Average Error: 63.0 → 0
Time: 15.7s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\frac{0.5}{n} - \mathsf{fma}\left(1, -\log n, \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right)
double f(double n) {
        double r69314 = n;
        double r69315 = 1.0;
        double r69316 = r69314 + r69315;
        double r69317 = log(r69316);
        double r69318 = r69316 * r69317;
        double r69319 = log(r69314);
        double r69320 = r69314 * r69319;
        double r69321 = r69318 - r69320;
        double r69322 = r69321 - r69315;
        return r69322;
}
↓
double f(double n) {
        double r69323 = 0.5;
        double r69324 = n;
        double r69325 = r69323 / r69324;
        double r69326 = 1.0;
        double r69327 = log(r69324);
        double r69328 = -r69327;
        double r69329 = 0.16666666666666669;
        double r69330 = r69324 * r69324;
        double r69331 = r69329 / r69330;
        double r69332 = fma(r69326, r69328, r69331);
        double r69333 = r69325 - r69332;
        return r69333;
}

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

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e15)

  :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))