logs (example 3.8)

Average Error: 63.0 → 0.1

Time: 3.7s

Precision: binary64

\[n > 6.8 \cdot 10^{+15}\]

Math

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

↓

\[\log \left(\frac{e \cdot n}{e}\right)\]

FPCore

(FPCore (n)
 :precision binary64
 (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))

↓

(FPCore (n) :precision binary64 (log (/ (* E n) E)))

C

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

↓

double code(double n) {
	return log((((double) M_E) * n) / ((double) M_E));
}

Error

Bits error versus n

Target

Original	63.0
Target	0.0
Herbie	0.1

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

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

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

3. Simplified 0.0

   \[\leadsto \color{blue}{\left(1 + \log n\right)} - 1\]
4. Using strategy rm

5. Applied add-log-exp_binary64 0.0

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

6. Applied add-log-exp_binary64 0.0

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

7. Applied sum-log_binary64 0.1

   \[\leadsto \color{blue}{\log \left(e^{1} \cdot n\right)} - \log \left(e^{1}\right)\]

8. Applied diff-log_binary64 0.1

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

9. Final simplification 0.1

   \[\leadsto \log \left(\frac{e \cdot n}{e}\right)\]

Alternatives

Reproduce

herbie shell --seed 2021110 
(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))