Average Error: 34.6 → 10.2
Time: 41.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.744031351412432972171902712116585209201 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.744031351412432972171902712116585209201 \cdot 10^{-142}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r845945 = b_2;
        double r845946 = -r845945;
        double r845947 = r845945 * r845945;
        double r845948 = a;
        double r845949 = c;
        double r845950 = r845948 * r845949;
        double r845951 = r845947 - r845950;
        double r845952 = sqrt(r845951);
        double r845953 = r845946 + r845952;
        double r845954 = r845953 / r845948;
        return r845954;
}
↓
double f(double a, double b_2, double c) {
        double r845955 = b_2;
        double r845956 = -1.2705286994550075e+152;
        bool r845957 = r845955 <= r845956;
        double r845958 = 0.5;
        double r845959 = c;
        double r845960 = r845959 / r845955;
        double r845961 = r845958 * r845960;
        double r845962 = a;
        double r845963 = r845955 / r845962;
        double r845964 = 2.0;
        double r845965 = r845963 * r845964;
        double r845966 = r845961 - r845965;
        double r845967 = 1.744031351412433e-142;
        bool r845968 = r845955 <= r845967;
        double r845969 = r845955 * r845955;
        double r845970 = r845959 * r845962;
        double r845971 = r845969 - r845970;
        double r845972 = sqrt(r845971);
        double r845973 = r845972 - r845955;
        double r845974 = r845973 / r845962;
        double r845975 = -0.5;
        double r845976 = r845960 * r845975;
        double r845977 = r845968 ? r845974 : r845976;
        double r845978 = r845957 ? r845966 : r845977;
        return r845978;
}

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 < -1.2705286994550075e+152

    1. Initial program 62.9

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

    2. Simplified62.9

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

    3. Taylor expanded around -inf 1.6

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

    if -1.2705286994550075e+152 < b_2 < 1.744031351412433e-142

    1. Initial program 10.4

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

    2. Simplified10.4

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

    4. Applied div-inv10.5

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]
    5. Using strategy rm

    6. Applied un-div-inv10.4

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

    if 1.744031351412433e-142 < b_2

    1. Initial program 50.7

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

    2. Simplified50.7

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

    3. Taylor expanded around inf 12.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.744031351412432972171902712116585209201 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))