Average Error: 33.9 → 10.0
Time: 5.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.7237121667270365 \cdot 10^{113}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.51349304478110011 \cdot 10^{-103}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.7237121667270365 \cdot 10^{113}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.51349304478110011 \cdot 10^{-103}:\\
\;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r16253 = b_2;
        double r16254 = -r16253;
        double r16255 = r16253 * r16253;
        double r16256 = a;
        double r16257 = c;
        double r16258 = r16256 * r16257;
        double r16259 = r16255 - r16258;
        double r16260 = sqrt(r16259);
        double r16261 = r16254 - r16260;
        double r16262 = r16261 / r16256;
        return r16262;
}
↓
double f(double a, double b_2, double c) {
        double r16263 = b_2;
        double r16264 = -8.723712166727036e+113;
        bool r16265 = r16263 <= r16264;
        double r16266 = -0.5;
        double r16267 = c;
        double r16268 = r16267 / r16263;
        double r16269 = r16266 * r16268;
        double r16270 = 1.5134930447811e-103;
        bool r16271 = r16263 <= r16270;
        double r16272 = 1.0;
        double r16273 = r16272 / r16267;
        double r16274 = r16263 * r16263;
        double r16275 = a;
        double r16276 = r16275 * r16267;
        double r16277 = r16274 - r16276;
        double r16278 = sqrt(r16277);
        double r16279 = r16278 - r16263;
        double r16280 = r16273 * r16279;
        double r16281 = r16272 / r16280;
        double r16282 = 0.5;
        double r16283 = r16282 * r16268;
        double r16284 = 2.0;
        double r16285 = r16263 / r16275;
        double r16286 = r16284 * r16285;
        double r16287 = r16283 - r16286;
        double r16288 = r16271 ? r16281 : r16287;
        double r16289 = r16265 ? r16269 : r16288;
        return r16289;
}

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 < -8.723712166727036e+113

    1. Initial program 60.6

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

    2. Taylor expanded around -inf 2.2

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

    if -8.723712166727036e+113 < b_2 < 1.5134930447811e-103

    1. Initial program 26.8

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

    3. Applied flip--28.5

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

    4. Simplified17.0

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

    5. Simplified17.0

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

    7. Applied *-un-lft-identity17.0

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

    8. Applied *-un-lft-identity17.0

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

    9. Applied times-frac17.0

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

    10. Applied associate-/l*17.2

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

    11. Simplified16.4

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

    13. Applied clear-num16.4

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

    14. Simplified12.1

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

    if 1.5134930447811e-103 < b_2

    1. Initial program 25.7

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

    2. Taylor expanded around inf 12.4

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.7237121667270365 \cdot 10^{113}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.51349304478110011 \cdot 10^{-103}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))