Average Error: 29.1 → 0.1
Time: 13.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 3775.896918214588367845863103866577148438:\\ \;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 3775.896918214588367845863103866577148438:\\
\;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\

\mathbf{else}:\\
\;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\

\end{array}
double f(double N) {
        double r59085 = N;
        double r59086 = 1.0;
        double r59087 = r59085 + r59086;
        double r59088 = log(r59087);
        double r59089 = log(r59085);
        double r59090 = r59088 - r59089;
        return r59090;
}
↓
double f(double N) {
        double r59091 = N;
        double r59092 = 3775.8969182145884;
        bool r59093 = r59091 <= r59092;
        double r59094 = 1.0;
        double r59095 = r59091 + r59094;
        double r59096 = log(r59095);
        double r59097 = log(r59096);
        double r59098 = exp(r59097);
        double r59099 = log(r59091);
        double r59100 = r59098 - r59099;
        double r59101 = 0.3333333333333333;
        double r59102 = 3.0;
        double r59103 = pow(r59091, r59102);
        double r59104 = r59101 / r59103;
        double r59105 = 0.5;
        double r59106 = r59105 / r59091;
        double r59107 = r59094 - r59106;
        double r59108 = r59107 / r59091;
        double r59109 = r59104 + r59108;
        double r59110 = r59093 ? r59100 : r59109;
        return r59110;
}

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 < 3775.8969182145884

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied add-exp-log0.1

      \[\leadsto \color{blue}{e^{\log \left(\log \left(N + 1\right)\right)}} - \log N\]

    if 3775.8969182145884 < N

    1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3775.896918214588367845863103866577148438:\\ \;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))