2atan (example 3.5)

Average Error: 14.8 → 0.4

Time: 10.7s

Precision: 64

Math

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

↓

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

C

double f(double N) {
        double r3792403 = N;
        double r3792404 = 1.0;
        double r3792405 = r3792403 + r3792404;
        double r3792406 = atan(r3792405);
        double r3792407 = atan(r3792403);
        double r3792408 = r3792406 - r3792407;
        return r3792408;
}

↓

double f(double N) {
        double r3792409 = 1.0;
        double r3792410 = N;
        double r3792411 = fma(r3792410, r3792410, r3792409);
        double r3792412 = r3792410 + r3792411;
        double r3792413 = atan2(r3792409, r3792412);
        return r3792413;
}

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 13.8

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

4. Simplified 0.4

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

5. Simplified 0.4

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

7. Applied fma-udef 0.4

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

8. Applied associate-+r+ 0.4

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

9. Simplified 0.4

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

10. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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