Average Error: 15.2 → 0.4
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 r6410082 = N;
        double r6410083 = 1.0;
        double r6410084 = r6410082 + r6410083;
        double r6410085 = atan(r6410084);
        double r6410086 = atan(r6410082);
        double r6410087 = r6410085 - r6410086;
        return r6410087;
}
↓
double f(double N) {
        double r6410088 = 1.0;
        double r6410089 = N;
        double r6410090 = r6410089 + r6410088;
        double r6410091 = fma(r6410089, r6410090, r6410088);
        double r6410092 = atan2(r6410088, r6410091);
        return r6410092;
}

Error

Bits error versus N

Target

Original15.2
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

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. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

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

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