Average Error: 14.9 → 0.4
Time: 9.3s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\sqrt{1 + \left(N + 1\right) \cdot N} \cdot \sqrt{1 + \left(N + 1\right) \cdot N}}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\sqrt{1 + \left(N + 1\right) \cdot N} \cdot \sqrt{1 + \left(N + 1\right) \cdot N}}
double f(double N) {
        double r55348 = N;
        double r55349 = 1.0;
        double r55350 = r55348 + r55349;
        double r55351 = atan(r55350);
        double r55352 = atan(r55348);
        double r55353 = r55351 - r55352;
        return r55353;
}
↓
double f(double N) {
        double r55354 = 1.0;
        double r55355 = 1.0;
        double r55356 = N;
        double r55357 = r55356 + r55354;
        double r55358 = r55357 * r55356;
        double r55359 = r55355 + r55358;
        double r55360 = sqrt(r55359);
        double r55361 = r55360 * r55360;
        double r55362 = atan2(r55354, r55361);
        return r55362;
}

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. Using strategy rm

  6. Applied add-sqr-sqrt0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019235 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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