Average Error: 33.7 → 10.9
Time: 7.8s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.98276540088900058 \cdot 10^{134}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r13660 = b_2;
        double r13661 = -r13660;
        double r13662 = r13660 * r13660;
        double r13663 = a;
        double r13664 = c;
        double r13665 = r13663 * r13664;
        double r13666 = r13662 - r13665;
        double r13667 = sqrt(r13666);
        double r13668 = r13661 + r13667;
        double r13669 = r13668 / r13663;
        return r13669;
}
↓
double f(double a, double b_2, double c) {
        double r13670 = b_2;
        double r13671 = -1.9827654008890006e+134;
        bool r13672 = r13670 <= r13671;
        double r13673 = 0.5;
        double r13674 = c;
        double r13675 = r13674 / r13670;
        double r13676 = r13673 * r13675;
        double r13677 = 2.0;
        double r13678 = a;
        double r13679 = r13670 / r13678;
        double r13680 = r13677 * r13679;
        double r13681 = r13676 - r13680;
        double r13682 = 1.1860189201379418e-161;
        bool r13683 = r13670 <= r13682;
        double r13684 = -r13670;
        double r13685 = r13670 * r13670;
        double r13686 = r13678 * r13674;
        double r13687 = r13685 - r13686;
        double r13688 = sqrt(r13687);
        double r13689 = r13684 + r13688;
        double r13690 = 1.0;
        double r13691 = r13690 / r13678;
        double r13692 = r13689 * r13691;
        double r13693 = -0.5;
        double r13694 = r13693 * r13675;
        double r13695 = r13683 ? r13692 : r13694;
        double r13696 = r13672 ? r13681 : r13695;
        return r13696;
}

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 < -1.9827654008890006e+134

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 3.1

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

    if -1.9827654008890006e+134 < b_2 < 1.1860189201379418e-161

    1. Initial program 10.3

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv10.4

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

    if 1.1860189201379418e-161 < b_2

    1. Initial program 49.6

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

    2. Taylor expanded around inf 13.6

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))