Average Error: 29.5 → 0.1
Time: 16.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5400.000739830978:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{\sqrt{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{3}}{N \cdot N}, \frac{1}{N}, \frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 5400.000739830978:\\
\;\;\;\;\log \left(\frac{\sqrt{1 + N}}{\sqrt{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\

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

\end{array}
double f(double N) {
        double r2235076 = N;
        double r2235077 = 1.0;
        double r2235078 = r2235076 + r2235077;
        double r2235079 = log(r2235078);
        double r2235080 = log(r2235076);
        double r2235081 = r2235079 - r2235080;
        return r2235081;
}
↓
double f(double N) {
        double r2235082 = N;
        double r2235083 = 5400.000739830978;
        bool r2235084 = r2235082 <= r2235083;
        double r2235085 = 1.0;
        double r2235086 = r2235085 + r2235082;
        double r2235087 = sqrt(r2235086);
        double r2235088 = sqrt(r2235082);
        double r2235089 = r2235087 / r2235088;
        double r2235090 = log(r2235089);
        double r2235091 = log(r2235087);
        double r2235092 = log(r2235088);
        double r2235093 = r2235091 - r2235092;
        double r2235094 = r2235090 + r2235093;
        double r2235095 = 0.3333333333333333;
        double r2235096 = r2235082 * r2235082;
        double r2235097 = r2235095 / r2235096;
        double r2235098 = r2235085 / r2235082;
        double r2235099 = -0.5;
        double r2235100 = r2235099 / r2235096;
        double r2235101 = r2235100 + r2235098;
        double r2235102 = fma(r2235097, r2235098, r2235101);
        double r2235103 = r2235084 ? r2235094 : r2235102;
        return r2235103;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 5400.000739830978

    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 log1p-udef0.1

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

    5. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity0.1

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

    8. Applied add-sqr-sqrt0.1

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

    9. Applied times-frac0.1

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

    10. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(\frac{\sqrt{1 + N}}{1}\right) + \log \left(\frac{\sqrt{1 + N}}{N}\right)}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt0.1

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

    13. Applied *-un-lft-identity0.1

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

    14. Applied sqrt-prod0.1

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

    15. Applied times-frac0.1

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

    16. Applied log-prod0.1

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

    17. Applied associate-+r+0.1

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

    18. Simplified0.1

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

    if 5400.000739830978 < 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. Using strategy rm

    4. Applied log1p-udef59.4

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

    5. Applied diff-log59.2

      \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5400.000739830978:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{\sqrt{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{3}}{N \cdot N}, \frac{1}{N}, \frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))