2atan (example 3.5)

Average Error: 14.8 → 0.4

Time: 10.2s

Precision: 64

Math

\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]

↓

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

C

double f(double N) {
        double r17507655 = N;
        double r17507656 = 1.0;
        double r17507657 = r17507655 + r17507656;
        double r17507658 = atan(r17507657);
        double r17507659 = atan(r17507655);
        double r17507660 = r17507658 - r17507659;
        return r17507660;
}

↓

double f(double N) {
        double r17507661 = 1.0;
        double r17507662 = N;
        double r17507663 = r17507662 + r17507661;
        double r17507664 = fma(r17507662, r17507663, r17507661);
        double r17507665 = atan2(r17507661, r17507664);
        return r17507665;
}

Error

Bits error versus N

Target

Original	14.8
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 14.8

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

3. Applied diff-atan 13.7

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

4. Simplified 0.4

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

5. Simplified 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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