Average Error: 15.0 → 0.4
Time: 10.6s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N + N \cdot N\right) + 1}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\left(N + N \cdot N\right) + 1}
double f(double N) {
        double r4123479 = N;
        double r4123480 = 1.0;
        double r4123481 = r4123479 + r4123480;
        double r4123482 = atan(r4123481);
        double r4123483 = atan(r4123479);
        double r4123484 = r4123482 - r4123483;
        return r4123484;
}
↓
double f(double N) {
        double r4123485 = 1.0;
        double r4123486 = N;
        double r4123487 = r4123486 * r4123486;
        double r4123488 = r4123486 + r4123487;
        double r4123489 = r4123488 + r4123485;
        double r4123490 = atan2(r4123485, r4123489);
        return r4123490;
}

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. Taylor expanded around inf 0.4

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(N + {N}^{2}\right)}}\]

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019133 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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