Average Error: 63.0 → 0.0
Time: 3.4s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.1666666666666666851703837437526090070605 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\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
double f(double n) {
        double r66196 = n;
        double r66197 = 1.0;
        double r66198 = r66196 + r66197;
        double r66199 = log(r66198);
        double r66200 = r66198 * r66199;
        double r66201 = log(r66196);
        double r66202 = r66196 * r66201;
        double r66203 = r66200 - r66202;
        double r66204 = r66203 - r66197;
        return r66204;
}
↓
double f(double n) {
        double r66205 = 1.0;
        double r66206 = 1.0;
        double r66207 = n;
        double r66208 = r66206 / r66207;
        double r66209 = log(r66208);
        double r66210 = r66205 * r66209;
        double r66211 = 0.16666666666666669;
        double r66212 = 2.0;
        double r66213 = pow(r66207, r66212);
        double r66214 = r66206 / r66213;
        double r66215 = r66211 * r66214;
        double r66216 = r66210 + r66215;
        double r66217 = r66205 - r66216;
        double r66218 = 0.5;
        double r66219 = r66218 / r66207;
        double r66220 = r66217 + r66219;
        double r66221 = r66220 - r66205;
        return r66221;
}

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

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019347 
(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))