Average Error: 3.6 → 2.2
Time: 2.1m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 4.244133814393053 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 4.244133814393053 \cdot 10^{+184}:\\
\;\;\;\;\frac{\frac{\frac{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta) {
        double r5400897 = alpha;
        double r5400898 = beta;
        double r5400899 = r5400897 + r5400898;
        double r5400900 = r5400898 * r5400897;
        double r5400901 = r5400899 + r5400900;
        double r5400902 = 1.0;
        double r5400903 = r5400901 + r5400902;
        double r5400904 = 2.0;
        double r5400905 = 1.0;
        double r5400906 = r5400904 * r5400905;
        double r5400907 = r5400899 + r5400906;
        double r5400908 = r5400903 / r5400907;
        double r5400909 = r5400908 / r5400907;
        double r5400910 = r5400907 + r5400902;
        double r5400911 = r5400909 / r5400910;
        return r5400911;
}
↓
double f(double alpha, double beta) {
        double r5400912 = beta;
        double r5400913 = 4.244133814393053e+184;
        bool r5400914 = r5400912 <= r5400913;
        double r5400915 = 1.0;
        double r5400916 = alpha;
        double r5400917 = fma(r5400912, r5400916, r5400916);
        double r5400918 = r5400917 + r5400912;
        double r5400919 = r5400915 + r5400918;
        double r5400920 = 2.0;
        double r5400921 = r5400912 + r5400916;
        double r5400922 = r5400920 + r5400921;
        double r5400923 = r5400919 / r5400922;
        double r5400924 = r5400923 / r5400922;
        double r5400925 = r5400922 + r5400915;
        double r5400926 = r5400924 / r5400925;
        double r5400927 = 0.0;
        double r5400928 = r5400914 ? r5400926 : r5400927;
        return r5400928;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 4.244133814393053e+184

    1. Initial program 1.6

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified1.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

    if 4.244133814393053e+184 < beta

    1. Initial program 16.5

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified16.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity16.5

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{1 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 1.0\right)}}\]

    5. Applied add-sqr-sqrt16.5

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}}{1 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 1.0\right)}\]

    6. Applied times-frac16.5

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity16.5

      \[\leadsto \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{\left(2 + \left(\beta + \alpha\right)\right) + \color{blue}{1 \cdot 1.0}}\]

    9. Applied *-un-lft-identity16.5

      \[\leadsto \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{\color{blue}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)} + 1 \cdot 1.0}\]

    10. Applied distribute-lft-out16.5

      \[\leadsto \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{\color{blue}{1 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 1.0\right)}}\]

    11. Applied add-sqr-sqrt16.5

      \[\leadsto \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{\color{blue}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}}}{1 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 1.0\right)}\]

    12. Applied div-inv16.5

      \[\leadsto \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\sqrt{\frac{\color{blue}{\left(1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}}{1 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 1.0\right)}\]

    13. Applied times-frac16.5

      \[\leadsto \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\sqrt{\color{blue}{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{1}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}}}{1 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 1.0\right)}\]

    14. Applied sqrt-prod16.5

      \[\leadsto \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\color{blue}{\sqrt{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{\sqrt{2 + \left(\beta + \alpha\right)}}} \cdot \sqrt{\frac{\frac{1}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}}}{1 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 1.0\right)}\]

    15. Applied times-frac16.7

      \[\leadsto \frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \color{blue}{\left(\frac{\sqrt{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{\sqrt{2 + \left(\beta + \alpha\right)}}}}{1} \cdot \frac{\sqrt{\frac{\frac{1}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right)}\]

    16. Applied associate-*r*16.7

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{1} \cdot \frac{\sqrt{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{\sqrt{2 + \left(\beta + \alpha\right)}}}}{1}\right) \cdot \frac{\sqrt{\frac{\frac{1}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

    17. Taylor expanded around -inf 5.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 4.244133814393053 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))