Average Error: 29.0 → 0.2
Time: 4.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7693.69820421591703:\\ \;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{{N}^{\frac{1}{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7693.69820421591703:\\
\;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{{N}^{\frac{1}{3}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r67404 = N;
        double r67405 = 1.0;
        double r67406 = r67404 + r67405;
        double r67407 = log(r67406);
        double r67408 = log(r67404);
        double r67409 = r67407 - r67408;
        return r67409;
}
↓
double f(double N) {
        double r67410 = N;
        double r67411 = 7693.698204215917;
        bool r67412 = r67410 <= r67411;
        double r67413 = 1.0;
        double r67414 = r67410 + r67413;
        double r67415 = sqrt(r67414);
        double r67416 = cbrt(r67410);
        double r67417 = r67416 * r67416;
        double r67418 = r67415 / r67417;
        double r67419 = log(r67418);
        double r67420 = 0.3333333333333333;
        double r67421 = pow(r67410, r67420);
        double r67422 = r67415 / r67421;
        double r67423 = log(r67422);
        double r67424 = r67419 + r67423;
        double r67425 = 1.0;
        double r67426 = 2.0;
        double r67427 = pow(r67410, r67426);
        double r67428 = r67425 / r67427;
        double r67429 = 0.3333333333333333;
        double r67430 = r67429 / r67410;
        double r67431 = 0.5;
        double r67432 = r67430 - r67431;
        double r67433 = r67413 / r67410;
        double r67434 = fma(r67428, r67432, r67433);
        double r67435 = r67412 ? r67424 : r67434;
        return r67435;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 7693.698204215917

    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 pow1/30.3

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

    if 7693.698204215917 < 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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7693.69820421591703:\\ \;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{{N}^{\frac{1}{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

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