Average Error: 63.0 → 0
Time: 17.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(-\log n, 1, \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \mathsf{fma}\left(-\log n, 1, \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)
double f(double n) {
        double r57103 = n;
        double r57104 = 1.0;
        double r57105 = r57103 + r57104;
        double r57106 = log(r57105);
        double r57107 = r57105 * r57106;
        double r57108 = log(r57103);
        double r57109 = r57103 * r57108;
        double r57110 = r57107 - r57109;
        double r57111 = r57110 - r57104;
        return r57111;
}
↓
double f(double n) {
        double r57112 = 0.5;
        double r57113 = n;
        double r57114 = r57112 / r57113;
        double r57115 = log(r57113);
        double r57116 = -r57115;
        double r57117 = 1.0;
        double r57118 = 0.16666666666666669;
        double r57119 = r57113 * r57113;
        double r57120 = r57118 / r57119;
        double r57121 = fma(r57116, r57117, r57120);
        double r57122 = r57114 - r57121;
        return r57122;
}

Error

Bits error versus n

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

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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))