Average Error: 33.4 → 15.3
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.919145962379566 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) + \left(-b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.585954367743496 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3 \cdot a}\\ \end{array}\]
\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 -8.919145962379566 \cdot 10^{+111}:\\
\;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) + \left(-b\right)}{3 \cdot a}\\

\mathbf{elif}\;b \le 2.585954367743496 \cdot 10^{-66}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} - b}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1740303 = b;
        double r1740304 = -r1740303;
        double r1740305 = r1740303 * r1740303;
        double r1740306 = 3.0;
        double r1740307 = a;
        double r1740308 = r1740306 * r1740307;
        double r1740309 = c;
        double r1740310 = r1740308 * r1740309;
        double r1740311 = r1740305 - r1740310;
        double r1740312 = sqrt(r1740311);
        double r1740313 = r1740304 + r1740312;
        double r1740314 = r1740313 / r1740308;
        return r1740314;
}
↓
double f(double a, double b, double c) {
        double r1740315 = b;
        double r1740316 = -8.919145962379566e+111;
        bool r1740317 = r1740315 <= r1740316;
        double r1740318 = 1.5;
        double r1740319 = a;
        double r1740320 = c;
        double r1740321 = r1740319 * r1740320;
        double r1740322 = r1740321 / r1740315;
        double r1740323 = r1740318 * r1740322;
        double r1740324 = r1740323 - r1740315;
        double r1740325 = -r1740315;
        double r1740326 = r1740324 + r1740325;
        double r1740327 = 3.0;
        double r1740328 = r1740327 * r1740319;
        double r1740329 = r1740326 / r1740328;
        double r1740330 = 2.585954367743496e-66;
        bool r1740331 = r1740315 <= r1740330;
        double r1740332 = -3.0;
        double r1740333 = r1740315 * r1740315;
        double r1740334 = fma(r1740332, r1740321, r1740333);
        double r1740335 = sqrt(r1740334);
        double r1740336 = r1740335 - r1740315;
        double r1740337 = r1740336 / r1740327;
        double r1740338 = r1740337 / r1740319;
        double r1740339 = -1.5;
        double r1740340 = r1740322 * r1740339;
        double r1740341 = r1740340 / r1740328;
        double r1740342 = r1740331 ? r1740338 : r1740341;
        double r1740343 = r1740317 ? r1740329 : r1740342;
        return r1740343;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -8.919145962379566e+111

    1. Initial program 47.4

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

    2. Taylor expanded around -inf 9.9

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

    if -8.919145962379566e+111 < b < 2.585954367743496e-66

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied associate-/r*13.2

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

    if 2.585954367743496e-66 < b

    1. Initial program 52.7

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

    2. Taylor expanded around inf 19.8

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.919145962379566 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) + \left(-b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.585954367743496 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))