Average Error: 34.2 → 8.6
Time: 1.5m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\
\;\;\;\;\frac{\frac{a \cdot c}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1152585 = b_2;
        double r1152586 = -r1152585;
        double r1152587 = r1152585 * r1152585;
        double r1152588 = a;
        double r1152589 = c;
        double r1152590 = r1152588 * r1152589;
        double r1152591 = r1152587 - r1152590;
        double r1152592 = sqrt(r1152591);
        double r1152593 = r1152586 - r1152592;
        double r1152594 = r1152593 / r1152588;
        return r1152594;
}
↓
double f(double a, double b_2, double c) {
        double r1152595 = b_2;
        double r1152596 = -5.999462245480892e+73;
        bool r1152597 = r1152595 <= r1152596;
        double r1152598 = -0.5;
        double r1152599 = c;
        double r1152600 = r1152599 / r1152595;
        double r1152601 = r1152598 * r1152600;
        double r1152602 = -4.296272708131499e-127;
        bool r1152603 = r1152595 <= r1152602;
        double r1152604 = a;
        double r1152605 = r1152604 * r1152599;
        double r1152606 = r1152605 / r1152604;
        double r1152607 = r1152595 * r1152595;
        double r1152608 = r1152607 - r1152605;
        double r1152609 = sqrt(r1152608);
        double r1152610 = r1152609 - r1152595;
        double r1152611 = r1152606 / r1152610;
        double r1152612 = 5.748502676649783e+143;
        bool r1152613 = r1152595 <= r1152612;
        double r1152614 = r1152595 / r1152604;
        double r1152615 = -r1152614;
        double r1152616 = r1152609 / r1152604;
        double r1152617 = r1152615 - r1152616;
        double r1152618 = -2.0;
        double r1152619 = r1152595 * r1152618;
        double r1152620 = r1152619 / r1152604;
        double r1152621 = r1152613 ? r1152617 : r1152620;
        double r1152622 = r1152603 ? r1152611 : r1152621;
        double r1152623 = r1152597 ? r1152601 : r1152622;
        return r1152623;
}

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 < -5.999462245480892e+73

    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 3.4

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

    if -5.999462245480892e+73 < b_2 < -4.296272708131499e-127

    1. Initial program 40.0

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

    3. Applied flip--40.0

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

    4. Simplified15.6

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

    5. Simplified15.6

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity15.6

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

    8. Applied times-frac13.7

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

    9. Simplified13.7

      \[\leadsto \frac{\color{blue}{a} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]
    10. Using strategy rm

    11. Applied div-inv13.8

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

    13. Applied associate-*r/15.7

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

    14. Applied associate-*l/14.8

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

    15. Simplified14.7

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

    if -4.296272708131499e-127 < b_2 < 5.748502676649783e+143

    1. Initial program 11.3

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

    3. Applied div-sub11.2

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

    if 5.748502676649783e+143 < b_2

    1. Initial program 59.9

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

    3. Applied flip--63.9

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

    4. Simplified62.7

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

    5. Simplified62.7

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

    6. Taylor expanded around 0 2.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))