Average Error: 14.8 → 0.7
Time: 2.8s
Precision: binary64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{N + 1}} \cdot \sqrt[3]{\sqrt[3]{N + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{N + 1}}\right) \cdot N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{N + 1}} \cdot \sqrt[3]{\sqrt[3]{N + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{N + 1}}\right) \cdot N\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) (1.0 + ((double) (((double) (((double) cbrt(((double) (N + 1.0)))) * ((double) cbrt(((double) (N + 1.0)))))) * ((double) (((double) (((double) (((double) cbrt(((double) cbrt(((double) (N + 1.0)))))) * ((double) cbrt(((double) cbrt(((double) (N + 1.0)))))))) * ((double) cbrt(((double) cbrt(((double) (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
Herbie0.7
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.8

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

  3. Applied diff-atan13.7

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

  4. Simplified0.4

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

  6. Applied add-cube-cbrt0.6

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

  7. Applied associate-*l*0.6

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

  9. Applied add-cube-cbrt0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2020147 
(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)))