Average Error: 15.7 → 0.4
Time: 3.5s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{N + \left(1 + N \cdot N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{N + \left(1 + N \cdot N\right)}
(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))
↓
(FPCore (N) :precision binary64 (atan2 1.0 (+ N (+ 1.0 (* N N)))))
double code(double N) {
	return atan(N + 1.0) - atan(N);
}
↓
double code(double N) {
	return atan2(1.0, (N + (1.0 + (N * N))));
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.7
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.7

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

  3. Applied diff-atan_binary64_228114.5

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Taylor expanded around 0 0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2021046 
(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)))