Average Error: 34.0 → 6.4
Time: 20.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.075503372554188894042518234952899429882 \cdot 10^{112}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.18637976627172627699786603285537458311 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\frac{1}{c}}\\ \mathbf{elif}\;b_2 \le 1.294583421915071462758620229042548737723 \cdot 10^{99}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.075503372554188894042518234952899429882 \cdot 10^{112}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -6.18637976627172627699786603285537458311 \cdot 10^{-309}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\frac{1}{c}}\\

\mathbf{elif}\;b_2 \le 1.294583421915071462758620229042548737723 \cdot 10^{99}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r64818 = b_2;
        double r64819 = -r64818;
        double r64820 = r64818 * r64818;
        double r64821 = a;
        double r64822 = c;
        double r64823 = r64821 * r64822;
        double r64824 = r64820 - r64823;
        double r64825 = sqrt(r64824);
        double r64826 = r64819 - r64825;
        double r64827 = r64826 / r64821;
        return r64827;
}
↓
double f(double a, double b_2, double c) {
        double r64828 = b_2;
        double r64829 = -6.075503372554189e+112;
        bool r64830 = r64828 <= r64829;
        double r64831 = -0.5;
        double r64832 = c;
        double r64833 = r64832 / r64828;
        double r64834 = r64831 * r64833;
        double r64835 = -6.186379766271726e-309;
        bool r64836 = r64828 <= r64835;
        double r64837 = 1.0;
        double r64838 = r64828 * r64828;
        double r64839 = a;
        double r64840 = r64832 * r64839;
        double r64841 = r64838 - r64840;
        double r64842 = sqrt(r64841);
        double r64843 = r64842 - r64828;
        double r64844 = r64837 / r64843;
        double r64845 = r64837 / r64832;
        double r64846 = r64844 / r64845;
        double r64847 = 1.2945834219150715e+99;
        bool r64848 = r64828 <= r64847;
        double r64849 = -r64828;
        double r64850 = r64839 * r64832;
        double r64851 = r64838 - r64850;
        double r64852 = sqrt(r64851);
        double r64853 = r64849 - r64852;
        double r64854 = r64853 / r64839;
        double r64855 = -2.0;
        double r64856 = r64855 * r64828;
        double r64857 = r64856 / r64839;
        double r64858 = r64848 ? r64854 : r64857;
        double r64859 = r64836 ? r64846 : r64858;
        double r64860 = r64830 ? r64834 : r64859;
        return r64860;
}

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 < -6.075503372554189e+112

    1. Initial program 60.2

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

    2. Taylor expanded around -inf 2.3

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

    if -6.075503372554189e+112 < b_2 < -6.186379766271726e-309

    1. Initial program 33.3

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

    3. Applied flip--33.3

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

    4. Simplified16.4

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

    5. Simplified16.4

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

    7. Applied *-un-lft-identity16.4

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

    8. Applied associate-/r*16.4

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

    9. Simplified14.3

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

    11. Applied div-inv14.3

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

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

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

    13. Applied times-frac16.5

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

    14. Applied associate-/l*15.5

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

    15. Simplified8.9

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

    if -6.186379766271726e-309 < b_2 < 1.2945834219150715e+99

    1. Initial program 8.3

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

    if 1.2945834219150715e+99 < b_2

    1. Initial program 46.4

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

    3. Applied flip--63.5

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

    4. Simplified62.6

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

    5. Simplified62.6

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

    6. Taylor expanded around 0 3.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.075503372554188894042518234952899429882 \cdot 10^{112}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.18637976627172627699786603285537458311 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\frac{1}{c}}\\ \mathbf{elif}\;b_2 \le 1.294583421915071462758620229042548737723 \cdot 10^{99}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))