0 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]

    3. Simplified64.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\color{blue}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(c \cdot 2.0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]

    4. Taylor expanded around -inf 11.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(c \cdot 2.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2.0 \cdot a}\\ \end{array}\]

    5. Simplified2.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(c \cdot 2.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b} \cdot 2.0, c, b \cdot -2\right)}{2.0 \cdot a}\\ \end{array}\]

    if -2.0519886074223697e+155 < b < 8.119950078133889e+99

    1. Initial program 8.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]

    4. Applied sqrt-prod8.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]

    if 8.119950078133889e+99 < b

    1. Initial program 29.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 6.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\color{blue}{2.0 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]

    3. Simplified2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\color{blue}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(c \cdot 2.0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]

    4. Taylor expanded around -inf 2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(c \cdot 2.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2.0 \cdot \frac{a \cdot c}{b} - b\right)}{2.0 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(2.0 \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2.0 \cdot \frac{a}{b}, c, b \cdot -2\right)}{a \cdot 2.0}\\ \end{array}\\ \mathbf{elif}\;b \le 8.119950078133889 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4.0 \cdot a\right)} + \left(-b\right)}{a \cdot 2.0}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\mathsf{fma}\left(b, -2, \frac{a}{b} \cdot \left(2.0 \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2.0 \cdot \frac{a \cdot c}{b} - b\right) + \left(-b\right)}{a \cdot 2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))