Average Error: 63.0 → 0
Time: 26.1s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} + 1\right) - \left(\left(\frac{0.1666666666666666851703837437526090070605}{n \cdot n} - \log n \cdot 1\right) + 1\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} + 1\right) - \left(\left(\frac{0.1666666666666666851703837437526090070605}{n \cdot n} - \log n \cdot 1\right) + 1\right)
double f(double n) {
        double r73002 = n;
        double r73003 = 1.0;
        double r73004 = r73002 + r73003;
        double r73005 = log(r73004);
        double r73006 = r73004 * r73005;
        double r73007 = log(r73002);
        double r73008 = r73002 * r73007;
        double r73009 = r73006 - r73008;
        double r73010 = r73009 - r73003;
        return r73010;
}
↓
double f(double n) {
        double r73011 = 0.5;
        double r73012 = n;
        double r73013 = r73011 / r73012;
        double r73014 = 1.0;
        double r73015 = r73013 + r73014;
        double r73016 = 0.16666666666666669;
        double r73017 = r73012 * r73012;
        double r73018 = r73016 / r73017;
        double r73019 = log(r73012);
        double r73020 = r73019 * r73014;
        double r73021 = r73018 - r73020;
        double r73022 = r73021 + r73014;
        double r73023 = r73015 - r73022;
        return r73023;
}

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

    \[\leadsto \color{blue}{\left(\left(\left(\frac{0.5}{n} + 1\right) - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\right)} - 1\]
  4. Using strategy rm

  5. Applied associate-+l-0.0

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

  6. Applied associate--l-0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2019323 
(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))