Average Error: 33.8 → 10.0
Time: 15.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.0629549698067062 \cdot 10^{105}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 4.42526676714982627 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.0629549698067062 \cdot 10^{105}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 4.42526676714982627 \cdot 10^{-76}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r23931 = b_2;
        double r23932 = -r23931;
        double r23933 = r23931 * r23931;
        double r23934 = a;
        double r23935 = c;
        double r23936 = r23934 * r23935;
        double r23937 = r23933 - r23936;
        double r23938 = sqrt(r23937);
        double r23939 = r23932 + r23938;
        double r23940 = r23939 / r23934;
        return r23940;
}
↓
double f(double a, double b_2, double c) {
        double r23941 = b_2;
        double r23942 = -5.062954969806706e+105;
        bool r23943 = r23941 <= r23942;
        double r23944 = 0.5;
        double r23945 = c;
        double r23946 = r23945 / r23941;
        double r23947 = r23944 * r23946;
        double r23948 = a;
        double r23949 = r23941 / r23948;
        double r23950 = r23947 - r23949;
        double r23951 = r23950 - r23949;
        double r23952 = 4.425266767149826e-76;
        bool r23953 = r23941 <= r23952;
        double r23954 = r23941 * r23941;
        double r23955 = r23948 * r23945;
        double r23956 = r23954 - r23955;
        double r23957 = sqrt(r23956);
        double r23958 = r23957 / r23948;
        double r23959 = r23958 - r23949;
        double r23960 = -0.5;
        double r23961 = r23960 * r23946;
        double r23962 = r23953 ? r23959 : r23961;
        double r23963 = r23943 ? r23951 : r23962;
        return r23963;
}

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 < -5.062954969806706e+105

    1. Initial program 49.1

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

    2. Simplified49.1

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

    4. Applied div-sub49.1

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

    5. Taylor expanded around -inf 2.8

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

    if -5.062954969806706e+105 < b_2 < 4.425266767149826e-76

    1. Initial program 12.9

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

    2. Simplified12.9

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

    4. Applied div-sub12.9

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

    if 4.425266767149826e-76 < b_2

    1. Initial program 53.0

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

    2. Simplified53.0

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.0629549698067062 \cdot 10^{105}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 4.42526676714982627 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))