Average Error: 29.1 → 0.1
Time: 5.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9256.959046884599956683814525604248046875:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9256.959046884599956683814525604248046875:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r57172 = N;
        double r57173 = 1.0;
        double r57174 = r57172 + r57173;
        double r57175 = log(r57174);
        double r57176 = log(r57172);
        double r57177 = r57175 - r57176;
        return r57177;
}
↓
double f(double N) {
        double r57178 = N;
        double r57179 = 9256.9590468846;
        bool r57180 = r57178 <= r57179;
        double r57181 = 1.0;
        double r57182 = r57178 + r57181;
        double r57183 = r57182 / r57178;
        double r57184 = log(r57183);
        double r57185 = 1.0;
        double r57186 = 2.0;
        double r57187 = pow(r57178, r57186);
        double r57188 = r57185 / r57187;
        double r57189 = 0.3333333333333333;
        double r57190 = r57189 / r57178;
        double r57191 = 0.5;
        double r57192 = r57190 - r57191;
        double r57193 = r57181 / r57178;
        double r57194 = fma(r57188, r57192, r57193);
        double r57195 = r57180 ? r57184 : r57194;
        return r57195;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 9256.9590468846

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    if 9256.9590468846 < N

    1. Initial program 59.7

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

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9256.959046884599956683814525604248046875:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))