Average Error: 29.6 → 0.1
Time: 7.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le \frac{1975991590850343}{274877906944}:\\ \;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{\frac{6004799503160661}{18014398509481984}}{N} - \frac{1}{2}\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le \frac{1975991590850343}{274877906944}:\\
\;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{\frac{6004799503160661}{18014398509481984}}{N} - \frac{1}{2}\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r44552 = N;
        double r44553 = 1.0;
        double r44554 = r44552 + r44553;
        double r44555 = log(r44554);
        double r44556 = log(r44552);
        double r44557 = r44555 - r44556;
        return r44557;
}
↓
double f(double N) {
        double r44558 = N;
        double r44559 = 1975991590850343.0;
        double r44560 = 274877906944.0;
        double r44561 = r44559 / r44560;
        bool r44562 = r44558 <= r44561;
        double r44563 = 1.0;
        double r44564 = r44558 + r44563;
        double r44565 = sqrt(r44564);
        double r44566 = sqrt(r44558);
        double r44567 = r44565 / r44566;
        double r44568 = log(r44567);
        double r44569 = r44568 + r44568;
        double r44570 = 1.0;
        double r44571 = 2.0;
        double r44572 = pow(r44558, r44571);
        double r44573 = r44570 / r44572;
        double r44574 = 6004799503160661.0;
        double r44575 = 18014398509481984.0;
        double r44576 = r44574 / r44575;
        double r44577 = r44576 / r44558;
        double r44578 = 2.0;
        double r44579 = r44563 / r44578;
        double r44580 = r44577 - r44579;
        double r44581 = r44573 * r44580;
        double r44582 = r44563 / r44558;
        double r44583 = r44581 + r44582;
        double r44584 = r44562 ? r44569 : r44583;
        return r44584;
}

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 N < 7188.615530504987

    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

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied times-frac0.1

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

    8. Applied log-prod0.1

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

    if 7188.615530504987 < N

    1. Initial program 59.6

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

    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

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{\frac{6004799503160661}{18014398509481984}}{N} - \frac{1}{2}\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le \frac{1975991590850343}{274877906944}:\\ \;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{\frac{6004799503160661}{18014398509481984}}{N} - \frac{1}{2}\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))