Average Error: 30.2 → 0.1
Time: 3.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9018.88736323637386:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9018.88736323637386:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r39096 = N;
        double r39097 = 1.0;
        double r39098 = r39096 + r39097;
        double r39099 = log(r39098);
        double r39100 = log(r39096);
        double r39101 = r39099 - r39100;
        return r39101;
}
↓
double f(double N) {
        double r39102 = N;
        double r39103 = 9018.887363236374;
        bool r39104 = r39102 <= r39103;
        double r39105 = 1.0;
        double r39106 = r39102 + r39105;
        double r39107 = r39106 / r39102;
        double r39108 = log(r39107);
        double r39109 = 1.0;
        double r39110 = 2.0;
        double r39111 = pow(r39102, r39110);
        double r39112 = r39109 / r39111;
        double r39113 = 0.3333333333333333;
        double r39114 = r39113 / r39102;
        double r39115 = 0.5;
        double r39116 = r39114 - r39115;
        double r39117 = r39112 * r39116;
        double r39118 = r39105 / r39102;
        double r39119 = r39117 + r39118;
        double r39120 = r39104 ? r39108 : r39119;
        return r39120;
}

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 < 9018.887363236374

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    if 9018.887363236374 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9018.88736323637386:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))