3frac (problem 3.3.3)

Average Error: 9.8 → 0.3

Time: 18.7s

Precision: 64

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -960.5059570047206:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 4091.211296656346:\\ \;\;\;\;\left(\left(1 - x\right) + x \cdot x\right) \cdot \frac{x \cdot \left(1 + x\right) + \left(x + -1\right) \cdot \left(x - 2 \cdot \left(1 + x\right)\right)}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(1 + x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

C

double f(double x) {
        double r2167675 = 1.0;
        double r2167676 = x;
        double r2167677 = r2167676 + r2167675;
        double r2167678 = r2167675 / r2167677;
        double r2167679 = 2.0;
        double r2167680 = r2167679 / r2167676;
        double r2167681 = r2167678 - r2167680;
        double r2167682 = r2167676 - r2167675;
        double r2167683 = r2167675 / r2167682;
        double r2167684 = r2167681 + r2167683;
        return r2167684;
}

↓

double f(double x) {
        double r2167685 = x;
        double r2167686 = -960.5059570047206;
        bool r2167687 = r2167685 <= r2167686;
        double r2167688 = 2.0;
        double r2167689 = 7.0;
        double r2167690 = pow(r2167685, r2167689);
        double r2167691 = r2167688 / r2167690;
        double r2167692 = r2167685 * r2167685;
        double r2167693 = r2167685 * r2167692;
        double r2167694 = r2167688 / r2167693;
        double r2167695 = 5.0;
        double r2167696 = pow(r2167685, r2167695);
        double r2167697 = r2167688 / r2167696;
        double r2167698 = r2167694 + r2167697;
        double r2167699 = r2167691 + r2167698;
        double r2167700 = 4091.211296656346;
        bool r2167701 = r2167685 <= r2167700;
        double r2167702 = 1.0;
        double r2167703 = r2167702 - r2167685;
        double r2167704 = r2167703 + r2167692;
        double r2167705 = r2167702 + r2167685;
        double r2167706 = r2167685 * r2167705;
        double r2167707 = -1.0;
        double r2167708 = r2167685 + r2167707;
        double r2167709 = r2167688 * r2167705;
        double r2167710 = r2167685 - r2167709;
        double r2167711 = r2167708 * r2167710;
        double r2167712 = r2167706 + r2167711;
        double r2167713 = r2167685 * r2167708;
        double r2167714 = r2167702 + r2167693;
        double r2167715 = r2167713 * r2167714;
        double r2167716 = r2167712 / r2167715;
        double r2167717 = r2167704 * r2167716;
        double r2167718 = r2167701 ? r2167717 : r2167699;
        double r2167719 = r2167687 ? r2167699 : r2167718;
        return r2167719;
}

Error

Bits error versus x

Target

Original	9.8
Target	0.3
Herbie	0.3

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

1. Split input into 2 regimes

2. if x < -960.5059570047206 or 4091.211296656346 < x

   1. Initial program 19.7

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied frac-sub 52.5

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

   4. Applied frac-add 51.3

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

   5. Simplified 51.3

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

   6. Simplified 51.3

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

   7. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

   8. Simplified 0.6

      \[\leadsto \color{blue}{\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{x \cdot \left(x \cdot x\right)}\right)}\]

   if -960.5059570047206 < x < 4091.211296656346

   1. Initial program 0.1

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied frac-sub 0.1

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

   4. Applied frac-add 0.0

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

   5. Simplified 0.0

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

   6. Simplified 0.0

      \[\leadsto \frac{\left(x + -1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + \left(x + 1\right) \cdot x}{\color{blue}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x + -1\right)}}\]
   7. Using strategy rm

   8. Applied flip3-+ 0.0

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

   9. Applied associate-*l/ 0.0

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

   10. Applied associate-*l/ 0.0

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

   11. Applied associate-/r/ 0.0

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

   12. Simplified 0.0

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -960.5059570047206:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 4091.211296656346:\\ \;\;\;\;\left(\left(1 - x\right) + x \cdot x\right) \cdot \frac{x \cdot \left(1 + x\right) + \left(x + -1\right) \cdot \left(x - 2 \cdot \left(1 + x\right)\right)}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(1 + x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))