Average Error: 63.0 → 0
Time: 14.3s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \log n \cdot 1
double f(double n) {
        double r2217137 = n;
        double r2217138 = 1.0;
        double r2217139 = r2217137 + r2217138;
        double r2217140 = log(r2217139);
        double r2217141 = r2217139 * r2217140;
        double r2217142 = log(r2217137);
        double r2217143 = r2217137 * r2217142;
        double r2217144 = r2217141 - r2217143;
        double r2217145 = r2217144 - r2217138;
        return r2217145;
}
↓
double f(double n) {
        double r2217146 = 0.5;
        double r2217147 = n;
        double r2217148 = r2217146 / r2217147;
        double r2217149 = 0.16666666666666669;
        double r2217150 = r2217147 * r2217147;
        double r2217151 = r2217149 / r2217150;
        double r2217152 = r2217148 - r2217151;
        double r2217153 = log(r2217147);
        double r2217154 = 1.0;
        double r2217155 = r2217153 * r2217154;
        double r2217156 = r2217152 + r2217155;
        return r2217156;
}

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. Simplified62.0

    \[\leadsto \color{blue}{\log \left(1 + n\right) \cdot \left(1 + n\right) - \mathsf{fma}\left(n, \log n, 1\right)}\]

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))