Average Error: 15.0 → 0.4
Time: 10.0s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}
double f(double N) {
        double r14886380 = N;
        double r14886381 = 1.0;
        double r14886382 = r14886380 + r14886381;
        double r14886383 = atan(r14886382);
        double r14886384 = atan(r14886380);
        double r14886385 = r14886383 - r14886384;
        return r14886385;
}
↓
double f(double N) {
        double r14886386 = 1.0;
        double r14886387 = N;
        double r14886388 = r14886387 + r14886386;
        double r14886389 = r14886388 * r14886387;
        double r14886390 = r14886389 + r14886386;
        double r14886391 = atan2(r14886386, r14886390);
        return r14886391;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.0

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  2. Using strategy rm

  3. Applied diff-atan13.9

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}}\]

  4. Simplified0.4

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{1 + \left(N + 1\right) \cdot N}\]

  5. Final simplification0.4

    \[\leadsto \tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (N)
  :name "2atan (example 3.5)"

  :herbie-target
  (atan (/ 1 (+ 1 (* N (+ N 1)))))

  (- (atan (+ N 1)) (atan N)))