Average Error: 63.0 → 0.0
Time: 13.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n + \left(\left(1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(1 \cdot \log n + \left(\left(1 + \frac{0.5}{n}\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\right) - 1
double f(double n) {
        double r4315012 = n;
        double r4315013 = 1.0;
        double r4315014 = r4315012 + r4315013;
        double r4315015 = log(r4315014);
        double r4315016 = r4315014 * r4315015;
        double r4315017 = log(r4315012);
        double r4315018 = r4315012 * r4315017;
        double r4315019 = r4315016 - r4315018;
        double r4315020 = r4315019 - r4315013;
        return r4315020;
}
↓
double f(double n) {
        double r4315021 = 1.0;
        double r4315022 = n;
        double r4315023 = log(r4315022);
        double r4315024 = r4315021 * r4315023;
        double r4315025 = 0.5;
        double r4315026 = r4315025 / r4315022;
        double r4315027 = r4315021 + r4315026;
        double r4315028 = 0.16666666666666669;
        double r4315029 = r4315022 * r4315022;
        double r4315030 = r4315028 / r4315029;
        double r4315031 = r4315027 - r4315030;
        double r4315032 = r4315024 + r4315031;
        double r4315033 = r4315032 - r4315021;
        return r4315033;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 
(FPCore (n)
  :name "logs (example 3.8)"
  :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))