Average Error: 63.0 → 0
Time: 4.8s
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 r68173 = n;
        double r68174 = 1.0;
        double r68175 = r68173 + r68174;
        double r68176 = log(r68175);
        double r68177 = r68175 * r68176;
        double r68178 = log(r68173);
        double r68179 = r68173 * r68178;
        double r68180 = r68177 - r68179;
        double r68181 = r68180 - r68174;
        return r68181;
}
↓
double f(double n) {
        double r68182 = 1.0;
        double r68183 = n;
        double r68184 = log(r68183);
        double r68185 = r68182 * r68184;
        double r68186 = 0.5;
        double r68187 = 1.0;
        double r68188 = r68187 / r68183;
        double r68189 = r68186 * r68188;
        double r68190 = 0.16666666666666669;
        double r68191 = 2.0;
        double r68192 = pow(r68183, r68191);
        double r68193 = r68190 / r68192;
        double r68194 = r68189 - r68193;
        double r68195 = r68185 + r68194;
        return r68195;
}

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 2019354 
(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))