Average Error: 32.9 → 14.1
Time: 18.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3599228730895225 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, \frac{\frac{3}{2} \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.602826357305206 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} + \left(-b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{c \cdot a}{b}}{a \cdot 3}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.3599228730895225 \cdot 10^{+90}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-2, b, \frac{\frac{3}{2} \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\

\mathbf{elif}\;b \le 3.602826357305206 \cdot 10^{-72}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} + \left(-b\right)}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-3}{2} \cdot \frac{c \cdot a}{b}}{a \cdot 3}\\

\end{array}
double f(double a, double b, double c) {
        double r3256253 = b;
        double r3256254 = -r3256253;
        double r3256255 = r3256253 * r3256253;
        double r3256256 = 3.0;
        double r3256257 = a;
        double r3256258 = r3256256 * r3256257;
        double r3256259 = c;
        double r3256260 = r3256258 * r3256259;
        double r3256261 = r3256255 - r3256260;
        double r3256262 = sqrt(r3256261);
        double r3256263 = r3256254 + r3256262;
        double r3256264 = r3256263 / r3256258;
        return r3256264;
}
↓
double f(double a, double b, double c) {
        double r3256265 = b;
        double r3256266 = -1.3599228730895225e+90;
        bool r3256267 = r3256265 <= r3256266;
        double r3256268 = -2.0;
        double r3256269 = 1.5;
        double r3256270 = a;
        double r3256271 = r3256269 * r3256270;
        double r3256272 = c;
        double r3256273 = r3256265 / r3256272;
        double r3256274 = r3256271 / r3256273;
        double r3256275 = fma(r3256268, r3256265, r3256274);
        double r3256276 = 3.0;
        double r3256277 = r3256270 * r3256276;
        double r3256278 = r3256275 / r3256277;
        double r3256279 = 3.602826357305206e-72;
        bool r3256280 = r3256265 <= r3256279;
        double r3256281 = r3256272 * r3256270;
        double r3256282 = -3.0;
        double r3256283 = r3256281 * r3256282;
        double r3256284 = fma(r3256265, r3256265, r3256283);
        double r3256285 = sqrt(r3256284);
        double r3256286 = -r3256265;
        double r3256287 = r3256285 + r3256286;
        double r3256288 = r3256287 / r3256277;
        double r3256289 = -1.5;
        double r3256290 = r3256281 / r3256265;
        double r3256291 = r3256289 * r3256290;
        double r3256292 = r3256291 / r3256277;
        double r3256293 = r3256280 ? r3256288 : r3256292;
        double r3256294 = r3256267 ? r3256278 : r3256293;
        return r3256294;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3599228730895225e+90

    1. Initial program 41.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around 0 41.8

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]

    3. Simplified41.8

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a}\]

    4. Taylor expanded around -inf 9.6

      \[\leadsto \frac{\color{blue}{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3 \cdot a}\]

    5. Simplified4.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, b, \frac{\frac{3}{2} \cdot a}{\frac{b}{c}}\right)}}{3 \cdot a}\]

    if -1.3599228730895225e+90 < b < 3.602826357305206e-72

    1. Initial program 12.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around 0 13.0

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]

    3. Simplified13.0

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a}\]

    if 3.602826357305206e-72 < b

    1. Initial program 52.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around inf 19.7

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3599228730895225 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, \frac{\frac{3}{2} \cdot a}{\frac{b}{c}}\right)}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.602826357305206 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} + \left(-b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{c \cdot a}{b}}{a \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))