Average Error: 29.2 → 0.1
Time: 9.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5567.039381079165:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 5567.039381079165:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r972781 = N;
        double r972782 = 1.0;
        double r972783 = r972781 + r972782;
        double r972784 = log(r972783);
        double r972785 = log(r972781);
        double r972786 = r972784 - r972785;
        return r972786;
}
↓
double f(double N) {
        double r972787 = N;
        double r972788 = 5567.039381079165;
        bool r972789 = r972787 <= r972788;
        double r972790 = log1p(r972787);
        double r972791 = sqrt(r972790);
        double r972792 = log(r972787);
        double r972793 = -r972792;
        double r972794 = fma(r972791, r972791, r972793);
        double r972795 = 1.0;
        double r972796 = r972795 / r972787;
        double r972797 = r972796 / r972787;
        double r972798 = 0.3333333333333333;
        double r972799 = r972798 / r972787;
        double r972800 = 0.5;
        double r972801 = r972799 - r972800;
        double r972802 = fma(r972797, r972801, r972796);
        double r972803 = r972789 ? r972794 : r972802;
        return r972803;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 5567.039381079165

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(N\right)} \cdot \sqrt{\mathsf{log1p}\left(N\right)}} - \log N\]

    5. Applied fma-neg0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)}\]

    if 5567.039381079165 < N

    1. Initial program 59.4

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

    2. Simplified59.4

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5567.039381079165:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))