Average Error: 3.8 → 1.4
Time: 21.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.714456171199346013842965338116236345794 \cdot 10^{159}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}{\frac{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(\beta, \alpha, 1\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \end{array}\]
\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}\;\alpha \le 4.714456171199346013842965338116236345794 \cdot 10^{159}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}{\frac{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(\beta, \alpha, 1\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

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

\end{array}
double f(double alpha, double beta) {
        double r80814 = alpha;
        double r80815 = beta;
        double r80816 = r80814 + r80815;
        double r80817 = r80815 * r80814;
        double r80818 = r80816 + r80817;
        double r80819 = 1.0;
        double r80820 = r80818 + r80819;
        double r80821 = 2.0;
        double r80822 = r80821 * r80819;
        double r80823 = r80816 + r80822;
        double r80824 = r80820 / r80823;
        double r80825 = r80824 / r80823;
        double r80826 = r80823 + r80819;
        double r80827 = r80825 / r80826;
        return r80827;
}
↓
double f(double alpha, double beta) {
        double r80828 = alpha;
        double r80829 = 4.714456171199346e+159;
        bool r80830 = r80828 <= r80829;
        double r80831 = beta;
        double r80832 = r80828 + r80831;
        double r80833 = r80831 * r80828;
        double r80834 = r80832 + r80833;
        double r80835 = 1.0;
        double r80836 = r80834 + r80835;
        double r80837 = sqrt(r80836);
        double r80838 = 2.0;
        double r80839 = r80838 * r80835;
        double r80840 = r80832 + r80839;
        double r80841 = sqrt(r80840);
        double r80842 = r80837 / r80841;
        double r80843 = fma(r80835, r80838, r80832);
        double r80844 = fma(r80831, r80828, r80835);
        double r80845 = r80832 + r80844;
        double r80846 = sqrt(r80845);
        double r80847 = sqrt(r80843);
        double r80848 = r80846 / r80847;
        double r80849 = r80843 / r80848;
        double r80850 = r80842 / r80849;
        double r80851 = r80840 + r80835;
        double r80852 = r80850 / r80851;
        double r80853 = 1.0;
        double r80854 = 2.0;
        double r80855 = r80831 / r80828;
        double r80856 = r80828 / r80831;
        double r80857 = r80855 + r80856;
        double r80858 = r80854 + r80857;
        double r80859 = r80858 * r80843;
        double r80860 = r80853 / r80859;
        double r80861 = r80830 ? r80852 : r80860;
        return r80861;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.714456171199346e+159

    1. Initial program 1.5

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

    3. Applied add-sqr-sqrt2.1

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

    4. Applied add-sqr-sqrt2.0

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

    5. Applied times-frac2.0

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

    6. Applied associate-/l*1.5

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

    7. Simplified1.5

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

    if 4.714456171199346e+159 < alpha

    1. Initial program 15.8

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

    3. Applied *-un-lft-identity15.8

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

    4. Applied *-un-lft-identity15.8

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

    5. Applied *-un-lft-identity15.8

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

    6. Applied times-frac15.8

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

    7. Applied times-frac15.8

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

    8. Applied associate-/l*15.9

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

    9. Simplified15.9

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

    10. Taylor expanded around inf 0.7

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.714456171199346013842965338116236345794 \cdot 10^{159}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}{\frac{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(\beta, \alpha, 1\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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)))