Average Error: 32.9 → 10.3
Time: 12.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.088000531423294 \cdot 10^{+152}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 9.354082991670835 \cdot 10^{-125}:\\
\;\;\;\;\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 r267595 = b_2;
        double r267596 = -r267595;
        double r267597 = r267595 * r267595;
        double r267598 = a;
        double r267599 = c;
        double r267600 = r267598 * r267599;
        double r267601 = r267597 - r267600;
        double r267602 = sqrt(r267601);
        double r267603 = r267596 + r267602;
        double r267604 = r267603 / r267598;
        return r267604;
}
↓
double f(double a, double b_2, double c) {
        double r267605 = b_2;
        double r267606 = -9.088000531423294e+152;
        bool r267607 = r267605 <= r267606;
        double r267608 = 0.5;
        double r267609 = c;
        double r267610 = r267609 / r267605;
        double r267611 = r267608 * r267610;
        double r267612 = a;
        double r267613 = r267605 / r267612;
        double r267614 = 2.0;
        double r267615 = r267613 * r267614;
        double r267616 = r267611 - r267615;
        double r267617 = 9.354082991670835e-125;
        bool r267618 = r267605 <= r267617;
        double r267619 = r267605 * r267605;
        double r267620 = r267609 * r267612;
        double r267621 = r267619 - r267620;
        double r267622 = sqrt(r267621);
        double r267623 = r267622 - r267605;
        double r267624 = r267623 / r267612;
        double r267625 = -0.5;
        double r267626 = r267610 * r267625;
        double r267627 = r267618 ? r267624 : r267626;
        double r267628 = r267607 ? r267616 : r267627;
        return r267628;
}

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 < -9.088000531423294e+152

    1. Initial program 60.5

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

    2. Simplified60.5

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

    3. Taylor expanded around -inf 1.5

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

    if -9.088000531423294e+152 < b_2 < 9.354082991670835e-125

    1. Initial program 10.9

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

    2. Simplified10.9

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

    4. Applied div-inv11.0

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

    6. Applied un-div-inv10.9

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

    if 9.354082991670835e-125 < b_2

    1. Initial program 49.8

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

    2. Simplified49.8

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

    3. Taylor expanded around inf 11.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))