Average Error: 14.7 → 0.4
Time: 2.2s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{{N}^{2} + \left(1 \cdot N + 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{{N}^{2} + \left(1 \cdot N + 1\right)}
double code(double N) {
	return ((double) (((double) atan(((double) (N + 1.0)))) - ((double) atan(N))));
}
↓
double code(double N) {
	return ((double) atan2(1.0, ((double) (((double) pow(N, 2.0)) + ((double) (((double) (1.0 * N)) + 1.0))))));
}

Error

Bits error versus N

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.7

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

  3. Applied diff-atan13.5

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

  4. Simplified0.3

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

  6. Applied add-cube-cbrt0.6

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot \color{blue}{\left(\left(\sqrt[3]{N} \cdot \sqrt[3]{N}\right) \cdot \sqrt[3]{N}\right)}}\]

  7. Applied associate-*r*0.6

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(\left(N + 1\right) \cdot \left(\sqrt[3]{N} \cdot \sqrt[3]{N}\right)\right) \cdot \sqrt[3]{N}}}\]

  8. Taylor expanded around 0 0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020168 
(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)))