Average Error: 63.0 → 0
Time: 1.5m
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\log \left(e^{1 + \left(\left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \left(\frac{0.5}{n} - 1\right)\right)}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\log \left(e^{1 + \left(\left(1 \cdot \log n - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \left(\frac{0.5}{n} - 1\right)\right)}\right)
double f(double n) {
        double r58641 = n;
        double r58642 = 1.0;
        double r58643 = r58641 + r58642;
        double r58644 = log(r58643);
        double r58645 = r58643 * r58644;
        double r58646 = log(r58641);
        double r58647 = r58641 * r58646;
        double r58648 = r58645 - r58647;
        double r58649 = r58648 - r58642;
        return r58649;
}
↓
double f(double n) {
        double r58650 = 1.0;
        double r58651 = n;
        double r58652 = log(r58651);
        double r58653 = r58650 * r58652;
        double r58654 = 0.16666666666666669;
        double r58655 = r58651 * r58651;
        double r58656 = r58654 / r58655;
        double r58657 = r58653 - r58656;
        double r58658 = 0.5;
        double r58659 = r58658 / r58651;
        double r58660 = r58659 - r58650;
        double r58661 = r58657 + r58660;
        double r58662 = r58650 + r58661;
        double r58663 = exp(r58662);
        double r58664 = log(r58663);
        return r58664;
}

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

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

  5. Applied add-log-exp0.0

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

  6. Applied add-log-exp0.0

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

  7. Applied add-log-exp0.0

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

  8. Applied add-log-exp0.0

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

  9. Applied sum-log0.0

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

  10. Applied add-log-exp0.0

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

  11. Applied diff-log0.1

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

  12. Applied sum-log0.1

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

  13. Applied diff-log0.1

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

  14. Simplified0

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

  15. Final simplification0

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

Reproduce

herbie shell --seed 2019323 
(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))