Average Error: 33.6 → 6.9
Time: 19.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.1942109607748042 \cdot 10^{+147}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.092790018801933 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 3.4243727332802643 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a \cdot \frac{1}{2}}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.1942109607748042 \cdot 10^{+147}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.092790018801933 \cdot 10^{-307}:\\
\;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\

\mathbf{elif}\;b_2 \le 3.4243727332802643 \cdot 10^{+80}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r3037251 = b_2;
        double r3037252 = -r3037251;
        double r3037253 = r3037251 * r3037251;
        double r3037254 = a;
        double r3037255 = c;
        double r3037256 = r3037254 * r3037255;
        double r3037257 = r3037253 - r3037256;
        double r3037258 = sqrt(r3037257);
        double r3037259 = r3037252 - r3037258;
        double r3037260 = r3037259 / r3037254;
        return r3037260;
}
↓
double f(double a, double b_2, double c) {
        double r3037261 = b_2;
        double r3037262 = -1.1942109607748042e+147;
        bool r3037263 = r3037261 <= r3037262;
        double r3037264 = -0.5;
        double r3037265 = c;
        double r3037266 = r3037265 / r3037261;
        double r3037267 = r3037264 * r3037266;
        double r3037268 = -1.092790018801933e-307;
        bool r3037269 = r3037261 <= r3037268;
        double r3037270 = 1.0;
        double r3037271 = r3037261 * r3037261;
        double r3037272 = a;
        double r3037273 = r3037272 * r3037265;
        double r3037274 = r3037271 - r3037273;
        double r3037275 = sqrt(r3037274);
        double r3037276 = r3037275 - r3037261;
        double r3037277 = r3037276 / r3037265;
        double r3037278 = r3037270 / r3037277;
        double r3037279 = 3.4243727332802643e+80;
        bool r3037280 = r3037261 <= r3037279;
        double r3037281 = -r3037261;
        double r3037282 = r3037281 - r3037275;
        double r3037283 = r3037270 / r3037272;
        double r3037284 = r3037282 * r3037283;
        double r3037285 = 0.5;
        double r3037286 = r3037285 * r3037266;
        double r3037287 = r3037272 * r3037285;
        double r3037288 = r3037261 / r3037287;
        double r3037289 = r3037286 - r3037288;
        double r3037290 = r3037280 ? r3037284 : r3037289;
        double r3037291 = r3037269 ? r3037278 : r3037290;
        double r3037292 = r3037263 ? r3037267 : r3037291;
        return r3037292;
}

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 < -1.1942109607748042e+147

    1. Initial program 62.2

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

    2. Taylor expanded around -inf 1.3

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

    if -1.1942109607748042e+147 < b_2 < -1.092790018801933e-307

    1. Initial program 34.1

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

    3. Applied flip--34.2

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

    4. Simplified15.8

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

    5. Simplified15.8

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

    7. Applied clear-num16.0

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

    8. Simplified8.7

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

    if -1.092790018801933e-307 < b_2 < 3.4243727332802643e+80

    1. Initial program 10.0

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

    3. Applied div-inv10.1

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

    if 3.4243727332802643e+80 < b_2

    1. Initial program 41.9

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.1942109607748042 \cdot 10^{+147}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.092790018801933 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 3.4243727332802643 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a \cdot \frac{1}{2}}\\ \end{array}\]

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))