Average Error: 18.6 → 6.1
Time: 20.1s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6955952414049898 \cdot 10^{+140}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 1.0278279293485846 \cdot 10^{+87}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(-2, \left(\frac{a}{\frac{b}{c}}\right), b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]
\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 -1.6955952414049898 \cdot 10^{+140}:\\
\;\;\;\;\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 1.0278279293485846 \cdot 10^{+87}:\\
\;\;\;\;\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{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(-2, \left(\frac{a}{\frac{b}{c}}\right), 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 r661274 = b;
        double r661275 = 0.0;
        bool r661276 = r661274 >= r661275;
        double r661277 = 2.0;
        double r661278 = c;
        double r661279 = r661277 * r661278;
        double r661280 = -r661274;
        double r661281 = r661274 * r661274;
        double r661282 = 4.0;
        double r661283 = a;
        double r661284 = r661282 * r661283;
        double r661285 = r661284 * r661278;
        double r661286 = r661281 - r661285;
        double r661287 = sqrt(r661286);
        double r661288 = r661280 - r661287;
        double r661289 = r661279 / r661288;
        double r661290 = r661280 + r661287;
        double r661291 = r661277 * r661283;
        double r661292 = r661290 / r661291;
        double r661293 = r661276 ? r661289 : r661292;
        return r661293;
}
↓
double f(double a, double b, double c) {
        double r661294 = b;
        double r661295 = -1.6955952414049898e+140;
        bool r661296 = r661294 <= r661295;
        double r661297 = 0.0;
        bool r661298 = r661294 >= r661297;
        double r661299 = 2.0;
        double r661300 = c;
        double r661301 = r661299 * r661300;
        double r661302 = -r661294;
        double r661303 = r661294 * r661294;
        double r661304 = 4.0;
        double r661305 = a;
        double r661306 = r661304 * r661305;
        double r661307 = r661306 * r661300;
        double r661308 = r661303 - r661307;
        double r661309 = sqrt(r661308);
        double r661310 = r661302 - r661309;
        double r661311 = r661301 / r661310;
        double r661312 = r661294 / r661305;
        double r661313 = r661300 / r661312;
        double r661314 = r661313 - r661294;
        double r661315 = r661299 * r661314;
        double r661316 = r661299 * r661305;
        double r661317 = r661315 / r661316;
        double r661318 = r661298 ? r661311 : r661317;
        double r661319 = 1.0278279293485846e+87;
        bool r661320 = r661294 <= r661319;
        double r661321 = sqrt(r661309);
        double r661322 = r661321 * r661321;
        double r661323 = r661302 + r661322;
        double r661324 = r661323 / r661316;
        double r661325 = r661298 ? r661311 : r661324;
        double r661326 = -2.0;
        double r661327 = r661294 / r661300;
        double r661328 = r661305 / r661327;
        double r661329 = fma(r661326, r661328, r661294);
        double r661330 = r661302 - r661329;
        double r661331 = r661301 / r661330;
        double r661332 = r661309 + r661302;
        double r661333 = r661332 / r661316;
        double r661334 = r661298 ? r661331 : r661333;
        double r661335 = r661320 ? r661325 : r661334;
        double r661336 = r661296 ? r661318 : r661335;
        return r661336;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.6955952414049898e+140

    1. Initial program 55.7

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

    3. Applied add-sqr-sqrt55.8

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

    4. Taylor expanded around -inf 11.1

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

    5. Simplified2.4

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

    if -1.6955952414049898e+140 < b < 1.0278279293485846e+87

    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

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

    if 1.0278279293485846e+87 < b

    1. Initial program 26.9

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

    3. Applied prod-diff27.0

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

    4. Taylor expanded around inf 6.1

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

    5. Simplified2.3

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6955952414049898 \cdot 10^{+140}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 1.0278279293485846 \cdot 10^{+87}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(-2, \left(\frac{a}{\frac{b}{c}}\right), b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +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))))