Average Error: 15.8 → 0.3
Time: 8.6s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{(N \cdot \left(N + 1\right) + 1)_*}\]
double f(double N) {
        double r8938441 = N;
        double r8938442 = 1.0;
        double r8938443 = r8938441 + r8938442;
        double r8938444 = atan(r8938443);
        double r8938445 = atan(r8938441);
        double r8938446 = r8938444 - r8938445;
        return r8938446;
}
↓
double f(double N) {
        double r8938447 = 1.0;
        double r8938448 = N;
        double r8938449 = r8938448 + r8938447;
        double r8938450 = fma(r8938448, r8938449, r8938447);
        double r8938451 = atan2(r8938447, r8938450);
        return r8938451;
}


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

Error

Bits error versus N

Target

Original15.8
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.8

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

  3. Applied diff-atan14.6

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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