Average Error: 30.2 → 0.1
Time: 4.5s
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 r59703 = N;
        double r59704 = 1.0;
        double r59705 = r59703 + r59704;
        double r59706 = log(r59705);
        double r59707 = log(r59703);
        double r59708 = r59706 - r59707;
        return r59708;
}
↓
double f(double N) {
        double r59709 = N;
        double r59710 = 9018.887363236374;
        bool r59711 = r59709 <= r59710;
        double r59712 = 1.0;
        double r59713 = r59709 + r59712;
        double r59714 = r59713 / r59709;
        double r59715 = log(r59714);
        double r59716 = 1.0;
        double r59717 = 2.0;
        double r59718 = pow(r59709, r59717);
        double r59719 = r59716 / r59718;
        double r59720 = 0.3333333333333333;
        double r59721 = r59720 / r59709;
        double r59722 = 0.5;
        double r59723 = r59721 - r59722;
        double r59724 = r59712 / r59709;
        double r59725 = fma(r59719, r59723, r59724);
        double r59726 = r59711 ? r59715 : r59725;
        return r59726;
}

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)))