Average Error: 14.7 → 0.5
Time: 2.9s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\left(\sqrt{\sqrt{\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}}} \cdot \sqrt{\sqrt{\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}}}\right) \cdot \sqrt{\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\left(\sqrt{\sqrt{\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}}} \cdot \sqrt{\sqrt{\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}}}\right) \cdot \sqrt{\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}}
double f(double N) {
        double r127073 = N;
        double r127074 = 1.0;
        double r127075 = r127073 + r127074;
        double r127076 = atan(r127075);
        double r127077 = atan(r127073);
        double r127078 = r127076 - r127077;
        return r127078;
}
↓
double f(double N) {
        double r127079 = 1.0;
        double r127080 = 1.0;
        double r127081 = N;
        double r127082 = r127081 + r127079;
        double r127083 = r127082 * r127081;
        double r127084 = r127080 + r127083;
        double r127085 = atan2(r127079, r127084);
        double r127086 = sqrt(r127085);
        double r127087 = sqrt(r127086);
        double r127088 = r127087 * r127087;
        double r127089 = r127088 * r127086;
        return r127089;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.3
Herbie0.5
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.7

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

  3. Applied diff-atan13.5

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

  4. Simplified0.3

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

  6. Applied add-sqr-sqrt0.9

    \[\leadsto \color{blue}{\sqrt{\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}} \cdot \sqrt{\tan^{-1}_* \frac{1}{1 + \left(N + 1\right) \cdot N}}}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.9

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

  9. Applied sqrt-prod0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2020024 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

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

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