Average Error: 33.2 → 10.1
Time: 19.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.4670828746192654 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot \frac{a}{b}, \frac{3}{2}, b \cdot -2\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 4.7587539095277834 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} + \left(-b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -6.4670828746192654 \cdot 10^{+153}:\\
\;\;\;\;\frac{\mathsf{fma}\left(c \cdot \frac{a}{b}, \frac{3}{2}, b \cdot -2\right)}{3 \cdot a}\\

\mathbf{elif}\;b \le 4.7587539095277834 \cdot 10^{-110}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} + \left(-b\right)}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3310233 = b;
        double r3310234 = -r3310233;
        double r3310235 = r3310233 * r3310233;
        double r3310236 = 3.0;
        double r3310237 = a;
        double r3310238 = r3310236 * r3310237;
        double r3310239 = c;
        double r3310240 = r3310238 * r3310239;
        double r3310241 = r3310235 - r3310240;
        double r3310242 = sqrt(r3310241);
        double r3310243 = r3310234 + r3310242;
        double r3310244 = r3310243 / r3310238;
        return r3310244;
}
↓
double f(double a, double b, double c) {
        double r3310245 = b;
        double r3310246 = -6.4670828746192654e+153;
        bool r3310247 = r3310245 <= r3310246;
        double r3310248 = c;
        double r3310249 = a;
        double r3310250 = r3310249 / r3310245;
        double r3310251 = r3310248 * r3310250;
        double r3310252 = 1.5;
        double r3310253 = -2.0;
        double r3310254 = r3310245 * r3310253;
        double r3310255 = fma(r3310251, r3310252, r3310254);
        double r3310256 = 3.0;
        double r3310257 = r3310256 * r3310249;
        double r3310258 = r3310255 / r3310257;
        double r3310259 = 4.7587539095277834e-110;
        bool r3310260 = r3310245 <= r3310259;
        double r3310261 = r3310248 * r3310249;
        double r3310262 = -3.0;
        double r3310263 = r3310261 * r3310262;
        double r3310264 = fma(r3310245, r3310245, r3310263);
        double r3310265 = sqrt(r3310264);
        double r3310266 = -r3310245;
        double r3310267 = r3310265 + r3310266;
        double r3310268 = r3310267 / r3310257;
        double r3310269 = -0.5;
        double r3310270 = r3310248 / r3310245;
        double r3310271 = r3310269 * r3310270;
        double r3310272 = r3310260 ? r3310268 : r3310271;
        double r3310273 = r3310247 ? r3310258 : r3310272;
        return r3310273;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -6.4670828746192654e+153

    1. Initial program 60.7

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

    2. Taylor expanded around -inf 11.7

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

    3. Simplified2.6

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

    if -6.4670828746192654e+153 < b < 4.7587539095277834e-110

    1. Initial program 11.0

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

    2. Taylor expanded around -inf 11.1

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

    3. Simplified11.1

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

    if 4.7587539095277834e-110 < b

    1. Initial program 51.0

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

    2. Taylor expanded around inf 10.8

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.4670828746192654 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot \frac{a}{b}, \frac{3}{2}, b \cdot -2\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 4.7587539095277834 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} + \left(-b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))