2log (problem 3.3.6)

Average Error: 29.2 → 0.0

Time: 30.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 9504.960256628374:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{3}, \frac{\frac{1}{N \cdot N}}{N}, \frac{1}{N}\right)\right)\\ \end{array}\]

C

double f(double N) {
        double r1004236 = N;
        double r1004237 = 1.0;
        double r1004238 = r1004236 + r1004237;
        double r1004239 = log(r1004238);
        double r1004240 = log(r1004236);
        double r1004241 = r1004239 - r1004240;
        return r1004241;
}

↓

double f(double N) {
        double r1004242 = N;
        double r1004243 = 9504.960256628374;
        bool r1004244 = r1004242 <= r1004243;
        double r1004245 = 1.0;
        double r1004246 = r1004245 + r1004242;
        double r1004247 = r1004242 / r1004246;
        double r1004248 = log(r1004247);
        double r1004249 = -r1004248;
        double r1004250 = r1004242 * r1004242;
        double r1004251 = r1004245 / r1004250;
        double r1004252 = -0.5;
        double r1004253 = 0.3333333333333333;
        double r1004254 = r1004251 / r1004242;
        double r1004255 = r1004245 / r1004242;
        double r1004256 = fma(r1004253, r1004254, r1004255);
        double r1004257 = fma(r1004251, r1004252, r1004256);
        double r1004258 = r1004244 ? r1004249 : r1004257;
        return r1004258;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 9504.960256628374

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
   3. Using strategy rm

   4. Applied log1p-udef 0.1

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

   5. Applied diff-log 0.1

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

   7. Applied *-un-lft-identity 0.1

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

   8. Applied associate-/l* 0.1

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

   10. Applied add-exp-log 0.1

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

   11. Applied rec-exp 0.1

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

   12. Applied rem-log-exp 0.1

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

   if 9504.960256628374 < N

   1. Initial program 59.6

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

   2. Simplified 59.6

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
   3. Using strategy rm

   4. Applied log1p-udef 59.6

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

   5. Applied diff-log 59.4

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

   7. Applied *-un-lft-identity 59.4

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

   8. Applied associate-/l* 59.4

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

   9. Taylor expanded around -inf 0.0

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

   10. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9504.960256628374:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{3}, \frac{\frac{1}{N \cdot N}}{N}, \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))