Average Error: 34.0 → 10.5
Time: 20.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\
\;\;\;\;\frac{\left(-b_2\right) - \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 r2609309 = b_2;
        double r2609310 = -r2609309;
        double r2609311 = r2609309 * r2609309;
        double r2609312 = a;
        double r2609313 = c;
        double r2609314 = r2609312 * r2609313;
        double r2609315 = r2609311 - r2609314;
        double r2609316 = sqrt(r2609315);
        double r2609317 = r2609310 - r2609316;
        double r2609318 = r2609317 / r2609312;
        return r2609318;
}
↓
double f(double a, double b_2, double c) {
        double r2609319 = b_2;
        double r2609320 = -9.332433396832084e-58;
        bool r2609321 = r2609319 <= r2609320;
        double r2609322 = -0.5;
        double r2609323 = c;
        double r2609324 = r2609323 / r2609319;
        double r2609325 = r2609322 * r2609324;
        double r2609326 = 3.038903409991338e+107;
        bool r2609327 = r2609319 <= r2609326;
        double r2609328 = -r2609319;
        double r2609329 = r2609319 * r2609319;
        double r2609330 = a;
        double r2609331 = r2609330 * r2609323;
        double r2609332 = r2609329 - r2609331;
        double r2609333 = sqrt(r2609332);
        double r2609334 = r2609328 - r2609333;
        double r2609335 = r2609334 / r2609330;
        double r2609336 = -2.0;
        double r2609337 = r2609319 * r2609336;
        double r2609338 = r2609337 / r2609330;
        double r2609339 = r2609327 ? r2609335 : r2609338;
        double r2609340 = r2609321 ? r2609325 : r2609339;
        return r2609340;
}

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 3 regimes

  2. if b_2 < -9.332433396832084e-58

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 8.7

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

    if -9.332433396832084e-58 < b_2 < 3.038903409991338e+107

    1. Initial program 14.0

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

    if 3.038903409991338e+107 < b_2

    1. Initial program 49.1

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

    3. Applied flip--63.2

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

    4. Simplified62.2

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

    5. Simplified62.2

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

    6. Taylor expanded around 0 3.7

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))