Average Error: 29.3 → 0.1
Time: 4.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5876.780919713783:\\ \;\;\;\;\log \left(\frac{1}{\sqrt{N}}\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 5876.780919713783:\\
\;\;\;\;\log \left(\frac{1}{\sqrt{N}}\right) + \log \left(\frac{N + 1}{\sqrt{N}}\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 r54457 = N;
        double r54458 = 1.0;
        double r54459 = r54457 + r54458;
        double r54460 = log(r54459);
        double r54461 = log(r54457);
        double r54462 = r54460 - r54461;
        return r54462;
}
↓
double f(double N) {
        double r54463 = N;
        double r54464 = 5876.780919713783;
        bool r54465 = r54463 <= r54464;
        double r54466 = 1.0;
        double r54467 = sqrt(r54463);
        double r54468 = r54466 / r54467;
        double r54469 = log(r54468);
        double r54470 = 1.0;
        double r54471 = r54463 + r54470;
        double r54472 = r54471 / r54467;
        double r54473 = log(r54472);
        double r54474 = r54469 + r54473;
        double r54475 = 2.0;
        double r54476 = pow(r54463, r54475);
        double r54477 = r54466 / r54476;
        double r54478 = 0.3333333333333333;
        double r54479 = r54478 / r54463;
        double r54480 = 0.5;
        double r54481 = r54479 - r54480;
        double r54482 = r54470 / r54463;
        double r54483 = fma(r54477, r54481, r54482);
        double r54484 = r54465 ? r54474 : r54483;
        return r54484;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 5876.780919713783

    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 *-un-lft-identity0.1

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

    7. Applied times-frac0.1

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

    8. Applied log-prod0.1

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

    if 5876.780919713783 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5876.780919713783:\\ \;\;\;\;\log \left(\frac{1}{\sqrt{N}}\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

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