Average Error: 15.0 → 0.4
Time: 11.6s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}
double f(double N) {
        double r7216314 = N;
        double r7216315 = 1.0;
        double r7216316 = r7216314 + r7216315;
        double r7216317 = atan(r7216316);
        double r7216318 = atan(r7216314);
        double r7216319 = r7216317 - r7216318;
        return r7216319;
}
↓
double f(double N) {
        double r7216320 = 1.0;
        double r7216321 = N;
        double r7216322 = r7216321 + r7216320;
        double r7216323 = r7216322 * r7216321;
        double r7216324 = r7216323 + r7216320;
        double r7216325 = atan2(r7216320, r7216324);
        return r7216325;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.0

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

  3. Applied diff-atan13.8

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

  4. Simplified0.4

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

  6. Applied *-commutative0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019158 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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