Average Error: 29.4 → 0.1
Time: 7.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le \frac{40885599}{281474976710656}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{\frac{6004799503160661}{18014398509481984}}{N} - \frac{1}{2}\right) + \frac{1}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}} \cdot \sqrt{\frac{N + 1}{N}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \le \frac{40885599}{281474976710656}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{\frac{6004799503160661}{18014398509481984}}{N} - \frac{1}{2}\right) + \frac{1}{N}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}} \cdot \sqrt{\frac{N + 1}{N}}\right)\\

\end{array}
double f(double N) {
        double r25151 = N;
        double r25152 = 1.0;
        double r25153 = r25151 + r25152;
        double r25154 = log(r25153);
        double r25155 = log(r25151);
        double r25156 = r25154 - r25155;
        return r25156;
}
↓
double f(double N) {
        double r25157 = N;
        double r25158 = 1.0;
        double r25159 = r25157 + r25158;
        double r25160 = log(r25159);
        double r25161 = log(r25157);
        double r25162 = r25160 - r25161;
        double r25163 = 40885599.0;
        double r25164 = 281474976710656.0;
        double r25165 = r25163 / r25164;
        bool r25166 = r25162 <= r25165;
        double r25167 = 1.0;
        double r25168 = 2.0;
        double r25169 = pow(r25157, r25168);
        double r25170 = r25167 / r25169;
        double r25171 = 6004799503160661.0;
        double r25172 = 18014398509481984.0;
        double r25173 = r25171 / r25172;
        double r25174 = r25173 / r25157;
        double r25175 = 2.0;
        double r25176 = r25158 / r25175;
        double r25177 = r25174 - r25176;
        double r25178 = r25170 * r25177;
        double r25179 = r25158 / r25157;
        double r25180 = r25178 + r25179;
        double r25181 = r25159 / r25157;
        double r25182 = sqrt(r25181);
        double r25183 = r25182 * r25182;
        double r25184 = log(r25183);
        double r25185 = r25166 ? r25180 : r25184;
        return r25185;
}

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 (- (log (+ N 1.0)) (log N)) < 1.4525482683325208e-07

    1. Initial program 60.0

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

    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

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{\frac{6004799503160661}{18014398509481984}}{N} - \frac{1}{2}\right) + \frac{1}{N}}\]

    if 1.4525482683325208e-07 < (- (log (+ N 1.0)) (log N))

    1. Initial program 0.3

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

    3. Applied diff-log0.2

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

    5. Applied add-sqr-sqrt0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le \frac{40885599}{281474976710656}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{\frac{6004799503160661}{18014398509481984}}{N} - \frac{1}{2}\right) + \frac{1}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}} \cdot \sqrt{\frac{N + 1}{N}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))