Average Error: 34.2 → 10.4
Time: 11.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r15283 = b_2;
        double r15284 = -r15283;
        double r15285 = r15283 * r15283;
        double r15286 = a;
        double r15287 = c;
        double r15288 = r15286 * r15287;
        double r15289 = r15285 - r15288;
        double r15290 = sqrt(r15289);
        double r15291 = r15284 + r15290;
        double r15292 = r15291 / r15286;
        return r15292;
}
↓
double f(double a, double b_2, double c) {
        double r15293 = b_2;
        double r15294 = -4.123103533644211e+95;
        bool r15295 = r15293 <= r15294;
        double r15296 = 0.5;
        double r15297 = c;
        double r15298 = r15297 / r15293;
        double r15299 = r15296 * r15298;
        double r15300 = 2.0;
        double r15301 = a;
        double r15302 = r15293 / r15301;
        double r15303 = r15300 * r15302;
        double r15304 = r15299 - r15303;
        double r15305 = 3.446447862996811e-75;
        bool r15306 = r15293 <= r15305;
        double r15307 = 1.0;
        double r15308 = r15293 * r15293;
        double r15309 = r15301 * r15297;
        double r15310 = r15308 - r15309;
        double r15311 = sqrt(r15310);
        double r15312 = r15311 - r15293;
        double r15313 = r15301 / r15312;
        double r15314 = r15307 / r15313;
        double r15315 = -0.5;
        double r15316 = r15315 * r15298;
        double r15317 = r15306 ? r15314 : r15316;
        double r15318 = r15295 ? r15304 : r15317;
        return r15318;
}

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 < -4.123103533644211e+95

    1. Initial program 47.3

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

    2. Simplified47.3

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

    3. Taylor expanded around -inf 3.8

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

    if -4.123103533644211e+95 < b_2 < 3.446447862996811e-75

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied clear-num13.4

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity13.4

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

    if 3.446447862996811e-75 < b_2

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 9.7

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))