Average Error: 63.0 → 0.0
Time: 11.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(\frac{\frac{1}{2}}{n} + 1\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(\frac{\frac{1}{2}}{n} + 1\right) - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\right) - 1
double f(double n) {
        double r982080 = n;
        double r982081 = 1.0;
        double r982082 = r982080 + r982081;
        double r982083 = log(r982082);
        double r982084 = r982082 * r982083;
        double r982085 = log(r982080);
        double r982086 = r982080 * r982085;
        double r982087 = r982084 - r982086;
        double r982088 = r982087 - r982081;
        return r982088;
}
↓
double f(double n) {
        double r982089 = 0.5;
        double r982090 = n;
        double r982091 = r982089 / r982090;
        double r982092 = 1.0;
        double r982093 = r982091 + r982092;
        double r982094 = 0.16666666666666666;
        double r982095 = r982090 * r982090;
        double r982096 = r982094 / r982095;
        double r982097 = log(r982090);
        double r982098 = r982096 - r982097;
        double r982099 = r982093 - r982098;
        double r982100 = r982099 - r982092;
        return r982100;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original63.0
Target0.0
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

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019154 
(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))