Average Error: 29.5 → 0.1
Time: 4.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7441.5629236213317:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7441.5629236213317:\\
\;\;\;\;\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 r65273 = N;
        double r65274 = 1.0;
        double r65275 = r65273 + r65274;
        double r65276 = log(r65275);
        double r65277 = log(r65273);
        double r65278 = r65276 - r65277;
        return r65278;
}
↓
double f(double N) {
        double r65279 = N;
        double r65280 = 7441.562923621332;
        bool r65281 = r65279 <= r65280;
        double r65282 = 1.0;
        double r65283 = r65279 + r65282;
        double r65284 = r65283 / r65279;
        double r65285 = log(r65284);
        double r65286 = 1.0;
        double r65287 = 2.0;
        double r65288 = pow(r65279, r65287);
        double r65289 = r65286 / r65288;
        double r65290 = 0.3333333333333333;
        double r65291 = r65290 / r65279;
        double r65292 = 0.5;
        double r65293 = r65291 - r65292;
        double r65294 = r65289 * r65293;
        double r65295 = r65282 / r65279;
        double r65296 = r65294 + r65295;
        double r65297 = r65281 ? r65285 : r65296;
        return r65297;
}

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 < 7441.562923621332

    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 7441.562923621332 < N

    1. Initial program 59.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7441.5629236213317:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))