Average Error: 63.0 → 0.0
Time: 17.9s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)
double f(double n) {
        double r3093702 = n;
        double r3093703 = 1.0;
        double r3093704 = r3093702 + r3093703;
        double r3093705 = log(r3093704);
        double r3093706 = r3093704 * r3093705;
        double r3093707 = log(r3093702);
        double r3093708 = r3093702 * r3093707;
        double r3093709 = r3093706 - r3093708;
        double r3093710 = r3093709 - r3093703;
        return r3093710;
}
↓
double f(double n) {
        double r3093711 = 0.5;
        double r3093712 = n;
        double r3093713 = r3093711 / r3093712;
        double r3093714 = 1.0;
        double r3093715 = log(r3093712);
        double r3093716 = -r3093715;
        double r3093717 = 0.16666666666666669;
        double r3093718 = r3093717 / r3093712;
        double r3093719 = r3093718 / r3093712;
        double r3093720 = fma(r3093714, r3093716, r3093719);
        double r3093721 = r3093713 - r3093720;
        return r3093721;
}

Error

Bits error versus n

Target

Original63.0
Target0
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. 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. Simplified0.0

    \[\leadsto \color{blue}{\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)}\]

  5. Final simplification0.0

    \[\leadsto \frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\right)\]

Reproduce

herbie shell --seed 2019192 +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))