Average Error: 34.5 → 9.7
Time: 4.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.9358923729233266 \cdot 10^{149}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 9.39036747108992214 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.9358923729233266 \cdot 10^{149}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 9.39036747108992214 \cdot 10^{-69}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r13718 = b_2;
        double r13719 = -r13718;
        double r13720 = r13718 * r13718;
        double r13721 = a;
        double r13722 = c;
        double r13723 = r13721 * r13722;
        double r13724 = r13720 - r13723;
        double r13725 = sqrt(r13724);
        double r13726 = r13719 + r13725;
        double r13727 = r13726 / r13721;
        return r13727;
}
↓
double f(double a, double b_2, double c) {
        double r13728 = b_2;
        double r13729 = -2.9358923729233266e+149;
        bool r13730 = r13728 <= r13729;
        double r13731 = 0.5;
        double r13732 = c;
        double r13733 = r13732 / r13728;
        double r13734 = r13731 * r13733;
        double r13735 = 2.0;
        double r13736 = a;
        double r13737 = r13728 / r13736;
        double r13738 = r13735 * r13737;
        double r13739 = r13734 - r13738;
        double r13740 = 9.390367471089922e-69;
        bool r13741 = r13728 <= r13740;
        double r13742 = -r13728;
        double r13743 = r13728 * r13728;
        double r13744 = r13736 * r13732;
        double r13745 = r13743 - r13744;
        double r13746 = sqrt(r13745);
        double r13747 = r13742 + r13746;
        double r13748 = r13747 / r13736;
        double r13749 = -0.5;
        double r13750 = r13749 * r13733;
        double r13751 = r13741 ? r13748 : r13750;
        double r13752 = r13730 ? r13739 : r13751;
        return r13752;
}

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 < -2.9358923729233266e+149

    1. Initial program 62.1

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

    2. Taylor expanded around -inf 1.7

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

    if -2.9358923729233266e+149 < b_2 < 9.390367471089922e-69

    1. Initial program 12.4

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

    if 9.390367471089922e-69 < b_2

    1. Initial program 53.5

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

    2. Taylor expanded around inf 8.7

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.9358923729233266 \cdot 10^{149}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 9.39036747108992214 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))