Average Error: 29.8 → 0.1
Time: 34.7s
Precision: 64
\[\log \left(N + 1.0\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 11332.770353358372:\\ \;\;\;\;\log \left(\sqrt{\frac{1.0 + N}{N}}\right) + \log \left(\sqrt{\frac{1.0 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{0.3333333333333333}{N}}{N \cdot N} + \frac{1.0}{N}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]
\log \left(N + 1.0\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 11332.770353358372:\\
\;\;\;\;\log \left(\sqrt{\frac{1.0 + N}{N}}\right) + \log \left(\sqrt{\frac{1.0 + N}{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{0.3333333333333333}{N}}{N \cdot N} + \frac{1.0}{N}\right) - \frac{0.5}{N \cdot N}\\

\end{array}
double f(double N) {
        double r3457921 = N;
        double r3457922 = 1.0;
        double r3457923 = r3457921 + r3457922;
        double r3457924 = log(r3457923);
        double r3457925 = log(r3457921);
        double r3457926 = r3457924 - r3457925;
        return r3457926;
}
↓
double f(double N) {
        double r3457927 = N;
        double r3457928 = 11332.770353358372;
        bool r3457929 = r3457927 <= r3457928;
        double r3457930 = 1.0;
        double r3457931 = r3457930 + r3457927;
        double r3457932 = r3457931 / r3457927;
        double r3457933 = sqrt(r3457932);
        double r3457934 = log(r3457933);
        double r3457935 = r3457934 + r3457934;
        double r3457936 = 0.3333333333333333;
        double r3457937 = r3457936 / r3457927;
        double r3457938 = r3457927 * r3457927;
        double r3457939 = r3457937 / r3457938;
        double r3457940 = r3457930 / r3457927;
        double r3457941 = r3457939 + r3457940;
        double r3457942 = 0.5;
        double r3457943 = r3457942 / r3457938;
        double r3457944 = r3457941 - r3457943;
        double r3457945 = r3457929 ? r3457935 : r3457944;
        return r3457945;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if N < 11332.770353358372

    1. Initial program 0.1

      \[\log \left(N + 1.0\right) - \log N\]
    2. Using strategy rm

    3. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{\log \left(N + 1.0\right) - \log N}\right)}\]

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\frac{1.0 + N}{N}\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.1

      \[\leadsto \log \color{blue}{\left(\sqrt{\frac{1.0 + N}{N}} \cdot \sqrt{\frac{1.0 + N}{N}}\right)}\]

    7. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{1.0 + N}{N}}\right) + \log \left(\sqrt{\frac{1.0 + N}{N}}\right)}\]

    if 11332.770353358372 < N

    1. Initial program 59.7

      \[\log \left(N + 1.0\right) - \log N\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + 1.0 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{\frac{0.3333333333333333}{N}}{N \cdot N} + \frac{1.0}{N}\right) - \frac{0.5}{N \cdot N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 11332.770353358372:\\ \;\;\;\;\log \left(\sqrt{\frac{1.0 + N}{N}}\right) + \log \left(\sqrt{\frac{1.0 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{0.3333333333333333}{N}}{N \cdot N} + \frac{1.0}{N}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))