Average Error: 33.6 → 9.1
Time: 18.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.2887136042886476 \cdot 10^{+71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.407079315314288 \cdot 10^{-176}:\\ \;\;\;\;\frac{a \cdot \frac{c}{{\left(\sqrt{e}\right)}^{\left(\log \left(b_2 \cdot b_2 - c \cdot a\right)\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 8.016779424032652 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.2887136042886476 \cdot 10^{+71}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.407079315314288 \cdot 10^{-176}:\\
\;\;\;\;\frac{a \cdot \frac{c}{{\left(\sqrt{e}\right)}^{\left(\log \left(b_2 \cdot b_2 - c \cdot a\right)\right)} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 8.016779424032652 \cdot 10^{+82}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r2922362 = b_2;
        double r2922363 = -r2922362;
        double r2922364 = r2922362 * r2922362;
        double r2922365 = a;
        double r2922366 = c;
        double r2922367 = r2922365 * r2922366;
        double r2922368 = r2922364 - r2922367;
        double r2922369 = sqrt(r2922368);
        double r2922370 = r2922363 - r2922369;
        double r2922371 = r2922370 / r2922365;
        return r2922371;
}
↓
double f(double a, double b_2, double c) {
        double r2922372 = b_2;
        double r2922373 = -4.2887136042886476e+71;
        bool r2922374 = r2922372 <= r2922373;
        double r2922375 = -0.5;
        double r2922376 = c;
        double r2922377 = r2922376 / r2922372;
        double r2922378 = r2922375 * r2922377;
        double r2922379 = -3.407079315314288e-176;
        bool r2922380 = r2922372 <= r2922379;
        double r2922381 = a;
        double r2922382 = exp(1.0);
        double r2922383 = sqrt(r2922382);
        double r2922384 = r2922372 * r2922372;
        double r2922385 = r2922376 * r2922381;
        double r2922386 = r2922384 - r2922385;
        double r2922387 = log(r2922386);
        double r2922388 = pow(r2922383, r2922387);
        double r2922389 = r2922388 - r2922372;
        double r2922390 = r2922376 / r2922389;
        double r2922391 = r2922381 * r2922390;
        double r2922392 = r2922391 / r2922381;
        double r2922393 = 8.016779424032652e+82;
        bool r2922394 = r2922372 <= r2922393;
        double r2922395 = 1.0;
        double r2922396 = -r2922372;
        double r2922397 = sqrt(r2922386);
        double r2922398 = r2922396 - r2922397;
        double r2922399 = r2922381 / r2922398;
        double r2922400 = r2922395 / r2922399;
        double r2922401 = 0.5;
        double r2922402 = r2922377 * r2922401;
        double r2922403 = 2.0;
        double r2922404 = r2922372 / r2922381;
        double r2922405 = r2922403 * r2922404;
        double r2922406 = r2922402 - r2922405;
        double r2922407 = r2922394 ? r2922400 : r2922406;
        double r2922408 = r2922380 ? r2922392 : r2922407;
        double r2922409 = r2922374 ? r2922378 : r2922408;
        return r2922409;
}

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 < -4.2887136042886476e+71

    1. Initial program 57.3

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

    2. Taylor expanded around -inf 3.3

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

    if -4.2887136042886476e+71 < b_2 < -3.407079315314288e-176

    1. Initial program 36.0

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

    3. Applied flip--36.1

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

    4. Simplified15.5

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

    5. Simplified15.5

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

    7. Applied *-un-lft-identity15.5

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

    8. Applied times-frac12.6

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

    9. Simplified12.6

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

    11. Applied add-exp-log15.3

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

    13. Applied pow1/215.3

      \[\leadsto \frac{a \cdot \frac{c}{e^{\log \color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\frac{1}{2}}\right)}} - b_2}}{a}\]

    14. Applied log-pow15.3

      \[\leadsto \frac{a \cdot \frac{c}{e^{\color{blue}{\frac{1}{2} \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}} - b_2}}{a}\]

    15. Applied exp-prod15.7

      \[\leadsto \frac{a \cdot \frac{c}{\color{blue}{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(b_2 \cdot b_2 - a \cdot c\right)\right)}} - b_2}}{a}\]

    16. Simplified15.7

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

    if -3.407079315314288e-176 < b_2 < 8.016779424032652e+82

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 8.016779424032652e+82 < b_2

    1. Initial program 42.4

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

    2. Taylor expanded around inf 3.6

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.2887136042886476 \cdot 10^{+71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.407079315314288 \cdot 10^{-176}:\\ \;\;\;\;\frac{a \cdot \frac{c}{{\left(\sqrt{e}\right)}^{\left(\log \left(b_2 \cdot b_2 - c \cdot a\right)\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 8.016779424032652 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))