Average Error: 15.0 → 0.4
Time: 13.1s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{(N \cdot \left(N + 1\right) + 1)_*}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{(N \cdot \left(N + 1\right) + 1)_*}
double f(double N) {
        double r7437449 = N;
        double r7437450 = 1.0;
        double r7437451 = r7437449 + r7437450;
        double r7437452 = atan(r7437451);
        double r7437453 = atan(r7437449);
        double r7437454 = r7437452 - r7437453;
        return r7437454;
}
↓
double f(double N) {
        double r7437455 = 1.0;
        double r7437456 = N;
        double r7437457 = r7437456 + r7437455;
        double r7437458 = fma(r7437456, r7437457, r7437455);
        double r7437459 = atan2(r7437455, r7437458);
        return r7437459;
}

Error

Bits error versus N

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-atan13.9

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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