Average Error: 33.1 → 10.9
Time: 19.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6519381339788066 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.990519652731023 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)\\ \mathbf{elif}\;b \le 2.9864531425864206 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.3117642643391765 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.6519381339788066 \cdot 10^{+37}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 1.990519652731023 \cdot 10^{-106}:\\
\;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)\\

\mathbf{elif}\;b \le 2.9864531425864206 \cdot 10^{-69}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.3117642643391765 \cdot 10^{-20}:\\
\;\;\;\;\frac{\sqrt{\sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} - b}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3431909 = b;
        double r3431910 = -r3431909;
        double r3431911 = r3431909 * r3431909;
        double r3431912 = 3.0;
        double r3431913 = a;
        double r3431914 = r3431912 * r3431913;
        double r3431915 = c;
        double r3431916 = r3431914 * r3431915;
        double r3431917 = r3431911 - r3431916;
        double r3431918 = sqrt(r3431917);
        double r3431919 = r3431910 + r3431918;
        double r3431920 = r3431919 / r3431914;
        return r3431920;
}
↓
double f(double a, double b, double c) {
        double r3431921 = b;
        double r3431922 = -1.6519381339788066e+37;
        bool r3431923 = r3431921 <= r3431922;
        double r3431924 = 0.5;
        double r3431925 = c;
        double r3431926 = r3431925 / r3431921;
        double r3431927 = r3431924 * r3431926;
        double r3431928 = a;
        double r3431929 = r3431921 / r3431928;
        double r3431930 = 0.6666666666666666;
        double r3431931 = r3431929 * r3431930;
        double r3431932 = r3431927 - r3431931;
        double r3431933 = 1.990519652731023e-106;
        bool r3431934 = r3431921 <= r3431933;
        double r3431935 = 1.0;
        double r3431936 = 3.0;
        double r3431937 = r3431936 * r3431928;
        double r3431938 = r3431935 / r3431937;
        double r3431939 = r3431921 * r3431921;
        double r3431940 = r3431937 * r3431925;
        double r3431941 = r3431939 - r3431940;
        double r3431942 = sqrt(r3431941);
        double r3431943 = r3431942 - r3431921;
        double r3431944 = r3431938 * r3431943;
        double r3431945 = 2.9864531425864206e-69;
        bool r3431946 = r3431921 <= r3431945;
        double r3431947 = -0.5;
        double r3431948 = r3431947 * r3431926;
        double r3431949 = 5.3117642643391765e-20;
        bool r3431950 = r3431921 <= r3431949;
        double r3431951 = cbrt(r3431941);
        double r3431952 = r3431951 * r3431951;
        double r3431953 = r3431951 * r3431952;
        double r3431954 = sqrt(r3431953);
        double r3431955 = r3431954 - r3431921;
        double r3431956 = r3431955 / r3431937;
        double r3431957 = r3431950 ? r3431956 : r3431948;
        double r3431958 = r3431946 ? r3431948 : r3431957;
        double r3431959 = r3431934 ? r3431944 : r3431958;
        double r3431960 = r3431923 ? r3431932 : r3431959;
        return r3431960;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6519381339788066e+37

    1. Initial program 33.7

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

    2. Simplified33.7

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

    3. Taylor expanded around -inf 6.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

    if -1.6519381339788066e+37 < b < 1.990519652731023e-106

    1. Initial program 12.9

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

    2. Simplified12.9

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

    4. Applied div-inv13.0

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

    if 1.990519652731023e-106 < b < 2.9864531425864206e-69 or 5.3117642643391765e-20 < b

    1. Initial program 53.1

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

    2. Simplified53.1

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

    3. Taylor expanded around inf 8.6

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

    if 2.9864531425864206e-69 < b < 5.3117642643391765e-20

    1. Initial program 35.5

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

    2. Simplified35.5

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

    4. Applied add-cube-cbrt35.9

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6519381339788066 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.990519652731023 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)\\ \mathbf{elif}\;b \le 2.9864531425864206 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.3117642643391765 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))