Average Error: 34.6 → 9.3
Time: 7.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -5.331911540015065182156270674462636313069 \cdot 10^{-301}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.493919351930657503244813394241198531074 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -5.331911540015065182156270674462636313069 \cdot 10^{-301}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 1.493919351930657503244813394241198531074 \cdot 10^{-69}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r28284 = b_2;
        double r28285 = -r28284;
        double r28286 = r28284 * r28284;
        double r28287 = a;
        double r28288 = c;
        double r28289 = r28287 * r28288;
        double r28290 = r28286 - r28289;
        double r28291 = sqrt(r28290);
        double r28292 = r28285 + r28291;
        double r28293 = r28292 / r28287;
        return r28293;
}
↓
double f(double a, double b_2, double c) {
        double r28294 = b_2;
        double r28295 = -6.371698442415157e+150;
        bool r28296 = r28294 <= r28295;
        double r28297 = 0.5;
        double r28298 = c;
        double r28299 = r28298 / r28294;
        double r28300 = r28297 * r28299;
        double r28301 = 2.0;
        double r28302 = a;
        double r28303 = r28294 / r28302;
        double r28304 = r28301 * r28303;
        double r28305 = r28300 - r28304;
        double r28306 = -5.331911540015065e-301;
        bool r28307 = r28294 <= r28306;
        double r28308 = -r28294;
        double r28309 = r28294 * r28294;
        double r28310 = r28302 * r28298;
        double r28311 = r28309 - r28310;
        double r28312 = sqrt(r28311);
        double r28313 = r28308 + r28312;
        double r28314 = 1.0;
        double r28315 = r28314 / r28302;
        double r28316 = r28313 * r28315;
        double r28317 = 1.4939193519306575e-69;
        bool r28318 = r28294 <= r28317;
        double r28319 = r28308 - r28312;
        double r28320 = r28319 / r28302;
        double r28321 = r28320 / r28298;
        double r28322 = r28314 / r28321;
        double r28323 = r28322 / r28302;
        double r28324 = -0.5;
        double r28325 = r28324 * r28299;
        double r28326 = r28318 ? r28323 : r28325;
        double r28327 = r28307 ? r28316 : r28326;
        double r28328 = r28296 ? r28305 : r28327;
        return r28328;
}

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 < -6.371698442415157e+150

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 2.5

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

    if -6.371698442415157e+150 < b_2 < -5.331911540015065e-301

    1. Initial program 9.0

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

    3. Applied div-inv9.1

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

    if -5.331911540015065e-301 < b_2 < 1.4939193519306575e-69

    1. Initial program 21.6

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

    3. Applied flip-+21.7

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

    4. Simplified18.1

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied clear-num18.2

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

    7. Simplified14.6

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

    if 1.4939193519306575e-69 < b_2

    1. Initial program 53.7

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

    2. Taylor expanded around inf 9.5

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -5.331911540015065182156270674462636313069 \cdot 10^{-301}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.493919351930657503244813394241198531074 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))