Octave 3.8, jcobi/3

Average Error: 3.6 → 2.4

Time: 1.3m

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.000811582078763680955781043863702464006 \cdot 10^{161}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{1}}{1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\sqrt{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}} \cdot \frac{\sqrt{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r389765 = alpha;
        double r389766 = beta;
        double r389767 = r389765 + r389766;
        double r389768 = r389766 * r389765;
        double r389769 = r389767 + r389768;
        double r389770 = 1.0;
        double r389771 = r389769 + r389770;
        double r389772 = 2.0;
        double r389773 = r389772 * r389770;
        double r389774 = r389767 + r389773;
        double r389775 = r389771 / r389774;
        double r389776 = r389775 / r389774;
        double r389777 = r389774 + r389770;
        double r389778 = r389776 / r389777;
        return r389778;
}

↓

double f(double alpha, double beta) {
        double r389779 = alpha;
        double r389780 = 1.0008115820787637e+161;
        bool r389781 = r389779 <= r389780;
        double r389782 = beta;
        double r389783 = r389779 + r389782;
        double r389784 = r389782 * r389779;
        double r389785 = r389783 + r389784;
        double r389786 = 1.0;
        double r389787 = r389785 + r389786;
        double r389788 = sqrt(r389787);
        double r389789 = 1.0;
        double r389790 = r389788 / r389789;
        double r389791 = r389790 / r389789;
        double r389792 = sqrt(r389791);
        double r389793 = 2.0;
        double r389794 = r389793 * r389786;
        double r389795 = r389783 + r389794;
        double r389796 = r389795 + r389786;
        double r389797 = r389789 / r389795;
        double r389798 = r389787 * r389797;
        double r389799 = sqrt(r389798);
        double r389800 = r389796 / r389799;
        double r389801 = r389792 / r389800;
        double r389802 = r389788 / r389795;
        double r389803 = r389802 / r389795;
        double r389804 = sqrt(r389803);
        double r389805 = sqrt(r389795);
        double r389806 = r389804 / r389805;
        double r389807 = r389801 * r389806;
        double r389808 = 0.0;
        double r389809 = r389808 / r389796;
        double r389810 = r389781 ? r389807 : r389809;
        return r389810;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 1.0008115820787637e+161

   1. Initial program 1.3

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

   3. Applied add-sqr-sqrt 1.4

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

   4. Applied associate-/l* 1.4

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

   6. Applied sqrt-div 1.3

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

   7. Applied associate-/r/ 1.3

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

   8. Applied *-un-lft-identity 1.3

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

   9. Applied *-un-lft-identity 1.3

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

   10. Applied add-sqr-sqrt 1.3

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

   11. Applied times-frac 1.3

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

   12. Applied times-frac 1.3

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

   13. Applied sqrt-prod 1.3

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

   14. Applied times-frac 1.4

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

   16. Applied div-inv 1.4

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

   if 1.0008115820787637e+161 < alpha

   1. Initial program 15.6

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

   2. Taylor expanded around inf 7.7

      \[\leadsto \frac{\color{blue}{0}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.000811582078763680955781043863702464006 \cdot 10^{161}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{1}}{1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\sqrt{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}} \cdot \frac{\sqrt{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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)))