2log (problem 3.3.6)

Average Error: 29.1 → 0.1

Time: 16.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 9772.093561518631759099662303924560546875:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, \sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\frac{3}{2}}} \cdot \sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\frac{3}{2}}}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]

C

double f(double N) {
        double r41342 = N;
        double r41343 = 1.0;
        double r41344 = r41342 + r41343;
        double r41345 = log(r41344);
        double r41346 = log(r41342);
        double r41347 = r41345 - r41346;
        return r41347;
}

↓

double f(double N) {
        double r41348 = N;
        double r41349 = 9772.093561518632;
        bool r41350 = r41348 <= r41349;
        double r41351 = 1.0;
        double r41352 = r41348 + r41351;
        double r41353 = r41352 / r41348;
        double r41354 = log(r41353);
        double r41355 = 1.0;
        double r41356 = r41355 / r41348;
        double r41357 = 0.5;
        double r41358 = r41357 / r41348;
        double r41359 = r41351 - r41358;
        double r41360 = 1.5;
        double r41361 = pow(r41359, r41360);
        double r41362 = cbrt(r41361);
        double r41363 = r41362 * r41362;
        double r41364 = 0.3333333333333333;
        double r41365 = 3.0;
        double r41366 = pow(r41348, r41365);
        double r41367 = r41364 / r41366;
        double r41368 = fma(r41356, r41363, r41367);
        double r41369 = r41350 ? r41354 : r41368;
        return r41369;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 9772.093561518632

   1. Initial program 0.1

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

   3. Applied diff-log 0.1

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

   if 9772.093561518632 < N

   1. Initial program 59.6

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

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{N}, \color{blue}{\sqrt[3]{\left(\left(1 - \frac{0.5}{N}\right) \cdot \left(1 - \frac{0.5}{N}\right)\right) \cdot \left(1 - \frac{0.5}{N}\right)}}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\]

   6. Simplified 0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{N}, \sqrt[3]{\color{blue}{{\left(1 - \frac{0.5}{N}\right)}^{3}}}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\]
   7. Using strategy rm

   8. Applied sqr-pow 0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{N}, \sqrt[3]{\color{blue}{{\left(1 - \frac{0.5}{N}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(1 - \frac{0.5}{N}\right)}^{\left(\frac{3}{2}\right)}}}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\]

   9. Applied cbrt-prod 0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{N}, \color{blue}{\sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\left(\frac{3}{2}\right)}}}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\]

   10. Simplified 0.0

       \[\leadsto \mathsf{fma}\left(\frac{1}{N}, \color{blue}{\sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\frac{3}{2}}}} \cdot \sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\left(\frac{3}{2}\right)}}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\]

   11. Simplified 0.0

       \[\leadsto \mathsf{fma}\left(\frac{1}{N}, \sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\frac{3}{2}}} \cdot \color{blue}{\sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\frac{3}{2}}}}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9772.093561518631759099662303924560546875:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, \sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\frac{3}{2}}} \cdot \sqrt[3]{{\left(1 - \frac{0.5}{N}\right)}^{\frac{3}{2}}}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))