Average Error: 19.0 → 6.5
Time: 23.5s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.942920241018838 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\left(-\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right), \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right), \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(\left(\sqrt{b}\right), \left(-\sqrt{b}\right), \left(\left(-\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \le 3.942920241018838 \cdot 10^{+125}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\left(-\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right), \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right), \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(\left(\sqrt{b}\right), \left(-\sqrt{b}\right), \left(\left(-\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r633276 = b;
        double r633277 = 0.0;
        bool r633278 = r633276 >= r633277;
        double r633279 = 2.0;
        double r633280 = c;
        double r633281 = r633279 * r633280;
        double r633282 = -r633276;
        double r633283 = r633276 * r633276;
        double r633284 = 4.0;
        double r633285 = a;
        double r633286 = r633284 * r633285;
        double r633287 = r633286 * r633280;
        double r633288 = r633283 - r633287;
        double r633289 = sqrt(r633288);
        double r633290 = r633282 - r633289;
        double r633291 = r633281 / r633290;
        double r633292 = r633282 + r633289;
        double r633293 = r633279 * r633285;
        double r633294 = r633292 / r633293;
        double r633295 = r633278 ? r633291 : r633294;
        return r633295;
}
↓
double f(double a, double b, double c) {
        double r633296 = b;
        double r633297 = -3.794505329565205e+146;
        bool r633298 = r633296 <= r633297;
        double r633299 = 0.0;
        bool r633300 = r633296 >= r633299;
        double r633301 = 2.0;
        double r633302 = c;
        double r633303 = r633301 * r633302;
        double r633304 = -r633296;
        double r633305 = r633296 * r633296;
        double r633306 = 4.0;
        double r633307 = a;
        double r633308 = r633306 * r633307;
        double r633309 = r633308 * r633302;
        double r633310 = r633305 - r633309;
        double r633311 = sqrt(r633310);
        double r633312 = r633304 - r633311;
        double r633313 = r633303 / r633312;
        double r633314 = r633296 / r633307;
        double r633315 = r633302 / r633314;
        double r633316 = r633315 - r633296;
        double r633317 = r633301 * r633316;
        double r633318 = r633301 * r633307;
        double r633319 = r633317 / r633318;
        double r633320 = r633300 ? r633313 : r633319;
        double r633321 = 3.942920241018838e+125;
        bool r633322 = r633296 <= r633321;
        double r633323 = sqrt(r633311);
        double r633324 = -r633323;
        double r633325 = r633323 * r633323;
        double r633326 = fma(r633324, r633323, r633325);
        double r633327 = sqrt(r633296);
        double r633328 = -r633327;
        double r633329 = r633324 * r633323;
        double r633330 = fma(r633327, r633328, r633329);
        double r633331 = r633326 + r633330;
        double r633332 = r633303 / r633331;
        double r633333 = r633311 + r633304;
        double r633334 = r633333 / r633318;
        double r633335 = r633300 ? r633332 : r633334;
        double r633336 = r633303 / r633317;
        double r633337 = r633300 ? r633336 : r633334;
        double r633338 = r633322 ? r633335 : r633337;
        double r633339 = r633298 ? r633320 : r633338;
        return r633339;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -3.794505329565205e+146

    1. Initial program 58.0

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-exp-log58.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\\ \end{array}\]

    4. Taylor expanded around -inf 11.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    5. Simplified3.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2 \cdot a}\\ \end{array}\]

    if -3.794505329565205e+146 < b < 3.942920241018838e+125

    1. Initial program 8.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt8.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    4. Applied add-sqr-sqrt8.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    5. Applied distribute-rgt-neg-in8.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\sqrt{b} \cdot \left(-\sqrt{b}\right)} - \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    6. Applied prod-diff8.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(\left(\sqrt{b}\right), \left(-\sqrt{b}\right), \left(-\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(\left(-\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right), \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right), \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    if 3.942920241018838e+125 < b

    1. Initial program 32.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 6.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    3. Simplified2.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.942920241018838 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\left(-\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right), \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right), \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(\left(\sqrt{b}\right), \left(-\sqrt{b}\right), \left(\left(-\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))