Average Error: 29.1 → 0.1
Time: 15.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4148.853859252381880651228129863739013672:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4148.853859252381880651228129863739013672:\\
\;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r54982 = N;
        double r54983 = 1.0;
        double r54984 = r54982 + r54983;
        double r54985 = log(r54984);
        double r54986 = log(r54982);
        double r54987 = r54985 - r54986;
        return r54987;
}
↓
double f(double N) {
        double r54988 = N;
        double r54989 = 4148.853859252382;
        bool r54990 = r54988 <= r54989;
        double r54991 = 1.0;
        double r54992 = r54988 + r54991;
        double r54993 = log(r54992);
        double r54994 = 3.0;
        double r54995 = pow(r54993, r54994);
        double r54996 = cbrt(r54995);
        double r54997 = log(r54988);
        double r54998 = r54996 - r54997;
        double r54999 = 0.5;
        double r55000 = r54999 / r54988;
        double r55001 = r54991 - r55000;
        double r55002 = r55001 / r54988;
        double r55003 = 0.3333333333333333;
        double r55004 = pow(r54988, r54994);
        double r55005 = r55003 / r55004;
        double r55006 = r55002 + r55005;
        double r55007 = r54990 ? r54998 : r55006;
        return r55007;
}

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 < 4148.853859252382

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied add-cbrt-cube0.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(N + 1\right) \cdot \log \left(N + 1\right)\right) \cdot \log \left(N + 1\right)}} - \log N\]

    4. Simplified0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\log \left(N + 1\right)\right)}^{3}}} - \log N\]

    if 4148.853859252382 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)}\]
    4. Using strategy rm

    5. Applied fma-udef0.0

      \[\leadsto \color{blue}{\frac{1}{N} \cdot \left(1 - \frac{0.5}{N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}}\]

    6. Simplified0.0

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N}} + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4148.853859252381880651228129863739013672:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))