Average Error: 63.0 → 0
Time: 20.3s
Precision: 64
\[n \gt 6.8 \cdot 10^{+15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)
double f(double n) {
        double r3800110 = n;
        double r3800111 = 1.0;
        double r3800112 = r3800110 + r3800111;
        double r3800113 = log(r3800112);
        double r3800114 = r3800112 * r3800113;
        double r3800115 = log(r3800110);
        double r3800116 = r3800110 * r3800115;
        double r3800117 = r3800114 - r3800116;
        double r3800118 = r3800117 - r3800111;
        return r3800118;
}
↓
double f(double n) {
        double r3800119 = 0.5;
        double r3800120 = n;
        double r3800121 = r3800119 / r3800120;
        double r3800122 = 0.16666666666666666;
        double r3800123 = r3800120 * r3800120;
        double r3800124 = r3800122 / r3800123;
        double r3800125 = log(r3800120);
        double r3800126 = r3800124 - r3800125;
        double r3800127 = r3800121 - r3800126;
        return r3800127;
}

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
\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

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

    \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{n}\right) - \left(\frac{1}{6} \cdot \frac{1}{{n}^{2}} + \log \left(\frac{1}{n}\right)\right)\right)} - 1\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{2}}{n} + \left(1 + \frac{\frac{-1}{6}}{n \cdot n}\right)\right) + \log n\right)} - 1\]

  4. Taylor expanded around 0 0

    \[\leadsto \color{blue}{\left(\log n + \frac{1}{2} \cdot \frac{1}{n}\right) - \frac{1}{6} \cdot \frac{1}{{n}^{2}}}\]

  5. Simplified0

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)}\]

  6. Final simplification0

    \[\leadsto \frac{\frac{1}{2}}{n} - \left(\frac{\frac{1}{6}}{n \cdot n} - \log n\right)\]

Reproduce

herbie shell --seed 2019162 
(FPCore (n)
  :name "logs (example 3.8)"
  :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))