Average Error: 29.3 → 0.1
Time: 6.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5113.2558232039937:\\ \;\;\;\;\log \left(\frac{\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt[3]{{N}^{3} + {1}^{3}}}{\sqrt{N} \cdot \sqrt[3]{N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 5113.2558232039937:\\
\;\;\;\;\log \left(\frac{\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt[3]{{N}^{3} + {1}^{3}}}{\sqrt{N} \cdot \sqrt[3]{N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)}}\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 r55411 = N;
        double r55412 = 1.0;
        double r55413 = r55411 + r55412;
        double r55414 = log(r55413);
        double r55415 = log(r55411);
        double r55416 = r55414 - r55415;
        return r55416;
}
↓
double f(double N) {
        double r55417 = N;
        double r55418 = 5113.255823203994;
        bool r55419 = r55417 <= r55418;
        double r55420 = 1.0;
        double r55421 = r55417 + r55420;
        double r55422 = cbrt(r55421);
        double r55423 = r55422 * r55422;
        double r55424 = sqrt(r55417);
        double r55425 = r55423 / r55424;
        double r55426 = log(r55425);
        double r55427 = 3.0;
        double r55428 = pow(r55417, r55427);
        double r55429 = pow(r55420, r55427);
        double r55430 = r55428 + r55429;
        double r55431 = cbrt(r55430);
        double r55432 = r55417 * r55417;
        double r55433 = r55420 * r55420;
        double r55434 = r55417 * r55420;
        double r55435 = r55433 - r55434;
        double r55436 = r55432 + r55435;
        double r55437 = cbrt(r55436);
        double r55438 = r55424 * r55437;
        double r55439 = r55431 / r55438;
        double r55440 = log(r55439);
        double r55441 = r55426 + r55440;
        double r55442 = 1.0;
        double r55443 = 2.0;
        double r55444 = pow(r55417, r55443);
        double r55445 = r55442 / r55444;
        double r55446 = 0.3333333333333333;
        double r55447 = r55446 / r55417;
        double r55448 = 0.5;
        double r55449 = r55447 - r55448;
        double r55450 = r55445 * r55449;
        double r55451 = r55420 / r55417;
        double r55452 = r55450 + r55451;
        double r55453 = r55419 ? r55441 : r55452;
        return r55453;
}

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 < 5113.255823203994

    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-cube-cbrt0.1

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

    7. Applied times-frac0.1

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

    8. Applied log-prod0.1

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

    10. Applied flip3-+0.1

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

    11. Applied cbrt-div0.1

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

    12. Applied associate-/l/0.1

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

    if 5113.255823203994 < N

    1. Initial program 59.5

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

    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 simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5113.2558232039937:\\ \;\;\;\;\log \left(\frac{\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt[3]{{N}^{3} + {1}^{3}}}{\sqrt{N} \cdot \sqrt[3]{N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))