Average Error: 32.9 → 10.3
Time: 14.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.711132829713123 \cdot 10^{+85}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 3.5014024016497154 \cdot 10^{-97}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.711132829713123 \cdot 10^{+85}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 3.5014024016497154 \cdot 10^{-97}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r336928 = b_2;
        double r336929 = -r336928;
        double r336930 = r336928 * r336928;
        double r336931 = a;
        double r336932 = c;
        double r336933 = r336931 * r336932;
        double r336934 = r336930 - r336933;
        double r336935 = sqrt(r336934);
        double r336936 = r336929 + r336935;
        double r336937 = r336936 / r336931;
        return r336937;
}
↓
double f(double a, double b_2, double c) {
        double r336938 = b_2;
        double r336939 = -9.711132829713123e+85;
        bool r336940 = r336938 <= r336939;
        double r336941 = 0.5;
        double r336942 = c;
        double r336943 = r336942 / r336938;
        double r336944 = r336941 * r336943;
        double r336945 = a;
        double r336946 = r336938 / r336945;
        double r336947 = 2.0;
        double r336948 = r336946 * r336947;
        double r336949 = r336944 - r336948;
        double r336950 = 3.5014024016497154e-97;
        bool r336951 = r336938 <= r336950;
        double r336952 = 1.0;
        double r336953 = r336938 * r336938;
        double r336954 = r336942 * r336945;
        double r336955 = r336953 - r336954;
        double r336956 = sqrt(r336955);
        double r336957 = r336956 - r336938;
        double r336958 = r336945 / r336957;
        double r336959 = r336952 / r336958;
        double r336960 = -0.5;
        double r336961 = r336960 * r336943;
        double r336962 = r336951 ? r336959 : r336961;
        double r336963 = r336940 ? r336949 : r336962;
        return r336963;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -9.711132829713123e+85

    1. Initial program 40.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified40.7

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf 3.9

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    if -9.711132829713123e+85 < b_2 < 3.5014024016497154e-97

    1. Initial program 12.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified12.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied clear-num12.4

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]

    if 3.5014024016497154e-97 < b_2

    1. Initial program 51.6

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified51.6

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 10.6

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.711132829713123 \cdot 10^{+85}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 3.5014024016497154 \cdot 10^{-97}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))