Average Error: 30.2 → 0.1
Time: 7.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9018.88736323637386:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\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}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r220 = N;
        double r221 = 1.0;
        double r222 = r220 + r221;
        double r223 = log(r222);
        double r224 = log(r220);
        double r225 = r223 - r224;
        return r225;
}
↓
double f(double N) {
        double r226 = N;
        double r227 = 9018.887363236374;
        bool r228 = r226 <= r227;
        double r229 = 1.0;
        double r230 = r226 + r229;
        double r231 = r230 / r226;
        double r232 = log(r231);
        double r233 = 1.0;
        double r234 = 2.0;
        double r235 = pow(r226, r234);
        double r236 = r233 / r235;
        double r237 = 0.3333333333333333;
        double r238 = r237 / r226;
        double r239 = 0.5;
        double r240 = r238 - r239;
        double r241 = r229 / r226;
        double r242 = fma(r236, r240, r241);
        double r243 = r228 ? r232 : r242;
        return r243;
}

Error

Bits error versus N

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

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

  4. Final simplification0.1

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

Reproduce

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