Average Error: 63.0 → 0
Time: 16.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{-0.1666666666666666851703837437526090070605}{n \cdot n} + \mathsf{fma}\left(1, \log n, \frac{0.5}{n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{-0.1666666666666666851703837437526090070605}{n \cdot n} + \mathsf{fma}\left(1, \log n, \frac{0.5}{n}\right)
double f(double n) {
        double r147310 = n;
        double r147311 = 1.0;
        double r147312 = r147310 + r147311;
        double r147313 = log(r147312);
        double r147314 = r147312 * r147313;
        double r147315 = log(r147310);
        double r147316 = r147310 * r147315;
        double r147317 = r147314 - r147316;
        double r147318 = r147317 - r147311;
        return r147318;
}
↓
double f(double n) {
        double r147319 = 0.16666666666666669;
        double r147320 = -r147319;
        double r147321 = n;
        double r147322 = r147321 * r147321;
        double r147323 = r147320 / r147322;
        double r147324 = 1.0;
        double r147325 = log(r147321);
        double r147326 = 0.5;
        double r147327 = r147326 / r147321;
        double r147328 = fma(r147324, r147325, r147327);
        double r147329 = r147323 + r147328;
        return r147329;
}

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. Simplified61.9

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

  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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019194 +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))