Average Error: 15.3 → 0.4
Time: 1.1m
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(N \cdot N + 1 \cdot N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(N \cdot N + 1 \cdot N\right)}
double f(double N) {
        double r7232038 = N;
        double r7232039 = 1.0;
        double r7232040 = r7232038 + r7232039;
        double r7232041 = atan(r7232040);
        double r7232042 = atan(r7232038);
        double r7232043 = r7232041 - r7232042;
        return r7232043;
}
↓
double f(double N) {
        double r7232044 = 1.0;
        double r7232045 = 1.0;
        double r7232046 = N;
        double r7232047 = r7232046 * r7232046;
        double r7232048 = r7232044 * r7232046;
        double r7232049 = r7232047 + r7232048;
        double r7232050 = r7232045 + r7232049;
        double r7232051 = atan2(r7232044, r7232050);
        return r7232051;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

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. Simplified0.4

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

  5. Taylor expanded around 0 0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 
(FPCore (N)
  :name "2atan (example 3.5)"

  :herbie-target
  (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0)))))

  (- (atan (+ N 1.0)) (atan N)))