Average Error: 34.4 → 8.3
Time: 6.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -8.61562639771237576 \cdot 10^{-228}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.2011474830795371 \cdot 10^{35}:\\ \;\;\;\;\frac{1 \cdot \frac{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.9644058459680186 \cdot 10^{71}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -8.61562639771237576 \cdot 10^{-228}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 1.2011474830795371 \cdot 10^{35}:\\
\;\;\;\;\frac{1 \cdot \frac{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r21472 = b_2;
        double r21473 = -r21472;
        double r21474 = r21472 * r21472;
        double r21475 = a;
        double r21476 = c;
        double r21477 = r21475 * r21476;
        double r21478 = r21474 - r21477;
        double r21479 = sqrt(r21478);
        double r21480 = r21473 + r21479;
        double r21481 = r21480 / r21475;
        return r21481;
}
↓
double f(double a, double b_2, double c) {
        double r21482 = b_2;
        double r21483 = -2.9644058459680186e+71;
        bool r21484 = r21482 <= r21483;
        double r21485 = 0.5;
        double r21486 = c;
        double r21487 = r21486 / r21482;
        double r21488 = r21485 * r21487;
        double r21489 = 2.0;
        double r21490 = a;
        double r21491 = r21482 / r21490;
        double r21492 = r21489 * r21491;
        double r21493 = r21488 - r21492;
        double r21494 = -8.615626397712376e-228;
        bool r21495 = r21482 <= r21494;
        double r21496 = -r21482;
        double r21497 = r21482 * r21482;
        double r21498 = r21490 * r21486;
        double r21499 = r21497 - r21498;
        double r21500 = sqrt(r21499);
        double r21501 = r21496 + r21500;
        double r21502 = 1.0;
        double r21503 = r21502 / r21490;
        double r21504 = r21501 * r21503;
        double r21505 = 1.2011474830795371e+35;
        bool r21506 = r21482 <= r21505;
        double r21507 = r21496 - r21500;
        double r21508 = r21490 / r21507;
        double r21509 = r21502 / r21486;
        double r21510 = r21508 / r21509;
        double r21511 = r21502 * r21510;
        double r21512 = r21511 / r21490;
        double r21513 = -0.5;
        double r21514 = r21513 * r21487;
        double r21515 = r21506 ? r21512 : r21514;
        double r21516 = r21495 ? r21504 : r21515;
        double r21517 = r21484 ? r21493 : r21516;
        return r21517;
}

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 < -2.9644058459680186e+71

    1. Initial program 42.3

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

    2. Taylor expanded around -inf 4.6

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

    if -2.9644058459680186e+71 < b_2 < -8.615626397712376e-228

    1. Initial program 9.2

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

    3. Applied div-inv9.4

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

    if -8.615626397712376e-228 < b_2 < 1.2011474830795371e+35

    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-+26.2

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

    4. Simplified16.7

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

    6. Applied *-un-lft-identity16.7

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

    7. Applied *-un-lft-identity16.7

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

    8. Applied times-frac16.7

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

    9. Simplified16.7

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

    10. Simplified13.8

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

    12. Applied div-inv13.8

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

    13. Applied associate-/r*13.9

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

    if 1.2011474830795371e+35 < b_2

    1. Initial program 57.1

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

    2. Taylor expanded around inf 4.0

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -8.61562639771237576 \cdot 10^{-228}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.2011474830795371 \cdot 10^{35}:\\ \;\;\;\;\frac{1 \cdot \frac{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))