Average Error: 33.8 → 6.7
Time: 18.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.56950087216670373365855698069146367898 \cdot 10^{75}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le -4.828568422313432590633246300328868217722 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}}\\ \mathbf{elif}\;b \le 3.987267970694484549003039208370469995616 \cdot 10^{133}:\\ \;\;\;\;\frac{4 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot c\right) \cdot a\right)}} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.56950087216670373365855698069146367898 \cdot 10^{75}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le -4.828568422313432590633246300328868217722 \cdot 10^{-262}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}}\\

\mathbf{elif}\;b \le 3.987267970694484549003039208370469995616 \cdot 10^{133}:\\
\;\;\;\;\frac{4 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot c\right) \cdot a\right)}} \cdot \frac{1}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r46275 = b;
        double r46276 = -r46275;
        double r46277 = r46275 * r46275;
        double r46278 = 4.0;
        double r46279 = a;
        double r46280 = r46278 * r46279;
        double r46281 = c;
        double r46282 = r46280 * r46281;
        double r46283 = r46277 - r46282;
        double r46284 = sqrt(r46283);
        double r46285 = r46276 + r46284;
        double r46286 = 2.0;
        double r46287 = r46286 * r46279;
        double r46288 = r46285 / r46287;
        return r46288;
}
↓
double f(double a, double b, double c) {
        double r46289 = b;
        double r46290 = -3.5695008721667037e+75;
        bool r46291 = r46289 <= r46290;
        double r46292 = c;
        double r46293 = r46292 / r46289;
        double r46294 = a;
        double r46295 = r46289 / r46294;
        double r46296 = r46293 - r46295;
        double r46297 = 1.0;
        double r46298 = r46296 * r46297;
        double r46299 = -4.8285684223134326e-262;
        bool r46300 = r46289 <= r46299;
        double r46301 = 1.0;
        double r46302 = 2.0;
        double r46303 = r46302 * r46294;
        double r46304 = r46294 * r46292;
        double r46305 = 4.0;
        double r46306 = -r46305;
        double r46307 = r46289 * r46289;
        double r46308 = fma(r46304, r46306, r46307);
        double r46309 = sqrt(r46308);
        double r46310 = r46309 - r46289;
        double r46311 = r46303 / r46310;
        double r46312 = r46301 / r46311;
        double r46313 = 3.9872679706944845e+133;
        bool r46314 = r46289 <= r46313;
        double r46315 = r46305 * r46292;
        double r46316 = -r46289;
        double r46317 = r46315 * r46294;
        double r46318 = -r46317;
        double r46319 = fma(r46289, r46289, r46318);
        double r46320 = sqrt(r46319);
        double r46321 = r46316 - r46320;
        double r46322 = r46315 / r46321;
        double r46323 = r46301 / r46302;
        double r46324 = r46322 * r46323;
        double r46325 = -1.0;
        double r46326 = r46293 * r46325;
        double r46327 = r46314 ? r46324 : r46326;
        double r46328 = r46300 ? r46312 : r46327;
        double r46329 = r46291 ? r46298 : r46328;
        return r46329;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -3.5695008721667037e+75

    1. Initial program 42.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around -inf 4.0

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

    3. Simplified4.0

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

    if -3.5695008721667037e+75 < b < -4.8285684223134326e-262

    1. Initial program 8.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied clear-num8.7

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\]

    4. Simplified8.7

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}}}\]

    if -4.8285684223134326e-262 < b < 3.9872679706944845e+133

    1. Initial program 32.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+32.1

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    4. Simplified16.6

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(4 \cdot c\right) \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]

    5. Simplified16.6

      \[\leadsto \frac{\frac{0 + \left(4 \cdot c\right) \cdot a}{\color{blue}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity16.6

      \[\leadsto \frac{\frac{0 + \left(4 \cdot c\right) \cdot a}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)}}}{2 \cdot a}\]

    8. Applied *-un-lft-identity16.6

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + \left(4 \cdot c\right) \cdot a\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)}}{2 \cdot a}\]

    9. Applied times-frac16.6

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + \left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}}{2 \cdot a}\]

    10. Applied times-frac16.6

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{0 + \left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{a}}\]

    11. Simplified16.6

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{0 + \left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{a}\]

    12. Simplified15.3

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{\left(4 \cdot c\right) \cdot a}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(4 \cdot c\right)\right)}}}\]
    13. Using strategy rm

    14. Applied *-un-lft-identity15.3

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{\left(4 \cdot c\right) \cdot a}{\color{blue}{1 \cdot a}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(4 \cdot c\right)\right)}}\]

    15. Applied times-frac9.3

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{\frac{4 \cdot c}{1} \cdot \frac{a}{a}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(4 \cdot c\right)\right)}}\]

    16. Simplified9.3

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{\left(4 \cdot c\right)} \cdot \frac{a}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(4 \cdot c\right)\right)}}\]

    17. Simplified9.3

      \[\leadsto \frac{1}{2} \cdot \frac{\left(4 \cdot c\right) \cdot \color{blue}{1}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(4 \cdot c\right)\right)}}\]

    if 3.9872679706944845e+133 < b

    1. Initial program 61.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around inf 1.7

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.56950087216670373365855698069146367898 \cdot 10^{75}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le -4.828568422313432590633246300328868217722 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}}\\ \mathbf{elif}\;b \le 3.987267970694484549003039208370469995616 \cdot 10^{133}:\\ \;\;\;\;\frac{4 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot c\right) \cdot a\right)}} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))