Average Error: 34.0 → 10.4
Time: 21.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.541338025369698 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.1094847447691107 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.541338025369698 \cdot 10^{+80}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.1094847447691107 \cdot 10^{-113}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1328937 = b;
        double r1328938 = -r1328937;
        double r1328939 = r1328937 * r1328937;
        double r1328940 = 4.0;
        double r1328941 = a;
        double r1328942 = r1328940 * r1328941;
        double r1328943 = c;
        double r1328944 = r1328942 * r1328943;
        double r1328945 = r1328939 - r1328944;
        double r1328946 = sqrt(r1328945);
        double r1328947 = r1328938 + r1328946;
        double r1328948 = 2.0;
        double r1328949 = r1328948 * r1328941;
        double r1328950 = r1328947 / r1328949;
        return r1328950;
}
↓
double f(double a, double b, double c) {
        double r1328951 = b;
        double r1328952 = -2.541338025369698e+80;
        bool r1328953 = r1328951 <= r1328952;
        double r1328954 = c;
        double r1328955 = r1328954 / r1328951;
        double r1328956 = a;
        double r1328957 = r1328951 / r1328956;
        double r1328958 = r1328955 - r1328957;
        double r1328959 = 2.0;
        double r1328960 = r1328958 * r1328959;
        double r1328961 = r1328960 / r1328959;
        double r1328962 = 1.1094847447691107e-113;
        bool r1328963 = r1328951 <= r1328962;
        double r1328964 = 1.0;
        double r1328965 = -4.0;
        double r1328966 = r1328965 * r1328956;
        double r1328967 = r1328966 * r1328954;
        double r1328968 = fma(r1328951, r1328951, r1328967);
        double r1328969 = sqrt(r1328968);
        double r1328970 = r1328969 - r1328951;
        double r1328971 = r1328956 / r1328970;
        double r1328972 = r1328964 / r1328971;
        double r1328973 = r1328972 / r1328959;
        double r1328974 = -2.0;
        double r1328975 = r1328974 * r1328955;
        double r1328976 = r1328975 / r1328959;
        double r1328977 = r1328963 ? r1328973 : r1328976;
        double r1328978 = r1328953 ? r1328961 : r1328977;
        return r1328978;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.541338025369698e+80

    1. Initial program 41.0

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

    2. Simplified41.0

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

    3. Taylor expanded around -inf 4.2

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified4.2

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

    if -2.541338025369698e+80 < b < 1.1094847447691107e-113

    1. Initial program 12.5

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

    2. Simplified12.5

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

    4. Applied clear-num12.7

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}}}{2}\]

    if 1.1094847447691107e-113 < b

    1. Initial program 51.6

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

    2. Simplified51.6

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

    3. Taylor expanded around inf 10.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.541338025369698 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.1094847447691107 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))