Average Error: 34.3 → 8.1
Time: 7.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.388209440671705791656215927803897929135 \cdot 10^{145}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.366472411559289056044502261742376769816 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 9.191824015256066577021611473136461497822 \cdot 10^{77}:\\ \;\;\;\;\frac{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.388209440671705791656215927803897929135 \cdot 10^{145}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 8.366472411559289056044502261742376769816 \cdot 10^{-215}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 9.191824015256066577021611473136461497822 \cdot 10^{77}:\\
\;\;\;\;\frac{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r23430 = b_2;
        double r23431 = -r23430;
        double r23432 = r23430 * r23430;
        double r23433 = a;
        double r23434 = c;
        double r23435 = r23433 * r23434;
        double r23436 = r23432 - r23435;
        double r23437 = sqrt(r23436);
        double r23438 = r23431 + r23437;
        double r23439 = r23438 / r23433;
        return r23439;
}
↓
double f(double a, double b_2, double c) {
        double r23440 = b_2;
        double r23441 = -1.3882094406717058e+145;
        bool r23442 = r23440 <= r23441;
        double r23443 = 0.5;
        double r23444 = c;
        double r23445 = r23444 / r23440;
        double r23446 = r23443 * r23445;
        double r23447 = 2.0;
        double r23448 = a;
        double r23449 = r23440 / r23448;
        double r23450 = r23447 * r23449;
        double r23451 = r23446 - r23450;
        double r23452 = 8.366472411559289e-215;
        bool r23453 = r23440 <= r23452;
        double r23454 = -r23440;
        double r23455 = r23440 * r23440;
        double r23456 = r23448 * r23444;
        double r23457 = r23455 - r23456;
        double r23458 = sqrt(r23457);
        double r23459 = r23454 + r23458;
        double r23460 = r23459 / r23448;
        double r23461 = 9.191824015256067e+77;
        bool r23462 = r23440 <= r23461;
        double r23463 = r23454 - r23458;
        double r23464 = r23463 / r23444;
        double r23465 = r23448 / r23464;
        double r23466 = r23465 / r23448;
        double r23467 = -0.5;
        double r23468 = r23467 * r23445;
        double r23469 = r23462 ? r23466 : r23468;
        double r23470 = r23453 ? r23460 : r23469;
        double r23471 = r23442 ? r23451 : r23470;
        return r23471;
}

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 4 regimes

  2. if b_2 < -1.3882094406717058e+145

    1. Initial program 60.2

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

    2. Taylor expanded around -inf 2.6

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

    if -1.3882094406717058e+145 < b_2 < 8.366472411559289e-215

    1. Initial program 9.6

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

    if 8.366472411559289e-215 < b_2 < 9.191824015256067e+77

    1. Initial program 35.4

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

    3. Applied flip-+35.4

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

    4. Simplified17.1

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied div-inv17.1

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}}\]
    7. Using strategy rm

    8. Applied associate-*r/17.1

      \[\leadsto \color{blue}{\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot 1}{a}}\]

    9. Simplified14.5

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

    if 9.191824015256067e+77 < b_2

    1. Initial program 58.3

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

    2. Taylor expanded around inf 2.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.388209440671705791656215927803897929135 \cdot 10^{145}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.366472411559289056044502261742376769816 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 9.191824015256066577021611473136461497822 \cdot 10^{77}:\\ \;\;\;\;\frac{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))