Average Error: 33.3 → 14.0
Time: 35.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.2400532982587204 \cdot 10^{+146}:\\ \;\;\;\;\frac{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(b \cdot -2\right))_*}{3 \cdot a}\\ \mathbf{elif}\;b \le 5.673337904073483 \cdot 10^{-43}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3 \cdot a}\\ \end{array}\]
\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 -3.2400532982587204 \cdot 10^{+146}:\\
\;\;\;\;\frac{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(b \cdot -2\right))_*}{3 \cdot a}\\

\mathbf{elif}\;b \le 5.673337904073483 \cdot 10^{-43}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}}}{3 \cdot 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 __attribute__((unused)) d) {
        double r10709283 = b;
        double r10709284 = -r10709283;
        double r10709285 = r10709283 * r10709283;
        double r10709286 = 3.0;
        double r10709287 = a;
        double r10709288 = r10709286 * r10709287;
        double r10709289 = c;
        double r10709290 = r10709288 * r10709289;
        double r10709291 = r10709285 - r10709290;
        double r10709292 = sqrt(r10709291);
        double r10709293 = r10709284 + r10709292;
        double r10709294 = r10709293 / r10709288;
        return r10709294;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r10709295 = b;
        double r10709296 = -3.2400532982587204e+146;
        bool r10709297 = r10709295 <= r10709296;
        double r10709298 = 1.5;
        double r10709299 = a;
        double r10709300 = c;
        double r10709301 = r10709295 / r10709300;
        double r10709302 = r10709299 / r10709301;
        double r10709303 = -2.0;
        double r10709304 = r10709295 * r10709303;
        double r10709305 = fma(r10709298, r10709302, r10709304);
        double r10709306 = 3.0;
        double r10709307 = r10709306 * r10709299;
        double r10709308 = r10709305 / r10709307;
        double r10709309 = 5.673337904073483e-43;
        bool r10709310 = r10709295 <= r10709309;
        double r10709311 = -r10709295;
        double r10709312 = r10709295 * r10709295;
        double r10709313 = cbrt(r10709300);
        double r10709314 = r10709313 * r10709313;
        double r10709315 = r10709307 * r10709314;
        double r10709316 = r10709315 * r10709313;
        double r10709317 = r10709312 - r10709316;
        double r10709318 = sqrt(r10709317);
        double r10709319 = r10709311 + r10709318;
        double r10709320 = r10709319 / r10709307;
        double r10709321 = r10709299 * r10709300;
        double r10709322 = r10709321 / r10709295;
        double r10709323 = -1.5;
        double r10709324 = r10709322 * r10709323;
        double r10709325 = r10709324 / r10709307;
        double r10709326 = r10709310 ? r10709320 : r10709325;
        double r10709327 = r10709297 ? r10709308 : r10709326;
        return r10709327;
}

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 < -3.2400532982587204e+146

    1. Initial program 57.0

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

    3. Applied add-cube-cbrt57.0

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

    4. Applied associate-*r*57.0

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

    6. Applied add-cube-cbrt57.0

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

    7. Applied cbrt-prod57.0

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

    8. Taylor expanded around -inf 10.6

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

    9. Simplified2.4

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

    if -3.2400532982587204e+146 < b < 5.673337904073483e-43

    1. Initial program 13.3

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

    3. Applied add-cube-cbrt13.6

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

    4. Applied associate-*r*13.6

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

    if 5.673337904073483e-43 < b

    1. Initial program 54.1

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

    3. Applied add-cube-cbrt54.1

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

    4. Applied associate-*r*54.1

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

    6. Applied add-cube-cbrt54.1

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

    7. Applied cbrt-prod54.1

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

    8. Taylor expanded around inf 18.1

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

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.2400532982587204 \cdot 10^{+146}:\\ \;\;\;\;\frac{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(b \cdot -2\right))_*}{3 \cdot a}\\ \mathbf{elif}\;b \le 5.673337904073483 \cdot 10^{-43}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))