Average Error: 52.6 → 5.7
Time: 14.6s
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \le -6345072.402882894:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right) \cdot \left(\left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right) \cdot \left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \le -6345072.402882894:\\
\;\;\;\;\frac{\sqrt[3]{\left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right) \cdot \left(\left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right) \cdot \left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right)\right)}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2234297 = b;
        double r2234298 = -r2234297;
        double r2234299 = r2234297 * r2234297;
        double r2234300 = 3.0;
        double r2234301 = a;
        double r2234302 = r2234300 * r2234301;
        double r2234303 = c;
        double r2234304 = r2234302 * r2234303;
        double r2234305 = r2234299 - r2234304;
        double r2234306 = sqrt(r2234305);
        double r2234307 = r2234298 + r2234306;
        double r2234308 = r2234307 / r2234302;
        return r2234308;
}
↓
double f(double a, double b, double c) {
        double r2234309 = b;
        double r2234310 = r2234309 * r2234309;
        double r2234311 = 3.0;
        double r2234312 = a;
        double r2234313 = r2234311 * r2234312;
        double r2234314 = c;
        double r2234315 = r2234313 * r2234314;
        double r2234316 = r2234310 - r2234315;
        double r2234317 = sqrt(r2234316);
        double r2234318 = -r2234309;
        double r2234319 = r2234317 + r2234318;
        double r2234320 = r2234319 / r2234313;
        double r2234321 = -6345072.402882894;
        bool r2234322 = r2234320 <= r2234321;
        double r2234323 = -3.0;
        double r2234324 = r2234323 * r2234312;
        double r2234325 = fma(r2234314, r2234324, r2234310);
        double r2234326 = sqrt(r2234325);
        double r2234327 = r2234326 - r2234309;
        double r2234328 = r2234327 * r2234327;
        double r2234329 = r2234327 * r2234328;
        double r2234330 = cbrt(r2234329);
        double r2234331 = r2234330 / r2234313;
        double r2234332 = r2234314 / r2234309;
        double r2234333 = -0.5;
        double r2234334 = r2234332 * r2234333;
        double r2234335 = r2234322 ? r2234331 : r2234334;
        return r2234335;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)) < -6345072.402882894

    1. Initial program 21.1

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

    3. Applied add-cbrt-cube21.1

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

    4. Simplified21.1

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

    if -6345072.402882894 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a))

    1. Initial program 54.4

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

    2. Taylor expanded around inf 4.8

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \le -6345072.402882894:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right) \cdot \left(\left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right) \cdot \left(\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} - b\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))