Average Error: 33.9 → 6.9
Time: 9.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.442369070306521925925945300368519903805 \cdot 10^{91}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le \frac{8176673959656255}{5.8652475225036720546649669496166069229 \cdot 10^{253}}:\\ \;\;\;\;\left(\frac{1}{a} \cdot 0 + c\right) \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.530904404386409423412434180578840981346 \cdot 10^{64}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.442369070306521925925945300368519903805 \cdot 10^{91}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le \frac{8176673959656255}{5.8652475225036720546649669496166069229 \cdot 10^{253}}:\\
\;\;\;\;\left(\frac{1}{a} \cdot 0 + c\right) \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 1.530904404386409423412434180578840981346 \cdot 10^{64}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r76422 = b_2;
        double r76423 = -r76422;
        double r76424 = r76422 * r76422;
        double r76425 = a;
        double r76426 = c;
        double r76427 = r76425 * r76426;
        double r76428 = r76424 - r76427;
        double r76429 = sqrt(r76428);
        double r76430 = r76423 - r76429;
        double r76431 = r76430 / r76425;
        return r76431;
}
↓
double f(double a, double b_2, double c) {
        double r76432 = b_2;
        double r76433 = -2.442369070306522e+91;
        bool r76434 = r76432 <= r76433;
        double r76435 = -0.5;
        double r76436 = c;
        double r76437 = r76436 / r76432;
        double r76438 = r76435 * r76437;
        double r76439 = 8176673959656255.0;
        double r76440 = 5.865247522503672e+253;
        double r76441 = r76439 / r76440;
        bool r76442 = r76432 <= r76441;
        double r76443 = 1.0;
        double r76444 = a;
        double r76445 = r76443 / r76444;
        double r76446 = 0.0;
        double r76447 = r76445 * r76446;
        double r76448 = r76447 + r76436;
        double r76449 = -r76432;
        double r76450 = r76432 * r76432;
        double r76451 = r76444 * r76436;
        double r76452 = r76450 - r76451;
        double r76453 = sqrt(r76452);
        double r76454 = r76449 + r76453;
        double r76455 = r76443 / r76454;
        double r76456 = r76448 * r76455;
        double r76457 = 1.5309044043864094e+64;
        bool r76458 = r76432 <= r76457;
        double r76459 = r76449 - r76453;
        double r76460 = r76459 / r76444;
        double r76461 = 0.5;
        double r76462 = r76461 * r76437;
        double r76463 = 2.0;
        double r76464 = r76432 / r76444;
        double r76465 = r76463 * r76464;
        double r76466 = r76462 - r76465;
        double r76467 = r76458 ? r76460 : r76466;
        double r76468 = r76442 ? r76456 : r76467;
        double r76469 = r76434 ? r76438 : r76468;
        return r76469;
}

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 < -2.442369070306522e+91

    1. Initial program 59.0

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

    2. Taylor expanded around -inf 2.7

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

    if -2.442369070306522e+91 < b_2 < 1.3940884725297859e-238

    1. Initial program 30.0

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

    3. Applied clear-num30.0

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

    5. Applied flip--30.1

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

    6. Applied associate-/r/30.2

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

    7. Applied *-un-lft-identity30.2

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

    8. Applied times-frac30.2

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

    9. Simplified15.8

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

    10. Taylor expanded around 0 9.9

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

    if 1.3940884725297859e-238 < b_2 < 1.5309044043864094e+64

    1. Initial program 8.1

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

    if 1.5309044043864094e+64 < b_2

    1. Initial program 39.8

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

    2. Taylor expanded around inf 5.5

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.442369070306521925925945300368519903805 \cdot 10^{91}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le \frac{8176673959656255}{5.8652475225036720546649669496166069229 \cdot 10^{253}}:\\ \;\;\;\;\left(\frac{1}{a} \cdot 0 + c\right) \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.530904404386409423412434180578840981346 \cdot 10^{64}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))