Average Error: 14.9 → 0.4
Time: 13.5s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{N + \left(1 + N \cdot N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{N + \left(1 + N \cdot N\right)}
double f(double N) {
        double r4789266 = N;
        double r4789267 = 1.0;
        double r4789268 = r4789266 + r4789267;
        double r4789269 = atan(r4789268);
        double r4789270 = atan(r4789266);
        double r4789271 = r4789269 - r4789270;
        return r4789271;
}
↓
double f(double N) {
        double r4789272 = 1.0;
        double r4789273 = N;
        double r4789274 = r4789273 * r4789273;
        double r4789275 = r4789272 + r4789274;
        double r4789276 = r4789273 + r4789275;
        double r4789277 = atan2(r4789272, r4789276);
        return r4789277;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  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-sqr-sqrt1.0

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

  8. Applied add-sqr-sqrt1.0

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

  9. Applied sqrt-prod0.6

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

  10. Taylor expanded around 0 0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(FPCore (N)
  :name "2atan (example 3.5)"

  :herbie-target
  (atan (/ 1 (+ 1 (* N (+ N 1)))))

  (- (atan (+ N 1)) (atan N)))