Average Error: 3.9 → 2.5
Time: 1.4m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.63006397465412179 \cdot 10^{170}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \left(1 + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\ \end{array}\]
\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 1.63006397465412179 \cdot 10^{170}:\\
\;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\alpha + \left(1 + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r210781 = alpha;
        double r210782 = beta;
        double r210783 = r210781 + r210782;
        double r210784 = r210782 * r210781;
        double r210785 = r210783 + r210784;
        double r210786 = 1.0;
        double r210787 = r210785 + r210786;
        double r210788 = 2.0;
        double r210789 = r210788 * r210786;
        double r210790 = r210783 + r210789;
        double r210791 = r210787 / r210790;
        double r210792 = r210791 / r210790;
        double r210793 = r210790 + r210786;
        double r210794 = r210792 / r210793;
        return r210794;
}
↓
double f(double alpha, double beta) {
        double r210795 = beta;
        double r210796 = 1.6300639746541218e+170;
        bool r210797 = r210795 <= r210796;
        double r210798 = 1.0;
        double r210799 = 1.0;
        double r210800 = 2.0;
        double r210801 = alpha;
        double r210802 = r210801 + r210795;
        double r210803 = fma(r210799, r210800, r210802);
        double r210804 = r210803 + r210799;
        double r210805 = fma(r210801, r210795, r210802);
        double r210806 = r210799 + r210805;
        double r210807 = r210806 / r210803;
        double r210808 = r210807 / r210803;
        double r210809 = r210804 / r210808;
        double r210810 = r210798 / r210809;
        double r210811 = r210799 + r210795;
        double r210812 = r210801 + r210811;
        double r210813 = r210812 / r210803;
        double r210814 = r210813 / r210803;
        double r210815 = r210814 / r210804;
        double r210816 = r210797 ? r210810 : r210815;
        return r210816;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.6300639746541218e+170

    1. Initial program 1.4

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

    2. Simplified1.4

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

    4. Applied *-un-lft-identity1.4

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

    5. Applied *-un-lft-identity1.4

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

    6. Applied *-un-lft-identity1.4

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

    7. Applied times-frac1.4

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

    8. Applied times-frac1.4

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

    9. Applied associate-/l*1.7

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

    if 1.6300639746541218e+170 < beta

    1. Initial program 18.2

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

    2. Simplified18.2

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

    3. Taylor expanded around 0 7.2

      \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha + \left(1 + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.63006397465412179 \cdot 10^{170}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \left(1 + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))