Average Error: 33.5 → 10.2
Time: 26.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.957271491558129 \cdot 10^{+130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{a}{b} \cdot c\right), \frac{3}{2}, \left(b \cdot -2\right)\right)}{a \cdot 3}\\ \mathbf{elif}\;b \le 1.8714479467616624 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(c \cdot -3\right), \left(b \cdot b\right)\right)} - b}{\sqrt[3]{3}}}{a} \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\\ \mathbf{elif}\;b \le 4.215358931972171 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.4134669192958513 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{1}{3}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(c \cdot -3\right), \left(b \cdot b\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.957271491558129 \cdot 10^{+130}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{a}{b} \cdot c\right), \frac{3}{2}, \left(b \cdot -2\right)\right)}{a \cdot 3}\\

\mathbf{elif}\;b \le 1.8714479467616624 \cdot 10^{-83}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(c \cdot -3\right), \left(b \cdot b\right)\right)} - b}{\sqrt[3]{3}}}{a} \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\\

\mathbf{elif}\;b \le 4.215358931972171 \cdot 10^{-77}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.4134669192958513 \cdot 10^{-44}:\\
\;\;\;\;\frac{\frac{1}{3}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(c \cdot -3\right), \left(b \cdot b\right)\right)} - b\right)\\

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

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r3780270 = b;
        double r3780271 = -r3780270;
        double r3780272 = r3780270 * r3780270;
        double r3780273 = 3.0;
        double r3780274 = a;
        double r3780275 = r3780273 * r3780274;
        double r3780276 = c;
        double r3780277 = r3780275 * r3780276;
        double r3780278 = r3780272 - r3780277;
        double r3780279 = sqrt(r3780278);
        double r3780280 = r3780271 + r3780279;
        double r3780281 = r3780280 / r3780275;
        return r3780281;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r3780282 = b;
        double r3780283 = -2.957271491558129e+130;
        bool r3780284 = r3780282 <= r3780283;
        double r3780285 = a;
        double r3780286 = r3780285 / r3780282;
        double r3780287 = c;
        double r3780288 = r3780286 * r3780287;
        double r3780289 = 1.5;
        double r3780290 = -2.0;
        double r3780291 = r3780282 * r3780290;
        double r3780292 = fma(r3780288, r3780289, r3780291);
        double r3780293 = 3.0;
        double r3780294 = r3780285 * r3780293;
        double r3780295 = r3780292 / r3780294;
        double r3780296 = 1.8714479467616624e-83;
        bool r3780297 = r3780282 <= r3780296;
        double r3780298 = -3.0;
        double r3780299 = r3780287 * r3780298;
        double r3780300 = r3780282 * r3780282;
        double r3780301 = fma(r3780285, r3780299, r3780300);
        double r3780302 = sqrt(r3780301);
        double r3780303 = r3780302 - r3780282;
        double r3780304 = cbrt(r3780293);
        double r3780305 = r3780303 / r3780304;
        double r3780306 = r3780305 / r3780285;
        double r3780307 = 1.0;
        double r3780308 = r3780304 * r3780304;
        double r3780309 = r3780307 / r3780308;
        double r3780310 = r3780306 * r3780309;
        double r3780311 = 4.215358931972171e-77;
        bool r3780312 = r3780282 <= r3780311;
        double r3780313 = -0.5;
        double r3780314 = r3780287 / r3780282;
        double r3780315 = r3780313 * r3780314;
        double r3780316 = 2.4134669192958513e-44;
        bool r3780317 = r3780282 <= r3780316;
        double r3780318 = 0.3333333333333333;
        double r3780319 = r3780318 / r3780285;
        double r3780320 = r3780319 * r3780303;
        double r3780321 = r3780317 ? r3780320 : r3780315;
        double r3780322 = r3780312 ? r3780315 : r3780321;
        double r3780323 = r3780297 ? r3780310 : r3780322;
        double r3780324 = r3780284 ? r3780295 : r3780323;
        return r3780324;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 4 regimes

  2. if b < -2.957271491558129e+130

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 10.4

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

    3. Simplified3.2

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

    if -2.957271491558129e+130 < b < 1.8714479467616624e-83

    1. Initial program 12.6

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

    3. Applied associate-/r*12.7

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

    4. Simplified12.7

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

    5. Taylor expanded around -inf 12.7

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

    6. Simplified12.7

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

    8. Applied *-un-lft-identity12.7

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(-3 \cdot c\right), \left(b \cdot b\right)\right)} - b}{3}}{\color{blue}{1 \cdot a}}\]

    9. Applied add-cube-cbrt12.7

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(-3 \cdot c\right), \left(b \cdot b\right)\right)} - b}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}}{1 \cdot a}\]

    10. Applied *-un-lft-identity12.7

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(-3 \cdot c\right), \left(b \cdot b\right)\right)} - b\right)}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}{1 \cdot a}\]

    11. Applied times-frac12.8

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\mathsf{fma}\left(a, \left(-3 \cdot c\right), \left(b \cdot b\right)\right)} - b}{\sqrt[3]{3}}}}{1 \cdot a}\]

    12. Applied times-frac12.8

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{1} \cdot \frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(-3 \cdot c\right), \left(b \cdot b\right)\right)} - b}{\sqrt[3]{3}}}{a}}\]

    13. Simplified12.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(-3 \cdot c\right), \left(b \cdot b\right)\right)} - b}{\sqrt[3]{3}}}{a}\]

    if 1.8714479467616624e-83 < b < 4.215358931972171e-77 or 2.4134669192958513e-44 < b

    1. Initial program 53.4

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

    2. Taylor expanded around inf 7.6

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]

    if 4.215358931972171e-77 < b < 2.4134669192958513e-44

    1. Initial program 34.4

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

    3. Applied associate-/r*34.5

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

    4. Simplified34.5

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

    6. Applied *-un-lft-identity34.5

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -3, \left(b \cdot b\right)\right)} - b}{3}}{\color{blue}{1 \cdot a}}\]

    7. Applied div-inv34.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -3, \left(b \cdot b\right)\right)} - b\right) \cdot \frac{1}{3}}}{1 \cdot a}\]

    8. Applied times-frac34.5

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -3, \left(b \cdot b\right)\right)} - b}{1} \cdot \frac{\frac{1}{3}}{a}}\]

    9. Simplified34.5

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, \left(-3 \cdot c\right), \left(b \cdot b\right)\right)} - b\right)} \cdot \frac{\frac{1}{3}}{a}\]

    10. Simplified34.5

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(a, \left(-3 \cdot c\right), \left(b \cdot b\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.957271491558129 \cdot 10^{+130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{a}{b} \cdot c\right), \frac{3}{2}, \left(b \cdot -2\right)\right)}{a \cdot 3}\\ \mathbf{elif}\;b \le 1.8714479467616624 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(c \cdot -3\right), \left(b \cdot b\right)\right)} - b}{\sqrt[3]{3}}}{a} \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\\ \mathbf{elif}\;b \le 4.215358931972171 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.4134669192958513 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{1}{3}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(c \cdot -3\right), \left(b \cdot b\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))