Average Error: 34.3 → 8.5
Time: 7.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.76415189671242326 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.4667701057073822 \cdot 10^{-270}:\\ \;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.76415189671242326 \cdot 10^{-27}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.4667701057073822 \cdot 10^{-270}:\\
\;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\
\;\;\;\;\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 r104436 = b_2;
        double r104437 = -r104436;
        double r104438 = r104436 * r104436;
        double r104439 = a;
        double r104440 = c;
        double r104441 = r104439 * r104440;
        double r104442 = r104438 - r104441;
        double r104443 = sqrt(r104442);
        double r104444 = r104437 - r104443;
        double r104445 = r104444 / r104439;
        return r104445;
}
↓
double f(double a, double b_2, double c) {
        double r104446 = b_2;
        double r104447 = -2.7641518967124233e-27;
        bool r104448 = r104446 <= r104447;
        double r104449 = -0.5;
        double r104450 = c;
        double r104451 = r104450 / r104446;
        double r104452 = r104449 * r104451;
        double r104453 = 1.4667701057073822e-270;
        bool r104454 = r104446 <= r104453;
        double r104455 = a;
        double r104456 = r104446 * r104446;
        double r104457 = r104455 * r104450;
        double r104458 = r104456 - r104457;
        double r104459 = sqrt(r104458);
        double r104460 = r104459 - r104446;
        double r104461 = r104460 / r104450;
        double r104462 = r104455 / r104461;
        double r104463 = r104462 / r104455;
        double r104464 = 2.8046284917653458e+91;
        bool r104465 = r104446 <= r104464;
        double r104466 = -r104446;
        double r104467 = r104466 - r104459;
        double r104468 = r104467 / r104455;
        double r104469 = 0.5;
        double r104470 = r104469 * r104451;
        double r104471 = 2.0;
        double r104472 = r104446 / r104455;
        double r104473 = r104471 * r104472;
        double r104474 = r104470 - r104473;
        double r104475 = r104465 ? r104468 : r104474;
        double r104476 = r104454 ? r104463 : r104475;
        double r104477 = r104448 ? r104452 : r104476;
        return r104477;
}

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 < -2.7641518967124233e-27

    1. Initial program 55.1

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

    2. Taylor expanded around -inf 6.5

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

    if -2.7641518967124233e-27 < b_2 < 1.4667701057073822e-270

    1. Initial program 23.5

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

    3. Applied flip--23.5

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

    4. Simplified17.5

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

    5. Simplified17.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity17.5

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

    8. Applied associate-/r*17.5

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

    9. Simplified14.4

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

    if 1.4667701057073822e-270 < b_2 < 2.8046284917653458e+91

    1. Initial program 9.0

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

    if 2.8046284917653458e+91 < b_2

    1. Initial program 45.3

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.76415189671242326 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.4667701057073822 \cdot 10^{-270}:\\ \;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))