Average Error: 3.7 → 2.3
Time: 9.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\ \;\;\;\;\frac{\frac{1 \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\
\;\;\;\;\frac{\frac{1 \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}\\

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

\end{array}
double f(double alpha, double beta) {
        double r151049 = alpha;
        double r151050 = beta;
        double r151051 = r151049 + r151050;
        double r151052 = r151050 * r151049;
        double r151053 = r151051 + r151052;
        double r151054 = 1.0;
        double r151055 = r151053 + r151054;
        double r151056 = 2.0;
        double r151057 = r151056 * r151054;
        double r151058 = r151051 + r151057;
        double r151059 = r151055 / r151058;
        double r151060 = r151059 / r151058;
        double r151061 = r151058 + r151054;
        double r151062 = r151060 / r151061;
        return r151062;
}
↓
double f(double alpha, double beta) {
        double r151063 = beta;
        double r151064 = 8.48564295701033e+177;
        bool r151065 = r151063 <= r151064;
        double r151066 = 1.0;
        double r151067 = alpha;
        double r151068 = r151067 + r151063;
        double r151069 = r151063 * r151067;
        double r151070 = r151068 + r151069;
        double r151071 = 1.0;
        double r151072 = r151070 + r151071;
        double r151073 = 2.0;
        double r151074 = fma(r151071, r151073, r151068);
        double r151075 = r151072 / r151074;
        double r151076 = r151073 * r151071;
        double r151077 = r151068 - r151076;
        double r151078 = r151075 * r151077;
        double r151079 = r151078 / r151074;
        double r151080 = r151066 * r151079;
        double r151081 = fma(r151071, r151073, r151071);
        double r151082 = r151067 + r151081;
        double r151083 = r151063 + r151082;
        double r151084 = r151080 / r151083;
        double r151085 = r151084 / r151077;
        double r151086 = 0.0;
        double r151087 = r151065 ? r151085 : r151086;
        return r151087;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 8.48564295701033e+177

    1. Initial program 1.6

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity1.6

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

    4. Applied flip-+2.5

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}{\left(\alpha + \beta\right) - 2 \cdot 1}}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    5. Applied associate-/r/2.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    6. Applied times-frac2.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}}{1} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    7. Simplified1.6

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

    9. Applied clear-num2.0

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

    11. Applied associate-*l/2.0

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

    12. Applied associate-/r/1.9

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

    13. Applied associate-/r*1.6

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

    14. Simplified1.6

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

    if 8.48564295701033e+177 < beta

    1. Initial program 16.3

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

    2. Taylor expanded around inf 6.4

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\ \;\;\;\;\frac{\frac{1 \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))