Average Error: 29.8 → 0.1
Time: 33.8s
Precision: 64
\[\log \left(N + 1.0\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9848.91410832408:\\ \;\;\;\;\log \left(\frac{1.0 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{N \cdot N}, 0.5, \frac{\frac{0.3333333333333333}{N}}{N \cdot N}\right) + \frac{1.0}{N}\\ \end{array}\]
\log \left(N + 1.0\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9848.91410832408:\\
\;\;\;\;\log \left(\frac{1.0 + N}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{N \cdot N}, 0.5, \frac{\frac{0.3333333333333333}{N}}{N \cdot N}\right) + \frac{1.0}{N}\\

\end{array}
double f(double N) {
        double r3420990 = N;
        double r3420991 = 1.0;
        double r3420992 = r3420990 + r3420991;
        double r3420993 = log(r3420992);
        double r3420994 = log(r3420990);
        double r3420995 = r3420993 - r3420994;
        return r3420995;
}
↓
double f(double N) {
        double r3420996 = N;
        double r3420997 = 9848.91410832408;
        bool r3420998 = r3420996 <= r3420997;
        double r3420999 = 1.0;
        double r3421000 = r3420999 + r3420996;
        double r3421001 = r3421000 / r3420996;
        double r3421002 = log(r3421001);
        double r3421003 = -1.0;
        double r3421004 = r3420996 * r3420996;
        double r3421005 = r3421003 / r3421004;
        double r3421006 = 0.5;
        double r3421007 = 0.3333333333333333;
        double r3421008 = r3421007 / r3420996;
        double r3421009 = r3421008 / r3421004;
        double r3421010 = fma(r3421005, r3421006, r3421009);
        double r3421011 = r3420999 / r3420996;
        double r3421012 = r3421010 + r3421011;
        double r3421013 = r3420998 ? r3421002 : r3421012;
        return r3421013;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 9848.91410832408

    1. Initial program 0.1

      \[\log \left(N + 1.0\right) - \log N\]
    2. Using strategy rm

    3. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1.0}{N}\right)}\]

    if 9848.91410832408 < N

    1. Initial program 59.7

      \[\log \left(N + 1.0\right) - \log N\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + 1.0 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{N \cdot N}, 0.5, \frac{\frac{0.3333333333333333}{N}}{N \cdot N}\right) + \frac{1.0}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9848.91410832408:\\ \;\;\;\;\log \left(\frac{1.0 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{N \cdot N}, 0.5, \frac{\frac{0.3333333333333333}{N}}{N \cdot N}\right) + \frac{1.0}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))