Average Error: 63.0 → 0
Time: 9.5s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(1 \cdot \log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}
double f(double n) {
        double r321084 = n;
        double r321085 = 1.0;
        double r321086 = r321084 + r321085;
        double r321087 = log(r321086);
        double r321088 = r321086 * r321087;
        double r321089 = log(r321084);
        double r321090 = r321084 * r321089;
        double r321091 = r321088 - r321090;
        double r321092 = r321091 - r321085;
        return r321092;
}
↓
double f(double n) {
        double r321093 = 1.0;
        double r321094 = n;
        double r321095 = log(r321094);
        double r321096 = r321093 * r321095;
        double r321097 = 2.0;
        double r321098 = r321093 / r321097;
        double r321099 = r321098 / r321094;
        double r321100 = r321096 + r321099;
        double r321101 = 3002399751580331.0;
        double r321102 = 18014398509481984.0;
        double r321103 = r321101 / r321102;
        double r321104 = 2.0;
        double r321105 = pow(r321094, r321104);
        double r321106 = r321103 / r321105;
        double r321107 = r321100 - r321106;
        return r321107;
}

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

    \[\leadsto \color{blue}{\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + \frac{3002399751580331}{18014398509481984} \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{\frac{1}{2}}{n}\right)} - 1\]
  4. Using strategy rm

  5. Applied associate-+l-0.0

    \[\leadsto \color{blue}{\left(1 - \left(\left(1 \cdot \log \left(\frac{1}{n}\right) + \frac{3002399751580331}{18014398509481984} \cdot \frac{1}{{n}^{2}}\right) - \frac{\frac{1}{2}}{n}\right)\right)} - 1\]

  6. Applied associate--l-0.0

    \[\leadsto \color{blue}{1 - \left(\left(\left(1 \cdot \log \left(\frac{1}{n}\right) + \frac{3002399751580331}{18014398509481984} \cdot \frac{1}{{n}^{2}}\right) - \frac{\frac{1}{2}}{n}\right) + 1\right)}\]

  7. Simplified0.0

    \[\leadsto 1 - \color{blue}{\left(\left(1 + \left(\frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}} + 1 \cdot \log \left(\frac{1}{n}\right)\right)\right) - \frac{\frac{1}{2}}{n}\right)}\]

  8. Final simplification0

    \[\leadsto \left(1 \cdot \log n + \frac{\frac{1}{2}}{n}\right) - \frac{\frac{3002399751580331}{18014398509481984}}{{n}^{2}}\]

Reproduce

herbie shell --seed 350497007 
(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))