Average Error: 15.2 → 0.4
Time: 14.0s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{N \cdot N + \left(N + 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{N \cdot N + \left(N + 1\right)}
double f(double N) {
        double r4594301 = N;
        double r4594302 = 1.0;
        double r4594303 = r4594301 + r4594302;
        double r4594304 = atan(r4594303);
        double r4594305 = atan(r4594301);
        double r4594306 = r4594304 - r4594305;
        return r4594306;
}
↓
double f(double N) {
        double r4594307 = 1.0;
        double r4594308 = N;
        double r4594309 = r4594308 * r4594308;
        double r4594310 = r4594308 + r4594307;
        double r4594311 = r4594309 + r4594310;
        double r4594312 = atan2(r4594307, r4594311);
        return r4594312;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.2
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.2

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

  3. Applied diff-atan14.1

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

  4. Simplified0.4

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

  5. Taylor expanded around 0 0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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