Average Error: 33.6 → 10.2
Time: 15.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\
\;\;\;\;\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 r22675 = b_2;
        double r22676 = -r22675;
        double r22677 = r22675 * r22675;
        double r22678 = a;
        double r22679 = c;
        double r22680 = r22678 * r22679;
        double r22681 = r22677 - r22680;
        double r22682 = sqrt(r22681);
        double r22683 = r22676 + r22682;
        double r22684 = r22683 / r22678;
        return r22684;
}
↓
double f(double a, double b_2, double c) {
        double r22685 = b_2;
        double r22686 = -1.2609617020890706e+118;
        bool r22687 = r22685 <= r22686;
        double r22688 = 0.5;
        double r22689 = c;
        double r22690 = r22689 / r22685;
        double r22691 = r22688 * r22690;
        double r22692 = 2.0;
        double r22693 = a;
        double r22694 = r22685 / r22693;
        double r22695 = r22692 * r22694;
        double r22696 = r22691 - r22695;
        double r22697 = 5.81843322574321e-115;
        bool r22698 = r22685 <= r22697;
        double r22699 = -r22685;
        double r22700 = r22685 * r22685;
        double r22701 = r22693 * r22689;
        double r22702 = r22700 - r22701;
        double r22703 = sqrt(r22702);
        double r22704 = r22699 + r22703;
        double r22705 = r22704 / r22693;
        double r22706 = -0.5;
        double r22707 = r22706 * r22690;
        double r22708 = r22698 ? r22705 : r22707;
        double r22709 = r22687 ? r22696 : r22708;
        return r22709;
}

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 < -1.2609617020890706e+118

    1. Initial program 51.6

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

    2. Taylor expanded around -inf 2.7

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

    if -1.2609617020890706e+118 < b_2 < 5.81843322574321e-115

    1. Initial program 11.5

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

    if 5.81843322574321e-115 < b_2

    1. Initial program 51.3

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

    2. Taylor expanded around inf 11.3

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))