Average Error: 34.3 → 12.8
Time: 12.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.327347707720873 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -8.2053558757086961 \cdot 10^{-149}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8586636444574517 \cdot 10^{123}:\\ \;\;\;\;\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -1.327347707720873 \cdot 10^{154}:\\
\;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\

\mathbf{elif}\;b \le -8.2053558757086961 \cdot 10^{-149}:\\
\;\;\;\;\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{elif}\;b \le 1.8586636444574517 \cdot 10^{123}:\\
\;\;\;\;\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r94036 = b;
        double r94037 = -r94036;
        double r94038 = r94036 * r94036;
        double r94039 = 3.0;
        double r94040 = a;
        double r94041 = r94039 * r94040;
        double r94042 = c;
        double r94043 = r94041 * r94042;
        double r94044 = r94038 - r94043;
        double r94045 = sqrt(r94044);
        double r94046 = r94037 + r94045;
        double r94047 = r94046 / r94041;
        return r94047;
}
↓
double f(double a, double b, double c) {
        double r94048 = b;
        double r94049 = -1.327347707720873e+154;
        bool r94050 = r94048 <= r94049;
        double r94051 = -r94048;
        double r94052 = 1.5;
        double r94053 = a;
        double r94054 = c;
        double r94055 = r94053 * r94054;
        double r94056 = r94055 / r94048;
        double r94057 = r94052 * r94056;
        double r94058 = r94057 - r94048;
        double r94059 = r94051 + r94058;
        double r94060 = 3.0;
        double r94061 = r94060 * r94053;
        double r94062 = r94059 / r94061;
        double r94063 = -8.205355875708696e-149;
        bool r94064 = r94048 <= r94063;
        double r94065 = r94048 * r94048;
        double r94066 = cbrt(r94054);
        double r94067 = r94066 * r94066;
        double r94068 = r94061 * r94067;
        double r94069 = r94068 * r94066;
        double r94070 = r94065 - r94069;
        double r94071 = sqrt(r94070);
        double r94072 = r94051 + r94071;
        double r94073 = r94072 / r94061;
        double r94074 = 1.8586636444574517e+123;
        bool r94075 = r94048 <= r94074;
        double r94076 = r94054 * r94061;
        double r94077 = r94061 * r94054;
        double r94078 = r94065 - r94077;
        double r94079 = sqrt(r94078);
        double r94080 = r94051 - r94079;
        double r94081 = r94076 / r94080;
        double r94082 = r94081 / r94061;
        double r94083 = -1.5;
        double r94084 = r94083 * r94056;
        double r94085 = r94084 / r94061;
        double r94086 = r94075 ? r94082 : r94085;
        double r94087 = r94064 ? r94073 : r94086;
        double r94088 = r94050 ? r94062 : r94087;
        return r94088;
}

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 < -1.327347707720873e+154

    1. Initial program 64.0

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

    3. Applied add-cube-cbrt64.0

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

    4. Applied associate-*r*64.0

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

    5. Taylor expanded around -inf 11.9

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a}\]

    if -1.327347707720873e+154 < b < -8.205355875708696e-149

    1. Initial program 5.7

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

    3. Applied add-cube-cbrt5.9

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

    4. Applied associate-*r*5.9

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

    if -8.205355875708696e-149 < b < 1.8586636444574517e+123

    1. Initial program 28.9

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

    3. Applied add-cube-cbrt29.2

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

    4. Applied associate-*r*29.2

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

    6. Applied flip-+29.5

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

    7. Simplified16.6

      \[\leadsto \frac{\frac{\color{blue}{\left(3 \cdot a\right) \cdot c + 0}}{\left(-b\right) - \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}}}}{3 \cdot a}\]

    8. Simplified16.3

      \[\leadsto \frac{\frac{\left(3 \cdot a\right) \cdot c + 0}{\color{blue}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    if 1.8586636444574517e+123 < b

    1. Initial program 60.8

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

    2. Taylor expanded around inf 13.7

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.327347707720873 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -8.2053558757086961 \cdot 10^{-149}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8586636444574517 \cdot 10^{123}:\\ \;\;\;\;\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))