Average Error: 63.0 → 0
Time: 6.2s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[1 \cdot \log n + \left(0.5 \cdot \frac{1}{n} - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
1 \cdot \log n + \left(0.5 \cdot \frac{1}{n} - \frac{0.1666666666666666851703837437526090070605}{{n}^{2}}\right)
double f(double n) {
        double r79183 = n;
        double r79184 = 1.0;
        double r79185 = r79183 + r79184;
        double r79186 = log(r79185);
        double r79187 = r79185 * r79186;
        double r79188 = log(r79183);
        double r79189 = r79183 * r79188;
        double r79190 = r79187 - r79189;
        double r79191 = r79190 - r79184;
        return r79191;
}
↓
double f(double n) {
        double r79192 = 1.0;
        double r79193 = n;
        double r79194 = log(r79193);
        double r79195 = r79192 * r79194;
        double r79196 = 0.5;
        double r79197 = 1.0;
        double r79198 = r79197 / r79193;
        double r79199 = r79196 * r79198;
        double r79200 = 0.16666666666666669;
        double r79201 = 2.0;
        double r79202 = pow(r79193, r79201);
        double r79203 = r79200 / r79202;
        double r79204 = r79199 - r79203;
        double r79205 = r79195 + r79204;
        return r79205;
}

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

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

  4. Taylor expanded around 0 0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019318 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e15)

  :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))