Average Error: 33.3 → 10.3
Time: 49.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.373729884330869 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{3}{2}, \left(\frac{a}{\frac{b}{c}}\right), \left(b \cdot -2\right)\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 5.746416486233371 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -3, \left(b \cdot b\right)\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.373729884330869 \cdot 10^{+112}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{3}{2}, \left(\frac{a}{\frac{b}{c}}\right), \left(b \cdot -2\right)\right)}{3 \cdot a}\\

\mathbf{elif}\;b \le 5.746416486233371 \cdot 10^{-62}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -3, \left(b \cdot b\right)\right)} + \left(-b\right)}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r13001150 = b;
        double r13001151 = -r13001150;
        double r13001152 = r13001150 * r13001150;
        double r13001153 = 3.0;
        double r13001154 = a;
        double r13001155 = r13001153 * r13001154;
        double r13001156 = c;
        double r13001157 = r13001155 * r13001156;
        double r13001158 = r13001152 - r13001157;
        double r13001159 = sqrt(r13001158);
        double r13001160 = r13001151 + r13001159;
        double r13001161 = r13001160 / r13001155;
        return r13001161;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r13001162 = b;
        double r13001163 = -8.373729884330869e+112;
        bool r13001164 = r13001162 <= r13001163;
        double r13001165 = 1.5;
        double r13001166 = a;
        double r13001167 = c;
        double r13001168 = r13001162 / r13001167;
        double r13001169 = r13001166 / r13001168;
        double r13001170 = -2.0;
        double r13001171 = r13001162 * r13001170;
        double r13001172 = fma(r13001165, r13001169, r13001171);
        double r13001173 = 3.0;
        double r13001174 = r13001173 * r13001166;
        double r13001175 = r13001172 / r13001174;
        double r13001176 = 5.746416486233371e-62;
        bool r13001177 = r13001162 <= r13001176;
        double r13001178 = r13001166 * r13001167;
        double r13001179 = -3.0;
        double r13001180 = r13001162 * r13001162;
        double r13001181 = fma(r13001178, r13001179, r13001180);
        double r13001182 = sqrt(r13001181);
        double r13001183 = -r13001162;
        double r13001184 = r13001182 + r13001183;
        double r13001185 = r13001184 / r13001173;
        double r13001186 = r13001185 / r13001166;
        double r13001187 = -0.5;
        double r13001188 = r13001167 / r13001162;
        double r13001189 = r13001187 * r13001188;
        double r13001190 = r13001177 ? r13001186 : r13001189;
        double r13001191 = r13001164 ? r13001175 : r13001190;
        return r13001191;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < -8.373729884330869e+112

    1. Initial program 46.6

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

    2. Taylor expanded around -inf 9.4

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

    3. Simplified3.6

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

    if -8.373729884330869e+112 < b < 5.746416486233371e-62

    1. Initial program 13.5

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

    3. Applied associate-/r*13.6

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

    4. Taylor expanded around -inf 13.5

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

    5. Simplified13.5

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

    if 5.746416486233371e-62 < b

    1. Initial program 52.8

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

    3. Applied associate-/r*52.8

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

    4. Taylor expanded around -inf 52.8

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

    5. Simplified52.8

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

    6. Taylor expanded around inf 8.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.373729884330869 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{3}{2}, \left(\frac{a}{\frac{b}{c}}\right), \left(b \cdot -2\right)\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 5.746416486233371 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -3, \left(b \cdot b\right)\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))