Average Error: 33.2 → 10.0
Time: 21.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, \frac{3}{2}, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -7.397994825724217 \cdot 10^{+150}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, \frac{3}{2}, b \cdot -2\right)}{a \cdot 3}\\

\mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r3447346 = b;
        double r3447347 = -r3447346;
        double r3447348 = r3447346 * r3447346;
        double r3447349 = 3.0;
        double r3447350 = a;
        double r3447351 = r3447349 * r3447350;
        double r3447352 = c;
        double r3447353 = r3447351 * r3447352;
        double r3447354 = r3447348 - r3447353;
        double r3447355 = sqrt(r3447354);
        double r3447356 = r3447347 + r3447355;
        double r3447357 = r3447356 / r3447351;
        return r3447357;
}
↓
double f(double a, double b, double c) {
        double r3447358 = b;
        double r3447359 = -7.397994825724217e+150;
        bool r3447360 = r3447358 <= r3447359;
        double r3447361 = a;
        double r3447362 = c;
        double r3447363 = r3447358 / r3447362;
        double r3447364 = r3447361 / r3447363;
        double r3447365 = 1.5;
        double r3447366 = -2.0;
        double r3447367 = r3447358 * r3447366;
        double r3447368 = fma(r3447364, r3447365, r3447367);
        double r3447369 = 3.0;
        double r3447370 = r3447361 * r3447369;
        double r3447371 = r3447368 / r3447370;
        double r3447372 = 1.2158870426682226e-82;
        bool r3447373 = r3447358 <= r3447372;
        double r3447374 = -3.0;
        double r3447375 = r3447361 * r3447374;
        double r3447376 = r3447358 * r3447358;
        double r3447377 = fma(r3447375, r3447362, r3447376);
        double r3447378 = sqrt(r3447377);
        double r3447379 = r3447378 - r3447358;
        double r3447380 = r3447379 / r3447369;
        double r3447381 = r3447380 / r3447361;
        double r3447382 = -0.5;
        double r3447383 = r3447362 / r3447358;
        double r3447384 = r3447382 * r3447383;
        double r3447385 = r3447373 ? r3447381 : r3447384;
        double r3447386 = r3447360 ? r3447371 : r3447385;
        return r3447386;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -7.397994825724217e+150

    1. Initial program 59.1

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

    2. Taylor expanded around -inf 11.5

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

    3. Simplified2.5

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

    if -7.397994825724217e+150 < b < 1.2158870426682226e-82

    1. Initial program 11.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*11.9

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

    4. Simplified11.9

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

    if 1.2158870426682226e-82 < 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 9.9

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, \frac{3}{2}, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))