Average Error: 63.0 → 0
Time: 4.3s
Precision: binary64
\[n > 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\log \left(n \cdot e^{\frac{0.5}{n} + \frac{-0.16666666666666666}{n \cdot n}}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log \left(n \cdot e^{\frac{0.5}{n} + \frac{-0.16666666666666666}{n \cdot n}}\right)
(FPCore (n)
 :precision binary64
 (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))
↓
(FPCore (n)
 :precision binary64
 (log (* n (exp (+ (/ 0.5 n) (/ -0.16666666666666666 (* n n)))))))
double code(double n) {
	return (((n + 1.0) * log(n + 1.0)) - (n * log(n))) - 1.0;
}
↓
double code(double n) {
	return log(n * exp((0.5 / n) + (-0.16666666666666666 / (n * n))));
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0
Herbie0
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(\log \left(\frac{1}{n}\right) + 0.16666666666666666 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(1 + \frac{0.5}{n}\right) - \left(\frac{0.16666666666666666}{n \cdot n} - \log n\right)\right)} - 1\]
  4. Using strategy rm

  5. Applied add-log-exp_binary64_4580.0

    \[\leadsto \left(\left(1 + \frac{0.5}{n}\right) - \left(\frac{0.16666666666666666}{n \cdot n} - \log n\right)\right) - \color{blue}{\log \left(e^{1}\right)}\]

  6. Applied add-log-exp_binary64_4580.0

    \[\leadsto \left(\left(1 + \frac{0.5}{n}\right) - \left(\color{blue}{\log \left(e^{\frac{0.16666666666666666}{n \cdot n}}\right)} - \log n\right)\right) - \log \left(e^{1}\right)\]

  7. Applied diff-log_binary64_5110.0

    \[\leadsto \left(\left(1 + \frac{0.5}{n}\right) - \color{blue}{\log \left(\frac{e^{\frac{0.16666666666666666}{n \cdot n}}}{n}\right)}\right) - \log \left(e^{1}\right)\]

  8. Applied add-log-exp_binary64_4580.0

    \[\leadsto \left(\left(1 + \color{blue}{\log \left(e^{\frac{0.5}{n}}\right)}\right) - \log \left(\frac{e^{\frac{0.16666666666666666}{n \cdot n}}}{n}\right)\right) - \log \left(e^{1}\right)\]

  9. Applied add-log-exp_binary64_4580.0

    \[\leadsto \left(\left(\color{blue}{\log \left(e^{1}\right)} + \log \left(e^{\frac{0.5}{n}}\right)\right) - \log \left(\frac{e^{\frac{0.16666666666666666}{n \cdot n}}}{n}\right)\right) - \log \left(e^{1}\right)\]

  10. Applied sum-log_binary64_5100.0

    \[\leadsto \left(\color{blue}{\log \left(e^{1} \cdot e^{\frac{0.5}{n}}\right)} - \log \left(\frac{e^{\frac{0.16666666666666666}{n \cdot n}}}{n}\right)\right) - \log \left(e^{1}\right)\]

  11. Applied diff-log_binary64_5110.1

    \[\leadsto \color{blue}{\log \left(\frac{e^{1} \cdot e^{\frac{0.5}{n}}}{\frac{e^{\frac{0.16666666666666666}{n \cdot n}}}{n}}\right)} - \log \left(e^{1}\right)\]

  12. Applied diff-log_binary64_5110.1

    \[\leadsto \color{blue}{\log \left(\frac{\frac{e^{1} \cdot e^{\frac{0.5}{n}}}{\frac{e^{\frac{0.16666666666666666}{n \cdot n}}}{n}}}{e^{1}}\right)}\]

  13. Simplified0

    \[\leadsto \log \color{blue}{\left(n \cdot e^{\frac{0.5}{n} + \frac{-0.16666666666666666}{n \cdot n}}\right)}\]

  14. Final simplification0

    \[\leadsto \log \left(n \cdot e^{\frac{0.5}{n} + \frac{-0.16666666666666666}{n \cdot n}}\right)\]

Reproduce

herbie shell --seed 2020355 
(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))