Average Error: 14.8 → 0.4
Time: 2.2s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\sqrt{1 + N \cdot \left(1 + N\right)} \cdot \sqrt{1 + N \cdot \left(1 + N\right)}}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\sqrt{1 + N \cdot \left(1 + N\right)} \cdot \sqrt{1 + N \cdot \left(1 + N\right)}}
(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))
↓
(FPCore (N)
 :precision binary64
 (atan2 1.0 (* (sqrt (+ 1.0 (* N (+ 1.0 N)))) (sqrt (+ 1.0 (* N (+ 1.0 N)))))))
double code(double N) {
	return atan(N + 1.0) - atan(N);
}
↓
double code(double N) {
	return atan2(1.0, (sqrt(1.0 + (N * (1.0 + N))) * sqrt(1.0 + (N * (1.0 + N)))));
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.8

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

  3. Applied diff-atan_binary64_262213.7

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

  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. Using strategy rm

  7. Applied add-sqr-sqrt_binary64_24870.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020356 
(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)))