Average Error: 63.0 → 0
Time: 13.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\log n - \frac{\frac{\frac{1}{6}}{n}}{n}\right) + \frac{\frac{1}{2}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\log n - \frac{\frac{\frac{1}{6}}{n}}{n}\right) + \frac{\frac{1}{2}}{n}
double f(double n) {
        double r641076 = n;
        double r641077 = 1.0;
        double r641078 = r641076 + r641077;
        double r641079 = log(r641078);
        double r641080 = r641078 * r641079;
        double r641081 = log(r641076);
        double r641082 = r641076 * r641081;
        double r641083 = r641080 - r641082;
        double r641084 = r641083 - r641077;
        return r641084;
}
↓
double f(double n) {
        double r641085 = n;
        double r641086 = log(r641085);
        double r641087 = 0.16666666666666666;
        double r641088 = r641087 / r641085;
        double r641089 = r641088 / r641085;
        double r641090 = r641086 - r641089;
        double r641091 = 0.5;
        double r641092 = r641091 / r641085;
        double r641093 = r641090 + r641092;
        return r641093;
}

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

    \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{n}\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(1 + \frac{\frac{1}{2}}{n}\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right)} - 1\]

  4. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)}\]

  5. Simplified0

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{n} + \left(\log n - \frac{\frac{\frac{1}{6}}{n}}{n}\right)}\]

  6. Final simplification0

    \[\leadsto \left(\log n - \frac{\frac{\frac{1}{6}}{n}}{n}\right) + \frac{\frac{1}{2}}{n}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (n)
  :name "logs (example 3.8)"
  :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))