Average Error: 29.0 → 0.2
Time: 4.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4600.68399624307585:\\ \;\;\;\;\left(\log \left(\sqrt{N + 1}\right) - \left(\frac{2}{3} \cdot \log N + \log \left(\sqrt[3]{1}\right)\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4600.68399624307585:\\
\;\;\;\;\left(\log \left(\sqrt{N + 1}\right) - \left(\frac{2}{3} \cdot \log N + \log \left(\sqrt[3]{1}\right)\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r56094 = N;
        double r56095 = 1.0;
        double r56096 = r56094 + r56095;
        double r56097 = log(r56096);
        double r56098 = log(r56094);
        double r56099 = r56097 - r56098;
        return r56099;
}
↓
double f(double N) {
        double r56100 = N;
        double r56101 = 4600.683996243076;
        bool r56102 = r56100 <= r56101;
        double r56103 = 1.0;
        double r56104 = r56100 + r56103;
        double r56105 = sqrt(r56104);
        double r56106 = log(r56105);
        double r56107 = 0.6666666666666666;
        double r56108 = log(r56100);
        double r56109 = r56107 * r56108;
        double r56110 = 1.0;
        double r56111 = cbrt(r56110);
        double r56112 = log(r56111);
        double r56113 = r56109 + r56112;
        double r56114 = r56106 - r56113;
        double r56115 = cbrt(r56100);
        double r56116 = r56105 / r56115;
        double r56117 = log(r56116);
        double r56118 = r56114 + r56117;
        double r56119 = 2.0;
        double r56120 = pow(r56100, r56119);
        double r56121 = r56110 / r56120;
        double r56122 = 0.3333333333333333;
        double r56123 = r56122 / r56100;
        double r56124 = 0.5;
        double r56125 = r56123 - r56124;
        double r56126 = r56121 * r56125;
        double r56127 = r56103 / r56100;
        double r56128 = r56126 + r56127;
        double r56129 = r56102 ? r56118 : r56128;
        return r56129;
}

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 < 4600.683996243076

    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-cube-cbrt0.2

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

    6. Applied add-sqr-sqrt0.2

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

    7. Applied times-frac0.2

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

    8. Applied log-prod0.4

      \[\leadsto \color{blue}{\log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity0.4

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

    11. Applied cbrt-prod0.4

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

    12. Applied *-un-lft-identity0.4

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

    13. Applied sqrt-prod0.4

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

    14. Applied times-frac0.4

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

    15. Applied log-prod0.4

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

    16. Applied associate-+r+0.4

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

    17. Simplified0.3

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

    if 4600.683996243076 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4600.68399624307585:\\ \;\;\;\;\left(\log \left(\sqrt{N + 1}\right) - \left(\frac{2}{3} \cdot \log N + \log \left(\sqrt[3]{1}\right)\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))