Average Error: 29.2 → 0.1
Time: 4.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8090.07577261243296:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8090.07577261243296:\\
\;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\

\end{array}
double f(double N) {
        double r53207 = N;
        double r53208 = 1.0;
        double r53209 = r53207 + r53208;
        double r53210 = log(r53209);
        double r53211 = log(r53207);
        double r53212 = r53210 - r53211;
        return r53212;
}
↓
double f(double N) {
        double r53213 = N;
        double r53214 = 8090.075772612433;
        bool r53215 = r53213 <= r53214;
        double r53216 = 1.0;
        double r53217 = r53213 + r53216;
        double r53218 = r53217 / r53213;
        double r53219 = sqrt(r53218);
        double r53220 = log(r53219);
        double r53221 = r53220 + r53220;
        double r53222 = 1.0;
        double r53223 = r53222 / r53213;
        double r53224 = 0.5;
        double r53225 = r53224 / r53213;
        double r53226 = r53216 - r53225;
        double r53227 = 0.3333333333333333;
        double r53228 = 3.0;
        double r53229 = pow(r53213, r53228);
        double r53230 = r53227 / r53229;
        double r53231 = fma(r53223, r53226, r53230);
        double r53232 = r53215 ? r53221 : r53231;
        return r53232;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8090.075772612433

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied log-prod0.1

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

    if 8090.075772612433 < N

    1. Initial program 59.4

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

    3. Applied diff-log59.2

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

    4. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    5. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8090.07577261243296:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))