Average Error: 14.9 → 0.4
Time: 14.8s
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 r94485 = N;
        double r94486 = 1.0;
        double r94487 = r94485 + r94486;
        double r94488 = atan(r94487);
        double r94489 = atan(r94485);
        double r94490 = r94488 - r94489;
        return r94490;
}
↓
double f(double N) {
        double r94491 = 1.0;
        double r94492 = 1.0;
        double r94493 = N;
        double r94494 = r94493 + r94491;
        double r94495 = r94494 * r94493;
        double r94496 = r94492 + r94495;
        double r94497 = atan2(r94491, r94496);
        return r94497;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.7

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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