Average Error: 33.5 → 6.8
Time: 7.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -4.0039604448318847 \cdot 10^{-272}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 4.12298741547621415 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.0323767944871679 \cdot 10^{127}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -4.0039604448318847 \cdot 10^{-272}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 4.12298741547621415 \cdot 10^{92}:\\
\;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r24519 = b_2;
        double r24520 = -r24519;
        double r24521 = r24519 * r24519;
        double r24522 = a;
        double r24523 = c;
        double r24524 = r24522 * r24523;
        double r24525 = r24521 - r24524;
        double r24526 = sqrt(r24525);
        double r24527 = r24520 + r24526;
        double r24528 = r24527 / r24522;
        return r24528;
}
↓
double f(double a, double b_2, double c) {
        double r24529 = b_2;
        double r24530 = -4.032376794487168e+127;
        bool r24531 = r24529 <= r24530;
        double r24532 = 0.5;
        double r24533 = c;
        double r24534 = r24533 / r24529;
        double r24535 = r24532 * r24534;
        double r24536 = 2.0;
        double r24537 = a;
        double r24538 = r24529 / r24537;
        double r24539 = r24536 * r24538;
        double r24540 = r24535 - r24539;
        double r24541 = -4.003960444831885e-272;
        bool r24542 = r24529 <= r24541;
        double r24543 = -r24529;
        double r24544 = r24529 * r24529;
        double r24545 = r24537 * r24533;
        double r24546 = r24544 - r24545;
        double r24547 = sqrt(r24546);
        double r24548 = r24543 + r24547;
        double r24549 = 1.0;
        double r24550 = r24549 / r24537;
        double r24551 = r24548 * r24550;
        double r24552 = 4.122987415476214e+92;
        bool r24553 = r24529 <= r24552;
        double r24554 = r24543 - r24547;
        double r24555 = r24549 / r24554;
        double r24556 = r24549 / r24533;
        double r24557 = r24555 / r24556;
        double r24558 = -0.5;
        double r24559 = r24558 * r24534;
        double r24560 = r24553 ? r24557 : r24559;
        double r24561 = r24542 ? r24551 : r24560;
        double r24562 = r24531 ? r24540 : r24561;
        return r24562;
}

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 < -4.032376794487168e+127

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 3.1

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

    if -4.032376794487168e+127 < b_2 < -4.003960444831885e-272

    1. Initial program 8.8

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

    3. Applied div-inv8.9

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

    if -4.003960444831885e-272 < b_2 < 4.122987415476214e+92

    1. Initial program 30.5

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

    3. Applied flip-+30.5

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

    4. Simplified16.3

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

    6. Applied *-un-lft-identity16.3

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

    7. Applied associate-/r*16.3

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

    8. Simplified14.2

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

    10. Applied div-inv14.2

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

    11. Applied *-un-lft-identity14.2

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

    12. Applied times-frac16.4

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

    13. Applied associate-/l*15.7

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

    14. Simplified9.3

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

    if 4.122987415476214e+92 < b_2

    1. Initial program 59.4

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

    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 simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -4.0039604448318847 \cdot 10^{-272}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 4.12298741547621415 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))