logs (example 3.8)

Average Error: 63.0 → 0.0

Time: 4.0s

Precision: binary64

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

Math

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

↓

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

Error

Bits error versus n

Target

Original	63.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 63.0

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

2. Taylor expanded around inf 0.0

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

4. Applied log-rec 0.0

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

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020168 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))