Average Error: 15.3 → 0.4
Time: 10.5s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + N \cdot \left(1 + N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + N \cdot \left(1 + N\right)}
double f(double N) {
        double r6401534 = N;
        double r6401535 = 1.0;
        double r6401536 = r6401534 + r6401535;
        double r6401537 = atan(r6401536);
        double r6401538 = atan(r6401534);
        double r6401539 = r6401537 - r6401538;
        return r6401539;
}
↓
double f(double N) {
        double r6401540 = 1.0;
        double r6401541 = 1.0;
        double r6401542 = N;
        double r6401543 = r6401540 + r6401542;
        double r6401544 = r6401542 * r6401543;
        double r6401545 = r6401541 + r6401544;
        double r6401546 = atan2(r6401540, r6401545);
        return r6401546;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.3

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

  3. Applied diff-atan14.2

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(FPCore (N)
  :name "2atan (example 3.5)"

  :herbie-target
  (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0)))))

  (- (atan (+ N 1.0)) (atan N)))