Average Error: 63.0 → 0
Time: 1.2m
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(\frac{0.5}{n} + 1 \cdot \log n\right) - \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(\frac{0.5}{n} + 1 \cdot \log n\right) - \frac{\frac{0.1666666666666666851703837437526090070605}{n}}{n}
double f(double n) {
        double r3366258 = n;
        double r3366259 = 1.0;
        double r3366260 = r3366258 + r3366259;
        double r3366261 = log(r3366260);
        double r3366262 = r3366260 * r3366261;
        double r3366263 = log(r3366258);
        double r3366264 = r3366258 * r3366263;
        double r3366265 = r3366262 - r3366264;
        double r3366266 = r3366265 - r3366259;
        return r3366266;
}
↓
double f(double n) {
        double r3366267 = 0.5;
        double r3366268 = n;
        double r3366269 = r3366267 / r3366268;
        double r3366270 = 1.0;
        double r3366271 = log(r3366268);
        double r3366272 = r3366270 * r3366271;
        double r3366273 = r3366269 + r3366272;
        double r3366274 = 0.16666666666666669;
        double r3366275 = r3366274 / r3366268;
        double r3366276 = r3366275 / r3366268;
        double r3366277 = r3366273 - r3366276;
        return r3366277;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Taylor expanded around inf 0.0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019200 
(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))