Average Error: 15.2 → 0.4
Time: 23.4s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N \cdot N + 1\right) + 1 \cdot N}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\left(N \cdot N + 1\right) + 1 \cdot N}
double f(double N) {
        double r5841396 = N;
        double r5841397 = 1.0;
        double r5841398 = r5841396 + r5841397;
        double r5841399 = atan(r5841398);
        double r5841400 = atan(r5841396);
        double r5841401 = r5841399 - r5841400;
        return r5841401;
}
↓
double f(double N) {
        double r5841402 = 1.0;
        double r5841403 = N;
        double r5841404 = r5841403 * r5841403;
        double r5841405 = 1.0;
        double r5841406 = r5841404 + r5841405;
        double r5841407 = r5841402 * r5841403;
        double r5841408 = r5841406 + r5841407;
        double r5841409 = atan2(r5841402, r5841408);
        return r5841409;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.2
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.2

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

  3. Applied diff-atan14.1

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

  4. Simplified0.4

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

  5. Taylor expanded around 0 0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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