Average Error: 33.8 → 7.2
Time: 26.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.565090470125855 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -5.587449545143923 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\sqrt[3]{a}} \cdot \left(c \cdot \sqrt[3]{a}\right)\\ \mathbf{elif}\;b_2 \le 9.336288915836175 \cdot 10^{+83}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.565090470125855 \cdot 10^{+141}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -5.587449545143923 \cdot 10^{-253}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\sqrt[3]{a}} \cdot \left(c \cdot \sqrt[3]{a}\right)\\

\mathbf{elif}\;b_2 \le 9.336288915836175 \cdot 10^{+83}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\end{array}
double f(double a, double b_2, double c) {
        double r871948 = b_2;
        double r871949 = -r871948;
        double r871950 = r871948 * r871948;
        double r871951 = a;
        double r871952 = c;
        double r871953 = r871951 * r871952;
        double r871954 = r871950 - r871953;
        double r871955 = sqrt(r871954);
        double r871956 = r871949 - r871955;
        double r871957 = r871956 / r871951;
        return r871957;
}
↓
double f(double a, double b_2, double c) {
        double r871958 = b_2;
        double r871959 = -6.565090470125855e+141;
        bool r871960 = r871958 <= r871959;
        double r871961 = -0.5;
        double r871962 = c;
        double r871963 = r871962 / r871958;
        double r871964 = r871961 * r871963;
        double r871965 = -5.587449545143923e-253;
        bool r871966 = r871958 <= r871965;
        double r871967 = 1.0;
        double r871968 = r871958 * r871958;
        double r871969 = a;
        double r871970 = r871962 * r871969;
        double r871971 = r871968 - r871970;
        double r871972 = sqrt(r871971);
        double r871973 = r871972 - r871958;
        double r871974 = r871967 / r871973;
        double r871975 = cbrt(r871969);
        double r871976 = r871974 / r871975;
        double r871977 = r871962 * r871975;
        double r871978 = r871976 * r871977;
        double r871979 = 9.336288915836175e+83;
        bool r871980 = r871958 <= r871979;
        double r871981 = -r871958;
        double r871982 = r871981 - r871972;
        double r871983 = r871967 / r871969;
        double r871984 = r871982 * r871983;
        double r871985 = 0.5;
        double r871986 = r871985 * r871963;
        double r871987 = r871958 / r871969;
        double r871988 = 2.0;
        double r871989 = r871987 * r871988;
        double r871990 = r871986 - r871989;
        double r871991 = r871980 ? r871984 : r871990;
        double r871992 = r871966 ? r871978 : r871991;
        double r871993 = r871960 ? r871964 : r871992;
        return r871993;
}

Error

Bits error versus a

Bits error versus b_2

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_2 < -6.565090470125855e+141

    1. Initial program 61.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 1.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -6.565090470125855e+141 < b_2 < -5.587449545143923e-253

    1. Initial program 36.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--36.8

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified16.9

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified16.9

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt17.6

      \[\leadsto \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

    8. Applied div-inv17.7

      \[\leadsto \frac{\color{blue}{\left(0 + a \cdot c\right) \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    9. Applied times-frac16.2

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{a}}}\]

    10. Simplified13.5

      \[\leadsto \color{blue}{\left(\frac{a}{\sqrt[3]{a}} \cdot \frac{c}{\sqrt[3]{a}}\right)} \cdot \frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{a}}\]

    11. Taylor expanded around inf 37.8

      \[\leadsto \color{blue}{\left({a}^{\frac{1}{3}} \cdot c\right)} \cdot \frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{a}}\]

    12. Simplified9.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{a} \cdot c\right)} \cdot \frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{a}}\]

    if -5.587449545143923e-253 < b_2 < 9.336288915836175e+83

    1. Initial program 9.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv9.7

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

    if 9.336288915836175e+83 < b_2

    1. Initial program 42.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 4.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.565090470125855 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -5.587449545143923 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\sqrt[3]{a}} \cdot \left(c \cdot \sqrt[3]{a}\right)\\ \mathbf{elif}\;b_2 \le 9.336288915836175 \cdot 10^{+83}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))