Average Error: 34.5 → 8.9
Time: 5.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.72893889301538444 \cdot 10^{27}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -9.19851418750357702 \cdot 10^{-275}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.174676155214135 \cdot 10^{112}:\\ \;\;\;\;\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.72893889301538444 \cdot 10^{27}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -9.19851418750357702 \cdot 10^{-275}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 5.174676155214135 \cdot 10^{112}:\\
\;\;\;\;\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}} \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r19242 = b_2;
        double r19243 = -r19242;
        double r19244 = r19242 * r19242;
        double r19245 = a;
        double r19246 = c;
        double r19247 = r19245 * r19246;
        double r19248 = r19244 - r19247;
        double r19249 = sqrt(r19248);
        double r19250 = r19243 + r19249;
        double r19251 = r19250 / r19245;
        return r19251;
}
↓
double f(double a, double b_2, double c) {
        double r19252 = b_2;
        double r19253 = -1.7289388930153844e+27;
        bool r19254 = r19252 <= r19253;
        double r19255 = 0.5;
        double r19256 = c;
        double r19257 = r19256 / r19252;
        double r19258 = r19255 * r19257;
        double r19259 = 2.0;
        double r19260 = a;
        double r19261 = r19252 / r19260;
        double r19262 = r19259 * r19261;
        double r19263 = r19258 - r19262;
        double r19264 = -9.198514187503577e-275;
        bool r19265 = r19252 <= r19264;
        double r19266 = -r19252;
        double r19267 = r19252 * r19252;
        double r19268 = r19260 * r19256;
        double r19269 = r19267 - r19268;
        double r19270 = sqrt(r19269);
        double r19271 = r19266 + r19270;
        double r19272 = 1.0;
        double r19273 = r19272 / r19260;
        double r19274 = r19271 * r19273;
        double r19275 = 5.174676155214135e+112;
        bool r19276 = r19252 <= r19275;
        double r19277 = r19266 - r19270;
        double r19278 = r19277 / r19256;
        double r19279 = r19260 / r19278;
        double r19280 = r19279 * r19273;
        double r19281 = -0.5;
        double r19282 = r19281 * r19257;
        double r19283 = r19276 ? r19280 : r19282;
        double r19284 = r19265 ? r19274 : r19283;
        double r19285 = r19254 ? r19263 : r19284;
        return r19285;
}

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 < -1.7289388930153844e+27

    1. Initial program 35.3

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

    2. Taylor expanded around -inf 6.6

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

    if -1.7289388930153844e+27 < b_2 < -9.198514187503577e-275

    1. Initial program 9.8

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

    3. Applied div-inv9.9

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

    if -9.198514187503577e-275 < b_2 < 5.174676155214135e+112

    1. Initial program 32.0

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

    3. Applied flip-+32.0

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

    4. Simplified16.7

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

    6. Applied div-inv16.8

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

    8. Applied *-un-lft-identity16.8

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

    9. Applied *-un-lft-identity16.8

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

    10. Applied times-frac16.8

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

    11. Applied associate-*r*16.8

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

    12. Simplified14.6

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

    if 5.174676155214135e+112 < b_2

    1. Initial program 60.3

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.72893889301538444 \cdot 10^{27}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -9.19851418750357702 \cdot 10^{-275}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.174676155214135 \cdot 10^{112}:\\ \;\;\;\;\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))