Average Error: 29.3 → 0.1
Time: 3.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7930.6783706997021:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7930.6783706997021:\\
\;\;\;\;\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 r34260 = N;
        double r34261 = 1.0;
        double r34262 = r34260 + r34261;
        double r34263 = log(r34262);
        double r34264 = log(r34260);
        double r34265 = r34263 - r34264;
        return r34265;
}
↓
double f(double N) {
        double r34266 = N;
        double r34267 = 7930.678370699702;
        bool r34268 = r34266 <= r34267;
        double r34269 = 1.0;
        double r34270 = r34266 + r34269;
        double r34271 = r34270 / r34266;
        double r34272 = log(r34271);
        double r34273 = 1.0;
        double r34274 = 2.0;
        double r34275 = pow(r34266, r34274);
        double r34276 = r34273 / r34275;
        double r34277 = 0.3333333333333333;
        double r34278 = r34277 / r34266;
        double r34279 = 0.5;
        double r34280 = r34278 - r34279;
        double r34281 = r34276 * r34280;
        double r34282 = r34269 / r34266;
        double r34283 = r34281 + r34282;
        double r34284 = r34268 ? r34272 : r34283;
        return r34284;
}

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 < 7930.678370699702

    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 7930.678370699702 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7930.6783706997021:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))