Average Error: 15.3 → 0.3
Time: 12.8s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \left(N + 1\right), 1\right)}
double f(double N) {
        double r3855271 = N;
        double r3855272 = 1.0;
        double r3855273 = r3855271 + r3855272;
        double r3855274 = atan(r3855273);
        double r3855275 = atan(r3855271);
        double r3855276 = r3855274 - r3855275;
        return r3855276;
}
↓
double f(double N) {
        double r3855277 = 1.0;
        double r3855278 = N;
        double r3855279 = r3855278 + r3855277;
        double r3855280 = fma(r3855278, r3855279, r3855277);
        double r3855281 = atan2(r3855277, r3855280);
        return r3855281;
}

Error

Bits error versus N

Target

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

Derivation

  1. Initial program 15.3

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

  3. Applied diff-atan14.1

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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