Average Error: 15.3 → 0.4
Time: 13.0s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N + 1, 1\right)}
double f(double N) {
        double r5728214 = N;
        double r5728215 = 1.0;
        double r5728216 = r5728214 + r5728215;
        double r5728217 = atan(r5728216);
        double r5728218 = atan(r5728214);
        double r5728219 = r5728217 - r5728218;
        return r5728219;
}
↓
double f(double N) {
        double r5728220 = 1.0;
        double r5728221 = N;
        double r5728222 = r5728221 + r5728220;
        double r5728223 = 1.0;
        double r5728224 = fma(r5728221, r5728222, r5728223);
        double r5728225 = atan2(r5728220, r5728224);
        return r5728225;
}

Error

Bits error versus N

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. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

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

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

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