Average Error: 34.2 → 7.0
Time: 20.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.195906284172791803059743272391121596524 \cdot 10^{-304}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 9.766388217822194338414589021212524462695 \cdot 10^{89}:\\ \;\;\;\;\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -1.195906284172791803059743272391121596524 \cdot 10^{-304}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 9.766388217822194338414589021212524462695 \cdot 10^{89}:\\
\;\;\;\;\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r84935 = b_2;
        double r84936 = -r84935;
        double r84937 = r84935 * r84935;
        double r84938 = a;
        double r84939 = c;
        double r84940 = r84938 * r84939;
        double r84941 = r84937 - r84940;
        double r84942 = sqrt(r84941);
        double r84943 = r84936 + r84942;
        double r84944 = r84943 / r84938;
        return r84944;
}
↓
double f(double a, double b_2, double c) {
        double r84945 = b_2;
        double r84946 = -4.739386840053889e+131;
        bool r84947 = r84945 <= r84946;
        double r84948 = 0.5;
        double r84949 = c;
        double r84950 = r84949 / r84945;
        double r84951 = r84948 * r84950;
        double r84952 = 2.0;
        double r84953 = a;
        double r84954 = r84945 / r84953;
        double r84955 = r84952 * r84954;
        double r84956 = r84951 - r84955;
        double r84957 = -1.1959062841727918e-304;
        bool r84958 = r84945 <= r84957;
        double r84959 = -r84945;
        double r84960 = r84945 * r84945;
        double r84961 = r84953 * r84949;
        double r84962 = r84960 - r84961;
        double r84963 = sqrt(r84962);
        double r84964 = r84959 + r84963;
        double r84965 = 1.0;
        double r84966 = r84965 / r84953;
        double r84967 = r84964 * r84966;
        double r84968 = 9.766388217822194e+89;
        bool r84969 = r84945 <= r84968;
        double r84970 = cbrt(r84953);
        double r84971 = r84970 * r84970;
        double r84972 = r84965 / r84971;
        double r84973 = r84971 * r84972;
        double r84974 = r84965 / r84973;
        double r84975 = r84959 - r84963;
        double r84976 = r84975 / r84970;
        double r84977 = r84976 / r84949;
        double r84978 = r84965 / r84977;
        double r84979 = r84978 / r84970;
        double r84980 = r84974 * r84979;
        double r84981 = -0.5;
        double r84982 = r84981 * r84950;
        double r84983 = r84969 ? r84980 : r84982;
        double r84984 = r84958 ? r84967 : r84983;
        double r84985 = r84947 ? r84956 : r84984;
        return r84985;
}

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 < -4.739386840053889e+131

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 2.4

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

    if -4.739386840053889e+131 < b_2 < -1.1959062841727918e-304

    1. Initial program 9.2

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

    3. Applied div-inv9.3

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

    if -1.1959062841727918e-304 < b_2 < 9.766388217822194e+89

    1. Initial program 31.7

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

    3. Applied flip-+31.7

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

    4. Simplified16.2

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

    6. Applied clear-num16.4

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

    7. Simplified15.2

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

    9. Applied add-cube-cbrt15.9

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

    10. Applied *-un-lft-identity15.9

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

    11. Applied add-cube-cbrt15.2

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

    12. Applied *-un-lft-identity15.2

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

    13. Applied times-frac15.3

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

    14. Applied times-frac14.2

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

    15. Applied add-sqr-sqrt14.2

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

    16. Applied times-frac13.7

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

    17. Applied times-frac9.8

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

    18. Simplified9.8

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

    19. Simplified9.8

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

    if 9.766388217822194e+89 < b_2

    1. Initial program 59.2

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.195906284172791803059743272391121596524 \cdot 10^{-304}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 9.766388217822194338414589021212524462695 \cdot 10^{89}:\\ \;\;\;\;\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))