Average Error: 15.1 → 0.4
Time: 2.2s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{{N}^{2} + \left(1 \cdot N + 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{{N}^{2} + \left(1 \cdot N + 1\right)}
double code(double N) {
	return ((double) (((double) atan(((double) (N + 1.0)))) - ((double) atan(N))));
}
↓
double code(double N) {
	return ((double) atan2(1.0, ((double) (((double) pow(N, 2.0)) + ((double) (((double) (1.0 * N)) + 1.0))))));
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 15.1

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

  3. Applied diff-atan14.0

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

  4. Simplified0.4

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

  6. Applied add-cube-cbrt0.7

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(\sqrt[3]{\left(N + 1\right) \cdot N} \cdot \sqrt[3]{\left(N + 1\right) \cdot N}\right) \cdot \sqrt[3]{\left(N + 1\right) \cdot N}}}\]

  7. Taylor expanded around 0 0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020128 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

  :herbie-target
  (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0)))))

  (- (atan (+ N 1.0)) (atan N)))