logs (example 3.8)

Average Error: 63.0 → 0

Time: 3.2s

Precision: 64

\[n \gt 6.8 \cdot 10^{15}\]

Math

\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

↓

\[\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)\]

C

double code(double n) {
	return ((double) (((double) (((double) (((double) (n + 1.0)) * ((double) log(((double) (n + 1.0)))))) - ((double) (n * ((double) log(n)))))) - 1.0));
}

↓

double code(double n) {
	return ((double) fma(((double) (1.0 / n)), ((double) (0.5 - ((double) (0.16666666666666669 / n)))), ((double) (((double) log(n)) * 1.0))));
}

Error

Bits error versus n

Target

Original	63.0
Target	0
Herbie	0

\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

1. Initial program 63.0

   \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

2. Simplified 61.9

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(n + 1\right), n + 1, -\mathsf{fma}\left(\log n, n, 1\right)\right)}\]

3. Taylor expanded around inf 0.0

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{n} - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)}\]

4. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)}\]

5. Final simplification 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{n}, 0.5 - \frac{0.16666666666666669}{n}, \log n \cdot 1\right)\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (* n n))) (/ 4 (pow n 3)))))

  (- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1))