Average Error: 34.4 → 10.5
Time: 20.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 3.142311858008121469027865121070306475283 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 3.142311858008121469027865121070306475283 \cdot 10^{-35}:\\
\;\;\;\;\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 r672440 = b_2;
        double r672441 = -r672440;
        double r672442 = r672440 * r672440;
        double r672443 = a;
        double r672444 = c;
        double r672445 = r672443 * r672444;
        double r672446 = r672442 - r672445;
        double r672447 = sqrt(r672446);
        double r672448 = r672441 + r672447;
        double r672449 = r672448 / r672443;
        return r672449;
}
↓
double f(double a, double b_2, double c) {
        double r672450 = b_2;
        double r672451 = -2.221067196710922e+149;
        bool r672452 = r672450 <= r672451;
        double r672453 = c;
        double r672454 = r672453 / r672450;
        double r672455 = 2.0;
        double r672456 = r672454 / r672455;
        double r672457 = a;
        double r672458 = r672450 / r672457;
        double r672459 = r672458 * r672455;
        double r672460 = r672456 - r672459;
        double r672461 = 3.1423118580081215e-35;
        bool r672462 = r672450 <= r672461;
        double r672463 = r672450 * r672450;
        double r672464 = r672453 * r672457;
        double r672465 = r672463 - r672464;
        double r672466 = sqrt(r672465);
        double r672467 = r672466 - r672450;
        double r672468 = r672467 / r672457;
        double r672469 = -0.5;
        double r672470 = r672454 * r672469;
        double r672471 = r672462 ? r672468 : r672470;
        double r672472 = r672452 ? r672460 : r672471;
        return r672472;
}

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 < -2.221067196710922e+149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    3. Taylor expanded around -inf 2.7

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

    4. Simplified2.7

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

    if -2.221067196710922e+149 < b_2 < 3.1423118580081215e-35

    1. Initial program 14.5

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

    2. Simplified14.5

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

    if 3.1423118580081215e-35 < b_2

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 3.142311858008121469027865121070306475283 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))