Average Error: 33.0 → 10.8
Time: 22.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\

\mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r842294 = b_2;
        double r842295 = -r842294;
        double r842296 = r842294 * r842294;
        double r842297 = a;
        double r842298 = c;
        double r842299 = r842297 * r842298;
        double r842300 = r842296 - r842299;
        double r842301 = sqrt(r842300);
        double r842302 = r842295 + r842301;
        double r842303 = r842302 / r842297;
        return r842303;
}
↓
double f(double a, double b_2, double c) {
        double r842304 = b_2;
        double r842305 = -9.348931433494438e+39;
        bool r842306 = r842304 <= r842305;
        double r842307 = 0.5;
        double r842308 = c;
        double r842309 = r842308 / r842304;
        double r842310 = r842307 * r842309;
        double r842311 = a;
        double r842312 = r842304 / r842311;
        double r842313 = 2.0;
        double r842314 = r842312 * r842313;
        double r842315 = r842310 - r842314;
        double r842316 = 1.3353078790738604e-121;
        bool r842317 = r842304 <= r842316;
        double r842318 = 1.0;
        double r842319 = r842318 / r842311;
        double r842320 = r842304 * r842304;
        double r842321 = r842308 * r842311;
        double r842322 = r842320 - r842321;
        double r842323 = sqrt(r842322);
        double r842324 = r842323 - r842304;
        double r842325 = r842319 * r842324;
        double r842326 = 1.6168702840263923e-79;
        bool r842327 = r842304 <= r842326;
        double r842328 = -0.5;
        double r842329 = r842328 * r842309;
        double r842330 = 1.546013236023957e-67;
        bool r842331 = r842304 <= r842330;
        double r842332 = r842331 ? r842325 : r842329;
        double r842333 = r842327 ? r842329 : r842332;
        double r842334 = r842317 ? r842325 : r842333;
        double r842335 = r842306 ? r842315 : r842334;
        return r842335;
}

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 < -9.348931433494438e+39

    1. Initial program 34.0

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

    2. Simplified34.0

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

    3. Taylor expanded around -inf 6.2

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

    if -9.348931433494438e+39 < b_2 < 1.3353078790738604e-121 or 1.6168702840263923e-79 < b_2 < 1.546013236023957e-67

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied div-inv13.0

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

    if 1.3353078790738604e-121 < b_2 < 1.6168702840263923e-79 or 1.546013236023957e-67 < b_2

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 11.2

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))