Average Error: 15.2 → 0.3
Time: 11.6s
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 r5139068 = N;
        double r5139069 = 1.0;
        double r5139070 = r5139068 + r5139069;
        double r5139071 = atan(r5139070);
        double r5139072 = atan(r5139068);
        double r5139073 = r5139071 - r5139072;
        return r5139073;
}
↓
double f(double N) {
        double r5139074 = 1.0;
        double r5139075 = N;
        double r5139076 = r5139075 + r5139074;
        double r5139077 = 1.0;
        double r5139078 = fma(r5139075, r5139076, r5139077);
        double r5139079 = atan2(r5139074, r5139078);
        return r5139079;
}

Error

Bits error versus N

Target

Original15.2
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.2

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

  3. Applied diff-atan14.0

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

  4. Taylor expanded around 0 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +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)))