2atan (example 3.5)

Average Error: 15.0 → 0.4

Time: 4.4s

Precision: binary64

Math

\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]

↓

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

FPCore

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))

↓

(FPCore (N) :precision binary64 (atan2 1.0 (+ 1.0 (+ N (* N N)))))

C

double code(double N) {
	return atan(N + 1.0) - atan(N);
}

↓

double code(double N) {
	return atan2(1.0, (1.0 + (N + (N * N))));
}

Error

Bits error versus N

Target

Original	15.0
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 15.0

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

3. Applied diff-atan_binary64_2622 13.8

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

4. Simplified 0.4

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

5. Simplified 0.4

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

7. Applied flip-+_binary64_2439 0.4

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

8. Applied associate-*r/_binary64_2407 5.5

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

9. Simplified 5.5

   \[\leadsto \tan^{-1}_* \frac{1}{1 + \frac{\color{blue}{{N}^{3} - N}}{N - 1}}\]

10. Taylor expanded around 0 0.4

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

11. Simplified 0.4

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

12. Final simplification 0.4

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

Reproduce

herbie shell --seed 2021075 
(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)))