Average Error: 34.2 → 8.3
Time: 23.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.004804434998936137867310325826299333972 \cdot 10^{57}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -4.603808643431251996219483142154919887261 \cdot 10^{-290}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 148410.0719543073500972241163253784179688:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.004804434998936137867310325826299333972 \cdot 10^{57}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -4.603808643431251996219483142154919887261 \cdot 10^{-290}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 148410.0719543073500972241163253784179688:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r42618 = b_2;
        double r42619 = -r42618;
        double r42620 = r42618 * r42618;
        double r42621 = a;
        double r42622 = c;
        double r42623 = r42621 * r42622;
        double r42624 = r42620 - r42623;
        double r42625 = sqrt(r42624);
        double r42626 = r42619 + r42625;
        double r42627 = r42626 / r42621;
        return r42627;
}
↓
double f(double a, double b_2, double c) {
        double r42628 = b_2;
        double r42629 = -5.004804434998936e+57;
        bool r42630 = r42628 <= r42629;
        double r42631 = 0.5;
        double r42632 = c;
        double r42633 = r42632 / r42628;
        double r42634 = r42631 * r42633;
        double r42635 = 2.0;
        double r42636 = a;
        double r42637 = r42628 / r42636;
        double r42638 = r42635 * r42637;
        double r42639 = r42634 - r42638;
        double r42640 = -4.603808643431252e-290;
        bool r42641 = r42628 <= r42640;
        double r42642 = 1.0;
        double r42643 = r42628 * r42628;
        double r42644 = r42636 * r42632;
        double r42645 = r42643 - r42644;
        double r42646 = sqrt(r42645);
        double r42647 = r42646 - r42628;
        double r42648 = r42636 / r42647;
        double r42649 = r42642 / r42648;
        double r42650 = 148410.07195430735;
        bool r42651 = r42628 <= r42650;
        double r42652 = -r42628;
        double r42653 = r42652 - r42646;
        double r42654 = r42653 / r42636;
        double r42655 = r42654 / r42632;
        double r42656 = r42642 / r42655;
        double r42657 = r42656 / r42636;
        double r42658 = -0.5;
        double r42659 = r42658 * r42633;
        double r42660 = r42651 ? r42657 : r42659;
        double r42661 = r42641 ? r42649 : r42660;
        double r42662 = r42630 ? r42639 : r42661;
        return r42662;
}

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 < -5.004804434998936e+57

    1. Initial program 38.7

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

    2. Taylor expanded around -inf 4.9

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

    if -5.004804434998936e+57 < b_2 < -4.603808643431252e-290

    1. Initial program 9.1

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

    3. Applied clear-num9.2

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

    4. Simplified9.2

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

    if -4.603808643431252e-290 < b_2 < 148410.07195430735

    1. Initial program 26.0

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

    3. Applied flip-+26.0

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

    4. Simplified17.5

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

    6. Applied clear-num17.5

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

    7. Simplified14.5

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

    if 148410.07195430735 < b_2

    1. Initial program 56.1

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

    2. Taylor expanded around inf 5.3

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.004804434998936137867310325826299333972 \cdot 10^{57}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -4.603808643431251996219483142154919887261 \cdot 10^{-290}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 148410.0719543073500972241163253784179688:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))