Average Error: 63.0 → 0.0
Time: 12.8s
Precision: 64
\[n \gt 6.8 \cdot 10^{15}\]
\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]
↓
\[\left(1 \cdot \log n + \left(\left(1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}\right)\right) - 1\]
\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1
↓
\left(1 \cdot \log n + \left(\left(1 - \frac{0.1666666666666666851703837437526090070605}{n \cdot n}\right) + \frac{0.5}{n}\right)\right) - 1
double f(double n) {
        double r3162936 = n;
        double r3162937 = 1.0;
        double r3162938 = r3162936 + r3162937;
        double r3162939 = log(r3162938);
        double r3162940 = r3162938 * r3162939;
        double r3162941 = log(r3162936);
        double r3162942 = r3162936 * r3162941;
        double r3162943 = r3162940 - r3162942;
        double r3162944 = r3162943 - r3162937;
        return r3162944;
}
↓
double f(double n) {
        double r3162945 = 1.0;
        double r3162946 = n;
        double r3162947 = log(r3162946);
        double r3162948 = r3162945 * r3162947;
        double r3162949 = 0.16666666666666669;
        double r3162950 = r3162946 * r3162946;
        double r3162951 = r3162949 / r3162950;
        double r3162952 = r3162945 - r3162951;
        double r3162953 = 0.5;
        double r3162954 = r3162953 / r3162946;
        double r3162955 = r3162952 + r3162954;
        double r3162956 = r3162948 + r3162955;
        double r3162957 = r3162956 - r3162945;
        return r3162957;
}

Error

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 
(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))