Average Error: 15.1 → 0.3
Time: 9.5s
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 r10473941 = N;
        double r10473942 = 1.0;
        double r10473943 = r10473941 + r10473942;
        double r10473944 = atan(r10473943);
        double r10473945 = atan(r10473941);
        double r10473946 = r10473944 - r10473945;
        return r10473946;
}
↓
double f(double N) {
        double r10473947 = 1.0;
        double r10473948 = N;
        double r10473949 = r10473948 + r10473947;
        double r10473950 = fma(r10473948, r10473949, r10473947);
        double r10473951 = atan2(r10473947, r10473950);
        return r10473951;
}


\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

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

Derivation

  1. Initial program 15.1

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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