Average Error: 63.0 → 0.0
Time: 4.6s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\left(1 - \left(1 \cdot \left(\log \left(\frac{1}{\sqrt{n}}\right) + \log \left(\frac{1}{\sqrt{n}}\right)\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\left(1 - \left(1 \cdot \left(\log \left(\frac{1}{\sqrt{n}}\right) + \log \left(\frac{1}{\sqrt{n}}\right)\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1
double f(double n) {
        double r71031 = n;
        double r71032 = 1.0;
        double r71033 = r71031 + r71032;
        double r71034 = log(r71033);
        double r71035 = r71033 * r71034;
        double r71036 = log(r71031);
        double r71037 = r71031 * r71036;
        double r71038 = r71035 - r71037;
        double r71039 = r71038 - r71032;
        return r71039;
}
↓
double f(double n) {
        double r71040 = 1.0;
        double r71041 = 1.0;
        double r71042 = n;
        double r71043 = sqrt(r71042);
        double r71044 = r71041 / r71043;
        double r71045 = log(r71044);
        double r71046 = r71045 + r71045;
        double r71047 = r71040 * r71046;
        double r71048 = 0.16666666666666669;
        double r71049 = 2.0;
        double r71050 = pow(r71042, r71049);
        double r71051 = r71041 / r71050;
        double r71052 = r71048 * r71051;
        double r71053 = r71047 + r71052;
        double r71054 = r71040 - r71053;
        double r71055 = 0.5;
        double r71056 = r71055 / r71042;
        double r71057 = r71054 + r71056;
        double r71058 = r71057 - r71040;
        return r71058;
}

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

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(1 - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right)} - 1\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \log \left(\frac{1}{\color{blue}{\sqrt{n} \cdot \sqrt{n}}}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  6. Applied add-cube-cbrt0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{n} \cdot \sqrt{n}}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  7. Applied times-frac0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \log \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{n}} \cdot \frac{\sqrt[3]{1}}{\sqrt{n}}\right)} + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  8. Applied log-prod0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{n}}\right) + \log \left(\frac{\sqrt[3]{1}}{\sqrt{n}}\right)\right)} + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  9. Simplified0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \left(\color{blue}{\log \left(\frac{1}{\sqrt{n}}\right)} + \log \left(\frac{\sqrt[3]{1}}{\sqrt{n}}\right)\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  10. Simplified0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \left(\log \left(\frac{1}{\sqrt{n}}\right) + \color{blue}{\log \left(\frac{1}{\sqrt{n}}\right)}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

  11. Final simplification0.0

    \[\leadsto \left(\left(1 - \left(1 \cdot \left(\log \left(\frac{1}{\sqrt{n}}\right) + \log \left(\frac{1}{\sqrt{n}}\right)\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)\right) + \frac{0.5}{n}\right) - 1\]

Reproduce

herbie shell --seed 2020033 
(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))