2atan (example 3.5)

Average Error: 14.7 → 0.3

Time: 2.6s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (N) :precision binary64 (atan2 1.0 (+ 1.0 (fma 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 + fma(N, N, N)));
}

Error

Bits error versus N

Target

Original	14.7
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 14.7

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

2. Applied diff-atan_binary64 13.6

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

3. Simplified 0.3

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

4. Simplified 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022067 
(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)))