Average Error: 34.2 → 8.6
Time: 8.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.14194017547317126 \cdot 10^{130}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.5120874391809866 \cdot 10^{-204}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 0.0231735748307204843:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.6383712677255495 \cdot 10^{30}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.29266152734135 \cdot 10^{122}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.14194017547317126 \cdot 10^{130}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -1.5120874391809866 \cdot 10^{-204}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 0.0231735748307204843:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 4.6383712677255495 \cdot 10^{30}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.29266152734135 \cdot 10^{122}:\\
\;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r28923 = b_2;
        double r28924 = -r28923;
        double r28925 = r28923 * r28923;
        double r28926 = a;
        double r28927 = c;
        double r28928 = r28926 * r28927;
        double r28929 = r28925 - r28928;
        double r28930 = sqrt(r28929);
        double r28931 = r28924 + r28930;
        double r28932 = r28931 / r28926;
        return r28932;
}
↓
double f(double a, double b_2, double c) {
        double r28933 = b_2;
        double r28934 = -2.1419401754731713e+130;
        bool r28935 = r28933 <= r28934;
        double r28936 = 0.5;
        double r28937 = c;
        double r28938 = r28937 / r28933;
        double r28939 = r28936 * r28938;
        double r28940 = 2.0;
        double r28941 = a;
        double r28942 = r28933 / r28941;
        double r28943 = r28940 * r28942;
        double r28944 = r28939 - r28943;
        double r28945 = -1.5120874391809866e-204;
        bool r28946 = r28933 <= r28945;
        double r28947 = -r28933;
        double r28948 = r28933 * r28933;
        double r28949 = r28941 * r28937;
        double r28950 = r28948 - r28949;
        double r28951 = sqrt(r28950);
        double r28952 = r28947 + r28951;
        double r28953 = 1.0;
        double r28954 = r28953 / r28941;
        double r28955 = r28952 * r28954;
        double r28956 = 0.023173574830720484;
        bool r28957 = r28933 <= r28956;
        double r28958 = r28947 - r28951;
        double r28959 = r28958 / r28941;
        double r28960 = r28959 / r28937;
        double r28961 = r28953 / r28960;
        double r28962 = r28961 / r28941;
        double r28963 = 4.6383712677255495e+30;
        bool r28964 = r28933 <= r28963;
        double r28965 = -0.5;
        double r28966 = r28965 * r28938;
        double r28967 = 2.292661527341346e+122;
        bool r28968 = r28933 <= r28967;
        double r28969 = 0.0;
        double r28970 = r28969 + r28949;
        double r28971 = r28970 / r28958;
        double r28972 = r28971 / r28941;
        double r28973 = r28968 ? r28972 : r28966;
        double r28974 = r28964 ? r28966 : r28973;
        double r28975 = r28957 ? r28962 : r28974;
        double r28976 = r28946 ? r28955 : r28975;
        double r28977 = r28935 ? r28944 : r28976;
        return r28977;
}

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 5 regimes

  2. if b_2 < -2.1419401754731713e+130

    1. Initial program 57.2

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

    2. Taylor expanded around -inf 3.2

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

    if -2.1419401754731713e+130 < b_2 < -1.5120874391809866e-204

    1. Initial program 7.0

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

    3. Applied div-inv7.2

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

    if -1.5120874391809866e-204 < b_2 < 0.023173574830720484

    1. Initial program 22.0

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

    3. Applied flip-+22.2

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

    4. Simplified17.0

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

    6. Applied clear-num17.0

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

    7. Simplified14.6

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

    if 0.023173574830720484 < b_2 < 4.6383712677255495e+30 or 2.292661527341346e+122 < b_2

    1. Initial program 59.4

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

    2. Taylor expanded around inf 4.1

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

    if 4.6383712677255495e+30 < b_2 < 2.292661527341346e+122

    1. Initial program 47.1

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

    3. Applied flip-+47.1

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

    4. Simplified13.9

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.14194017547317126 \cdot 10^{130}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.5120874391809866 \cdot 10^{-204}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 0.0231735748307204843:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.6383712677255495 \cdot 10^{30}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.29266152734135 \cdot 10^{122}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))