Average Error: 34.2 → 8.6
Time: 4.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 9.5508506461031991 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.8072455366389721 \cdot 10^{45}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + \frac{1}{a} \cdot \left(a \cdot c\right)}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 9.5508506461031991 \cdot 10^{-303}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 5.8072455366389721 \cdot 10^{45}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot 0 + \frac{1}{a} \cdot \left(a \cdot c\right)}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r13316 = b_2;
        double r13317 = -r13316;
        double r13318 = r13316 * r13316;
        double r13319 = a;
        double r13320 = c;
        double r13321 = r13319 * r13320;
        double r13322 = r13318 - r13321;
        double r13323 = sqrt(r13322);
        double r13324 = r13317 + r13323;
        double r13325 = r13324 / r13319;
        return r13325;
}
↓
double f(double a, double b_2, double c) {
        double r13326 = b_2;
        double r13327 = -1.0524969308880696e+141;
        bool r13328 = r13326 <= r13327;
        double r13329 = 0.5;
        double r13330 = c;
        double r13331 = r13330 / r13326;
        double r13332 = r13329 * r13331;
        double r13333 = 2.0;
        double r13334 = a;
        double r13335 = r13326 / r13334;
        double r13336 = r13333 * r13335;
        double r13337 = r13332 - r13336;
        double r13338 = 9.550850646103199e-303;
        bool r13339 = r13326 <= r13338;
        double r13340 = -r13326;
        double r13341 = r13326 * r13326;
        double r13342 = r13334 * r13330;
        double r13343 = r13341 - r13342;
        double r13344 = sqrt(r13343);
        double r13345 = r13340 + r13344;
        double r13346 = 1.0;
        double r13347 = r13346 / r13334;
        double r13348 = r13345 * r13347;
        double r13349 = 5.807245536638972e+45;
        bool r13350 = r13326 <= r13349;
        double r13351 = 0.0;
        double r13352 = r13347 * r13351;
        double r13353 = r13347 * r13342;
        double r13354 = r13352 + r13353;
        double r13355 = r13340 - r13344;
        double r13356 = r13354 / r13355;
        double r13357 = -0.5;
        double r13358 = r13357 * r13331;
        double r13359 = r13350 ? r13356 : r13358;
        double r13360 = r13339 ? r13348 : r13359;
        double r13361 = r13328 ? r13337 : r13360;
        return r13361;
}

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 < -1.0524969308880696e+141

    1. Initial program 58.5

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

    2. Taylor expanded around -inf 2.4

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

    if -1.0524969308880696e+141 < b_2 < 9.550850646103199e-303

    1. Initial program 9.1

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

    3. Applied div-inv9.2

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

    if 9.550850646103199e-303 < b_2 < 5.807245536638972e+45

    1. Initial program 28.7

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

    3. Applied div-inv28.7

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

    5. Applied flip-+28.7

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

    6. Applied associate-*l/28.8

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

    7. Simplified16.7

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

    if 5.807245536638972e+45 < b_2

    1. Initial program 56.8

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

    2. Taylor expanded around inf 3.8

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 9.5508506461031991 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.8072455366389721 \cdot 10^{45}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + \frac{1}{a} \cdot \left(a \cdot c\right)}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))