Average Error: 29.2 → 0.1
Time: 30.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9550.567671573803:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9550.567671573803:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r3570102 = N;
        double r3570103 = 1.0;
        double r3570104 = r3570102 + r3570103;
        double r3570105 = log(r3570104);
        double r3570106 = log(r3570102);
        double r3570107 = r3570105 - r3570106;
        return r3570107;
}
↓
double f(double N) {
        double r3570108 = N;
        double r3570109 = 9550.567671573803;
        bool r3570110 = r3570108 <= r3570109;
        double r3570111 = 1.0;
        double r3570112 = r3570111 + r3570108;
        double r3570113 = r3570112 / r3570108;
        double r3570114 = log(r3570113);
        double r3570115 = r3570111 / r3570108;
        double r3570116 = -0.5;
        double r3570117 = r3570108 * r3570108;
        double r3570118 = r3570116 / r3570117;
        double r3570119 = r3570115 + r3570118;
        double r3570120 = 0.3333333333333333;
        double r3570121 = r3570120 / r3570117;
        double r3570122 = r3570121 / r3570108;
        double r3570123 = r3570119 + r3570122;
        double r3570124 = r3570110 ? r3570114 : r3570123;
        return r3570124;
}

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 < 9550.567671573803

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Simplified0.1

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

    if 9550.567671573803 < N

    1. Initial program 59.6

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

    3. Applied add-log-exp59.6

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

    4. Simplified59.3

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

    6. Applied add-sqr-sqrt59.3

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

    7. Taylor expanded around inf 0.0

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

    8. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9550.567671573803:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))