Asymptote C

Average Error: 29.7 → 0.1

Time: 52.7s

Precision: 64

Math

\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -11018.2382034044:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + 3 \cdot \frac{\frac{-1}{x \cdot x}}{x}\\ \mathbf{elif}\;x \le 21779.16323258692:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(\left(1 - x\right) \cdot \left(\left(x \cdot x + 1\right) - x\right)\right)\right) - \left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 - x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x + 1\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)\right)}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 - x\right) \cdot \left(\left(x \cdot x + 1\right) - x\right)\right)}}{\left(\left(1 + x\right) \cdot \left(x - 1\right)\right) \cdot \left(\frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + 3 \cdot \frac{\frac{-1}{x \cdot x}}{x}\\ \end{array}\]

C

double f(double x) {
        double r5517653 = x;
        double r5517654 = 1.0;
        double r5517655 = r5517653 + r5517654;
        double r5517656 = r5517653 / r5517655;
        double r5517657 = r5517653 - r5517654;
        double r5517658 = r5517655 / r5517657;
        double r5517659 = r5517656 - r5517658;
        return r5517659;
}

↓

double f(double x) {
        double r5517660 = x;
        double r5517661 = -11018.2382034044;
        bool r5517662 = r5517660 <= r5517661;
        double r5517663 = -1.0;
        double r5517664 = r5517660 * r5517660;
        double r5517665 = r5517663 / r5517664;
        double r5517666 = 3.0;
        double r5517667 = r5517666 / r5517660;
        double r5517668 = r5517665 - r5517667;
        double r5517669 = r5517665 / r5517660;
        double r5517670 = r5517666 * r5517669;
        double r5517671 = r5517668 + r5517670;
        double r5517672 = 21779.16323258692;
        bool r5517673 = r5517660 <= r5517672;
        double r5517674 = r5517664 * r5517660;
        double r5517675 = 1.0;
        double r5517676 = r5517660 - r5517675;
        double r5517677 = r5517675 - r5517660;
        double r5517678 = r5517664 + r5517675;
        double r5517679 = r5517678 - r5517660;
        double r5517680 = r5517677 * r5517679;
        double r5517681 = r5517676 * r5517680;
        double r5517682 = r5517674 * r5517681;
        double r5517683 = r5517675 + r5517660;
        double r5517684 = r5517683 * r5517683;
        double r5517685 = r5517675 - r5517664;
        double r5517686 = r5517674 + r5517675;
        double r5517687 = r5517683 / r5517676;
        double r5517688 = r5517687 * r5517687;
        double r5517689 = r5517686 * r5517688;
        double r5517690 = r5517685 * r5517689;
        double r5517691 = r5517684 * r5517690;
        double r5517692 = r5517682 - r5517691;
        double r5517693 = r5517684 * r5517680;
        double r5517694 = r5517692 / r5517693;
        double r5517695 = r5517683 * r5517676;
        double r5517696 = r5517660 / r5517683;
        double r5517697 = r5517696 + r5517687;
        double r5517698 = r5517696 * r5517697;
        double r5517699 = r5517698 + r5517688;
        double r5517700 = r5517695 * r5517699;
        double r5517701 = r5517694 / r5517700;
        double r5517702 = r5517673 ? r5517701 : r5517671;
        double r5517703 = r5517662 ? r5517671 : r5517702;
        return r5517703;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -11018.2382034044 or 21779.16323258692 < x

   1. Initial program 59.3

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

   2. Taylor expanded around -inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\frac{-1}{x \cdot x}}{x} \cdot 3 + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right)}\]

   if -11018.2382034044 < x < 21779.16323258692

   1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip3-- 0.1

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1} + \frac{x}{x + 1} \cdot \frac{x + 1}{x - 1}\right)}}\]

   4. Simplified 0.1

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

   5. Simplified 0.1

      \[\leadsto \frac{\frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) - \frac{1 + x}{x - 1} \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}{\color{blue}{\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)}}\]
   6. Using strategy rm

