Average Error: 34.1 → 7.2
Time: 6.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.687169645099804665049460580076288185662 \cdot 10^{72}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.178383319658251659941996545038743748932 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\ \mathbf{elif}\;b_2 \le 9.981010889976272049519072583984808542877 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.687169645099804665049460580076288185662 \cdot 10^{72}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.178383319658251659941996545038743748932 \cdot 10^{-298}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\

\mathbf{elif}\;b_2 \le 9.981010889976272049519072583984808542877 \cdot 10^{53}:\\
\;\;\;\;\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 r82946 = b_2;
        double r82947 = -r82946;
        double r82948 = r82946 * r82946;
        double r82949 = a;
        double r82950 = c;
        double r82951 = r82949 * r82950;
        double r82952 = r82948 - r82951;
        double r82953 = sqrt(r82952);
        double r82954 = r82947 - r82953;
        double r82955 = r82954 / r82949;
        return r82955;
}
↓
double f(double a, double b_2, double c) {
        double r82956 = b_2;
        double r82957 = -8.687169645099805e+72;
        bool r82958 = r82956 <= r82957;
        double r82959 = -0.5;
        double r82960 = c;
        double r82961 = r82960 / r82956;
        double r82962 = r82959 * r82961;
        double r82963 = 2.1783833196582517e-298;
        bool r82964 = r82956 <= r82963;
        double r82965 = 1.0;
        double r82966 = r82956 * r82956;
        double r82967 = a;
        double r82968 = r82967 * r82960;
        double r82969 = r82966 - r82968;
        double r82970 = sqrt(r82969);
        double r82971 = r82970 - r82956;
        double r82972 = r82965 / r82971;
        double r82973 = r82965 / r82960;
        double r82974 = r82972 / r82973;
        double r82975 = 9.981010889976272e+53;
        bool r82976 = r82956 <= r82975;
        double r82977 = -r82956;
        double r82978 = r82977 - r82970;
        double r82979 = r82967 / r82978;
        double r82980 = r82965 / r82979;
        double r82981 = 0.5;
        double r82982 = r82981 * r82961;
        double r82983 = 2.0;
        double r82984 = r82956 / r82967;
        double r82985 = r82983 * r82984;
        double r82986 = r82982 - r82985;
        double r82987 = r82976 ? r82980 : r82986;
        double r82988 = r82964 ? r82974 : r82987;
        double r82989 = r82958 ? r82962 : r82988;
        return r82989;
}

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 < -8.687169645099805e+72

    1. Initial program 58.4

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

    2. Taylor expanded around -inf 3.5

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

    if -8.687169645099805e+72 < b_2 < 2.1783833196582517e-298

    1. Initial program 30.5

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

    3. Applied flip--30.6

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

    4. Simplified16.8

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

    5. Simplified16.8

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

    7. Applied *-un-lft-identity16.8

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

    8. Applied associate-/r*16.8

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

    9. Simplified14.3

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

    11. Applied div-inv14.3

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

    12. Applied *-un-lft-identity14.3

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

    13. Applied times-frac16.9

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

    14. Applied associate-/l*16.4

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

    15. Simplified9.7

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

    if 2.1783833196582517e-298 < b_2 < 9.981010889976272e+53

    1. Initial program 9.7

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

    3. Applied clear-num9.8

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

    if 9.981010889976272e+53 < b_2

    1. Initial program 37.9

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

    2. Taylor expanded around inf 5.0

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.687169645099804665049460580076288185662 \cdot 10^{72}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.178383319658251659941996545038743748932 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\ \mathbf{elif}\;b_2 \le 9.981010889976272049519072583984808542877 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))