Average Error: 63.0 → 0
Time: 16.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\log \left(e^{\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1} \cdot {n}^{1}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log \left(e^{\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1} \cdot {n}^{1}\right)
double f(double n) {
        double r65967 = n;
        double r65968 = 1.0;
        double r65969 = r65967 + r65968;
        double r65970 = log(r65969);
        double r65971 = r65969 * r65970;
        double r65972 = log(r65967);
        double r65973 = r65967 * r65972;
        double r65974 = r65971 - r65973;
        double r65975 = r65974 - r65968;
        return r65975;
}
↓
double f(double n) {
        double r65976 = 0.5;
        double r65977 = n;
        double r65978 = r65976 / r65977;
        double r65979 = 1.0;
        double r65980 = r65978 + r65979;
        double r65981 = 0.16666666666666669;
        double r65982 = r65977 * r65977;
        double r65983 = r65981 / r65982;
        double r65984 = r65980 - r65983;
        double r65985 = r65984 - r65979;
        double r65986 = exp(r65985);
        double r65987 = pow(r65977, r65979);
        double r65988 = r65986 * r65987;
        double r65989 = log(r65988);
        return r65989;
}

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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right)} - 1\]
  4. Using strategy rm

  5. Applied add-log-exp0.0

    \[\leadsto \left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right) - \color{blue}{\log \left(e^{1}\right)}\]

  6. Applied add-log-exp0.0

    \[\leadsto \left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \color{blue}{\log \left(e^{\log n \cdot 1}\right)}\right) - \log \left(e^{1}\right)\]

  7. Applied add-log-exp0.0

    \[\leadsto \left(\left(\left(\frac{0.5}{n} + 1\right) - \color{blue}{\log \left(e^{\frac{0.1666666666666666851703837437526090070605}{n \cdot n}}\right)}\right) + \log \left(e^{\log n \cdot 1}\right)\right) - \log \left(e^{1}\right)\]

  8. Applied add-log-exp0.0

    \[\leadsto \left(\left(\left(\frac{0.5}{n} + \color{blue}{\log \left(e^{1}\right)}\right) - \log \left(e^{\frac{0.1666666666666666851703837437526090070605}{n \cdot n}}\right)\right) + \log \left(e^{\log n \cdot 1}\right)\right) - \log \left(e^{1}\right)\]

  9. Applied add-log-exp0.0

    \[\leadsto \left(\left(\left(\color{blue}{\log \left(e^{\frac{0.5}{n}}\right)} + \log \left(e^{1}\right)\right) - \log \left(e^{\frac{0.1666666666666666851703837437526090070605}{n \cdot n}}\right)\right) + \log \left(e^{\log n \cdot 1}\right)\right) - \log \left(e^{1}\right)\]

  10. Applied sum-log0.0

    \[\leadsto \left(\left(\color{blue}{\log \left(e^{\frac{0.5}{n}} \cdot e^{1}\right)} - \log \left(e^{\frac{0.1666666666666666851703837437526090070605}{n \cdot n}}\right)\right) + \log \left(e^{\log n \cdot 1}\right)\right) - \log \left(e^{1}\right)\]

  11. Applied diff-log0.0

    \[\leadsto \left(\color{blue}{\log \left(\frac{e^{\frac{0.5}{n}} \cdot e^{1}}{e^{\frac{0.1666666666666666851703837437526090070605}{n \cdot n}}}\right)} + \log \left(e^{\log n \cdot 1}\right)\right) - \log \left(e^{1}\right)\]

  12. Applied sum-log0.1

    \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{0.5}{n}} \cdot e^{1}}{e^{\frac{0.1666666666666666851703837437526090070605}{n \cdot n}}} \cdot e^{\log n \cdot 1}\right)} - \log \left(e^{1}\right)\]

  13. Applied diff-log0.1

    \[\leadsto \color{blue}{\log \left(\frac{\frac{e^{\frac{0.5}{n}} \cdot e^{1}}{e^{\frac{0.1666666666666666851703837437526090070605}{n \cdot n}}} \cdot e^{\log n \cdot 1}}{e^{1}}\right)}\]

  14. Simplified0

    \[\leadsto \log \color{blue}{\left(e^{\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1} \cdot {n}^{1}\right)}\]

  15. Final simplification0

    \[\leadsto \log \left(e^{\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) - 1} \cdot {n}^{1}\right)\]

Reproduce

herbie shell --seed 2019351 
(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))