Average Error: 34.0 → 6.8
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.00500656176984215351659893827263540922 \cdot 10^{132}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.054528764146387149688914666009662801656 \cdot 10^{-247}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.00500656176984215351659893827263540922 \cdot 10^{132}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.054528764146387149688914666009662801656 \cdot 10^{-247}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r63881 = b_2;
        double r63882 = -r63881;
        double r63883 = r63881 * r63881;
        double r63884 = a;
        double r63885 = c;
        double r63886 = r63884 * r63885;
        double r63887 = r63883 - r63886;
        double r63888 = sqrt(r63887);
        double r63889 = r63882 - r63888;
        double r63890 = r63889 / r63884;
        return r63890;
}
↓
double f(double a, double b_2, double c) {
        double r63891 = b_2;
        double r63892 = -5.005006561769842e+132;
        bool r63893 = r63891 <= r63892;
        double r63894 = -0.5;
        double r63895 = c;
        double r63896 = r63895 / r63891;
        double r63897 = r63894 * r63896;
        double r63898 = 1.0545287641463871e-247;
        bool r63899 = r63891 <= r63898;
        double r63900 = -r63891;
        double r63901 = r63891 * r63891;
        double r63902 = a;
        double r63903 = r63902 * r63895;
        double r63904 = r63901 - r63903;
        double r63905 = sqrt(r63904);
        double r63906 = r63900 + r63905;
        double r63907 = r63895 / r63906;
        double r63908 = 1.0273828621120979e+63;
        bool r63909 = r63891 <= r63908;
        double r63910 = 1.0;
        double r63911 = r63900 - r63905;
        double r63912 = r63902 / r63911;
        double r63913 = r63910 / r63912;
        double r63914 = 0.5;
        double r63915 = r63914 * r63896;
        double r63916 = 2.0;
        double r63917 = r63891 / r63902;
        double r63918 = r63916 * r63917;
        double r63919 = r63915 - r63918;
        double r63920 = r63909 ? r63913 : r63919;
        double r63921 = r63899 ? r63907 : r63920;
        double r63922 = r63893 ? r63897 : r63921;
        return r63922;
}

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 < -5.005006561769842e+132

    1. Initial program 61.7

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

    2. Taylor expanded around -inf 1.7

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

    if -5.005006561769842e+132 < b_2 < 1.0545287641463871e-247

    1. Initial program 31.9

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

    3. Applied clear-num31.9

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

    5. Applied flip--32.0

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

    6. Applied associate-/r/32.0

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

    7. Applied associate-/r*32.0

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

    8. Simplified14.6

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c\right)}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    9. Taylor expanded around 0 9.2

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

    if 1.0545287641463871e-247 < b_2 < 1.0273828621120979e+63

    1. Initial program 8.1

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

    3. Applied clear-num8.3

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

    if 1.0273828621120979e+63 < b_2

    1. Initial program 39.8

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

    2. Taylor expanded around inf 5.4

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.00500656176984215351659893827263540922 \cdot 10^{132}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.054528764146387149688914666009662801656 \cdot 10^{-247}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))