Average Error: 15.0 → 0.3
Time: 2.1s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}\right)\right)\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}\right)\right)
double code(double N) {
	return (atan((N + 1.0)) - atan(N));
}
↓
double code(double N) {
	return log1p(expm1(atan2(1.0, fma(N, (N + 1.0), 1.0))));
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.0

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

  3. Applied diff-atan13.8

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  7. Applied log1p-expm1-u0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}\right)\right)}\]

  8. Final simplification0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}\right)\right)\]

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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