   7. Applied associate-*l/ 0.1

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

   8. Applied associate-*r/ 0.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{1 + x} \cdot \left(x \cdot \frac{x}{1 + x}\right)}{1 + x}} - \frac{1 + x}{x - 1} \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}{\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)}\]
   9. Using strategy rm

   10. Applied associate-*l/ 0.1

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

   11. Applied frac-sub 0.1

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

   12. Applied associate-/l/ 0.1

       \[\leadsto \color{blue}{\frac{\left(\frac{x}{1 + x} \cdot \left(x \cdot \frac{x}{1 + x}\right)\right) \cdot \left(x - 1\right) - \left(1 + x\right) \cdot \left(\left(1 + x\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}}\]
   13. Using strategy rm

   14. Applied flip3-+ 0.1

       \[\leadsto \frac{\left(\frac{x}{1 + x} \cdot \left(x \cdot \frac{x}{1 + x}\right)\right) \cdot \left(x - 1\right) - \left(1 + x\right) \cdot \left(\color{blue}{\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}} \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\]

   15. Applied associate-*l/ 0.1

       \[\leadsto \frac{\left(\frac{x}{1 + x} \cdot \left(x \cdot \frac{x}{1 + x}\right)\right) \cdot \left(x - 1\right) - \left(1 + x\right) \cdot \color{blue}{\frac{\left({1}^{3} + {x}^{3}\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\]

   16. Applied flip-+ 0.1

       \[\leadsto \frac{\left(\frac{x}{1 + x} \cdot \left(x \cdot \frac{x}{1 + x}\right)\right) \cdot \left(x - 1\right) - \color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}} \cdot \frac{\left({1}^{3} + {x}^{3}\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\]

   17. Applied frac-times 0.1

       \[\leadsto \frac{\left(\frac{x}{1 + x} \cdot \left(x \cdot \frac{x}{1 + x}\right)\right) \cdot \left(x - 1\right) - \color{blue}{\frac{\left(1 \cdot 1 - x \cdot x\right) \cdot \left(\left({1}^{3} + {x}^{3}\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)}{\left(1 - x\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\]

   18. Applied associate-*r/ 0.1

       \[\leadsto \frac{\left(\frac{x}{1 + x} \cdot \color{blue}{\frac{x \cdot x}{1 + x}}\right) \cdot \left(x - 1\right) - \frac{\left(1 \cdot 1 - x \cdot x\right) \cdot \left(\left({1}^{3} + {x}^{3}\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)}{\left(1 - x\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\]

   19. Applied frac-times 0.1

       \[\leadsto \frac{\color{blue}{\frac{x \cdot \left(x \cdot x\right)}{\left(1 + x\right) \cdot \left(1 + x\right)}} \cdot \left(x - 1\right) - \frac{\left(1 \cdot 1 - x \cdot x\right) \cdot \left(\left({1}^{3} + {x}^{3}\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)}{\left(1 - x\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\]

   20. Applied associate-*l/ 0.1

       \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x - 1\right)}{\left(1 + x\right) \cdot \left(1 + x\right)}} - \frac{\left(1 \cdot 1 - x \cdot x\right) \cdot \left(\left({1}^{3} + {x}^{3}\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)}{\left(1 - x\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\]

   21. Applied frac-sub 0.1

       \[\leadsto \frac{\color{blue}{\frac{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x - 1\right)\right) \cdot \left(\left(1 - x\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 \cdot 1 - x \cdot x\right) \cdot \left(\left({1}^{3} + {x}^{3}\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)\right)}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 - x\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)}}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\]

   22. Simplified 0.1

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

   23. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11018.2382034044:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + 3 \cdot \frac{\frac{-1}{x \cdot x}}{x}\\ \mathbf{elif}\;x \le 21779.16323258692:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(\left(1 - x\right) \cdot \left(\left(x \cdot x + 1\right) - x\right)\right)\right) - \left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 - x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x + 1\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)\right)}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 - x\right) \cdot \left(\left(x \cdot x + 1\right) - x\right)\right)}}{\left(\left(1 + x\right) \cdot \left(x - 1\right)\right) \cdot \left(\frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + 3 \cdot \frac{\frac{-1}{x \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))