Average Error: 34.2 → 8.6
Time: 29.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \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}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r3347445 = b_2;
        double r3347446 = -r3347445;
        double r3347447 = r3347445 * r3347445;
        double r3347448 = a;
        double r3347449 = c;
        double r3347450 = r3347448 * r3347449;
        double r3347451 = r3347447 - r3347450;
        double r3347452 = sqrt(r3347451);
        double r3347453 = r3347446 - r3347452;
        double r3347454 = r3347453 / r3347448;
        return r3347454;
}
↓
double f(double a, double b_2, double c) {
        double r3347455 = b_2;
        double r3347456 = -5.999462245480892e+73;
        bool r3347457 = r3347455 <= r3347456;
        double r3347458 = -0.5;
        double r3347459 = c;
        double r3347460 = r3347459 / r3347455;
        double r3347461 = r3347458 * r3347460;
        double r3347462 = -4.296272708131499e-127;
        bool r3347463 = r3347455 <= r3347462;
        double r3347464 = a;
        double r3347465 = r3347464 * r3347459;
        double r3347466 = r3347465 / r3347464;
        double r3347467 = r3347455 * r3347455;
        double r3347468 = r3347467 - r3347465;
        double r3347469 = sqrt(r3347468);
        double r3347470 = r3347469 - r3347455;
        double r3347471 = r3347466 / r3347470;
        double r3347472 = 5.748502676649783e+143;
        bool r3347473 = r3347455 <= r3347472;
        double r3347474 = -r3347455;
        double r3347475 = r3347474 / r3347464;
        double r3347476 = r3347469 / r3347464;
        double r3347477 = r3347475 - r3347476;
        double r3347478 = r3347455 / r3347464;
        double r3347479 = -2.0;
        double r3347480 = r3347478 * r3347479;
        double r3347481 = r3347473 ? r3347477 : r3347480;
        double r3347482 = r3347463 ? r3347471 : r3347481;
        double r3347483 = r3347457 ? r3347461 : r3347482;
        return r3347483;
}

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. Using strategy rm

    7. Applied *-un-lft-identity62.7

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

    8. Applied times-frac62.6

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

    9. Simplified62.6

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

    11. Applied div-inv62.6

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

    12. Taylor expanded around 0 2.2

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

  4. Final simplification8.6

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

Reproduce

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