Average Error: 15.6 → 0.5
Time: 2.2s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}
double f(double N) {
        double r131062 = N;
        double r131063 = 1.0;
        double r131064 = r131062 + r131063;
        double r131065 = atan(r131064);
        double r131066 = atan(r131062);
        double r131067 = r131065 - r131066;
        return r131067;
}
↓
double f(double N) {
        double r131068 = 1.0;
        double r131069 = 1.0;
        double r131070 = N;
        double r131071 = r131070 + r131068;
        double r131072 = r131071 * r131070;
        double r131073 = r131069 + r131072;
        double r131074 = atan2(r131068, r131073);
        return r131074;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.6
Target0.5
Herbie0.5
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.6

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

  3. Applied diff-atan14.3

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020025 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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