Average Error: 15.1 → 0.4
Time: 3.5s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
↓
\[\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)} \]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \mathsf{fma}\left(N, N, N\right)}

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))
↓
(FPCore (N) :precision binary64 (atan2 1.0 (+ 1.0 (fma N N N))))
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

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. Applied diff-atan_binary6414.0

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

  3. Simplified0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022103 
(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)))