Average Error: 34.4 → 6.2
Time: 7.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.593339349284273934392489776701937282956 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.310125818367260256329730829931519203575 \cdot 10^{-294}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 1.076384822638075226373023800566186486206 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.593339349284273934392489776701937282956 \cdot 10^{151}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -1.310125818367260256329730829931519203575 \cdot 10^{-294}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 1.076384822638075226373023800566186486206 \cdot 10^{126}:\\
\;\;\;\;\frac{1}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r27948 = b_2;
        double r27949 = -r27948;
        double r27950 = r27948 * r27948;
        double r27951 = a;
        double r27952 = c;
        double r27953 = r27951 * r27952;
        double r27954 = r27950 - r27953;
        double r27955 = sqrt(r27954);
        double r27956 = r27949 + r27955;
        double r27957 = r27956 / r27951;
        return r27957;
}
↓
double f(double a, double b_2, double c) {
        double r27958 = b_2;
        double r27959 = -8.593339349284274e+151;
        bool r27960 = r27958 <= r27959;
        double r27961 = 0.5;
        double r27962 = c;
        double r27963 = r27962 / r27958;
        double r27964 = r27961 * r27963;
        double r27965 = 2.0;
        double r27966 = a;
        double r27967 = r27958 / r27966;
        double r27968 = r27965 * r27967;
        double r27969 = r27964 - r27968;
        double r27970 = -1.3101258183672603e-294;
        bool r27971 = r27958 <= r27970;
        double r27972 = -r27958;
        double r27973 = r27958 * r27958;
        double r27974 = r27966 * r27962;
        double r27975 = r27973 - r27974;
        double r27976 = sqrt(r27975);
        double r27977 = r27972 + r27976;
        double r27978 = r27977 / r27966;
        double r27979 = 1.0763848226380752e+126;
        bool r27980 = r27958 <= r27979;
        double r27981 = 1.0;
        double r27982 = r27972 - r27976;
        double r27983 = r27981 * r27982;
        double r27984 = r27983 / r27962;
        double r27985 = r27981 / r27984;
        double r27986 = -0.5;
        double r27987 = r27986 * r27963;
        double r27988 = r27980 ? r27985 : r27987;
        double r27989 = r27971 ? r27978 : r27988;
        double r27990 = r27960 ? r27969 : r27989;
        return r27990;
}

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 < -8.593339349284274e+151

    1. Initial program 63.2

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

    2. Taylor expanded around -inf 2.2

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

    if -8.593339349284274e+151 < b_2 < -1.3101258183672603e-294

    1. Initial program 7.9

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

    if -1.3101258183672603e-294 < b_2 < 1.0763848226380752e+126

    1. Initial program 33.4

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

    3. Applied flip-+33.4

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

    4. Simplified16.3

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

    6. Applied *-un-lft-identity16.3

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

    7. Applied *-un-lft-identity16.3

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

    8. Applied times-frac16.3

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

    9. Simplified16.3

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

    10. Simplified14.4

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

    12. Applied clear-num14.4

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

    13. Simplified8.9

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

    if 1.0763848226380752e+126 < b_2

    1. Initial program 60.9

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.593339349284273934392489776701937282956 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.310125818367260256329730829931519203575 \cdot 10^{-294}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 1.076384822638075226373023800566186486206 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))