Average Error: 29.6 → 0.1
Time: 1.3m
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8072.727785295212:\\ \;\;\;\;\log \left(\frac{\mathsf{fma}\left(\left(N \cdot N\right), N, 1\right)}{\left(N \cdot N + \left(1 - N\right)\right) \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8072.727785295212:\\
\;\;\;\;\log \left(\frac{\mathsf{fma}\left(\left(N \cdot N\right), N, 1\right)}{\left(N \cdot N + \left(1 - N\right)\right) \cdot N}\right)\\

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

\end{array}
double f(double N) {
        double r3576231 = N;
        double r3576232 = 1.0;
        double r3576233 = r3576231 + r3576232;
        double r3576234 = log(r3576233);
        double r3576235 = log(r3576231);
        double r3576236 = r3576234 - r3576235;
        return r3576236;
}
↓
double f(double N) {
        double r3576237 = N;
        double r3576238 = 8072.727785295212;
        bool r3576239 = r3576237 <= r3576238;
        double r3576240 = r3576237 * r3576237;
        double r3576241 = 1.0;
        double r3576242 = fma(r3576240, r3576237, r3576241);
        double r3576243 = r3576241 - r3576237;
        double r3576244 = r3576240 + r3576243;
        double r3576245 = r3576244 * r3576237;
        double r3576246 = r3576242 / r3576245;
        double r3576247 = log(r3576246);
        double r3576248 = r3576241 / r3576237;
        double r3576249 = 0.3333333333333333;
        double r3576250 = r3576249 / r3576237;
        double r3576251 = r3576250 / r3576240;
        double r3576252 = 0.5;
        double r3576253 = r3576252 / r3576240;
        double r3576254 = r3576251 - r3576253;
        double r3576255 = r3576248 + r3576254;
        double r3576256 = r3576239 ? r3576247 : r3576255;
        return r3576256;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8072.727785295212

    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 flip3-+0.1

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

    6. Applied associate-/l/0.1

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

    7. Simplified0.1

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

    if 8072.727785295212 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8072.727785295212:\\ \;\;\;\;\log \left(\frac{\mathsf{fma}\left(\left(N \cdot N\right), N, 1\right)}{\left(N \cdot N + \left(1 - N\right)\right) \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))