Average Error: 34.3 → 8.6
Time: 18.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9661478263987.111328125:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.792446256118343513399316360203386878654 \cdot 10^{-145}:\\ \;\;\;\;\frac{c \cdot a}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \frac{-1}{a}\\ \mathbf{elif}\;b_2 \le 1.764973784792091233612894541626891615653 \cdot 10^{83}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9661478263987.111328125:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -4.792446256118343513399316360203386878654 \cdot 10^{-145}:\\
\;\;\;\;\frac{c \cdot a}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \frac{-1}{a}\\

\mathbf{elif}\;b_2 \le 1.764973784792091233612894541626891615653 \cdot 10^{83}:\\
\;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r53264 = b_2;
        double r53265 = -r53264;
        double r53266 = r53264 * r53264;
        double r53267 = a;
        double r53268 = c;
        double r53269 = r53267 * r53268;
        double r53270 = r53266 - r53269;
        double r53271 = sqrt(r53270);
        double r53272 = r53265 - r53271;
        double r53273 = r53272 / r53267;
        return r53273;
}
↓
double f(double a, double b_2, double c) {
        double r53274 = b_2;
        double r53275 = -9661478263987.111;
        bool r53276 = r53274 <= r53275;
        double r53277 = -0.5;
        double r53278 = c;
        double r53279 = r53278 / r53274;
        double r53280 = r53277 * r53279;
        double r53281 = -4.7924462561183435e-145;
        bool r53282 = r53274 <= r53281;
        double r53283 = a;
        double r53284 = r53278 * r53283;
        double r53285 = r53274 * r53274;
        double r53286 = r53285 - r53284;
        double r53287 = sqrt(r53286);
        double r53288 = r53274 - r53287;
        double r53289 = r53284 / r53288;
        double r53290 = -1.0;
        double r53291 = r53290 / r53283;
        double r53292 = r53289 * r53291;
        double r53293 = 1.7649737847920912e+83;
        bool r53294 = r53274 <= r53293;
        double r53295 = r53287 + r53274;
        double r53296 = r53295 / r53283;
        double r53297 = -r53296;
        double r53298 = 0.5;
        double r53299 = r53298 * r53278;
        double r53300 = r53299 / r53274;
        double r53301 = r53274 / r53283;
        double r53302 = 2.0;
        double r53303 = r53301 * r53302;
        double r53304 = r53300 - r53303;
        double r53305 = r53294 ? r53297 : r53304;
        double r53306 = r53282 ? r53292 : r53305;
        double r53307 = r53276 ? r53280 : r53306;
        return r53307;
}

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 < -9661478263987.111

    1. Initial program 56.4

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

    2. Simplified56.4

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

    3. Taylor expanded around -inf 5.0

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

    if -9661478263987.111 < b_2 < -4.7924462561183435e-145

    1. Initial program 36.3

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

    2. Simplified36.3

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

    4. Applied flip-+36.3

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

    5. Simplified17.4

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

    7. Applied div-inv17.5

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

    if -4.7924462561183435e-145 < b_2 < 1.7649737847920912e+83

    1. Initial program 11.1

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

    2. Simplified11.1

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

    if 1.7649737847920912e+83 < b_2

    1. Initial program 43.7

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

    2. Simplified43.7

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

    3. Taylor expanded around inf 3.6

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

    4. Simplified3.6

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9661478263987.111328125:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.792446256118343513399316360203386878654 \cdot 10^{-145}:\\ \;\;\;\;\frac{c \cdot a}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \frac{-1}{a}\\ \mathbf{elif}\;b_2 \le 1.764973784792091233612894541626891615653 \cdot 10^{83}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))