2frac (problem 3.3.1)

Average Error: 14.6 → 0.1

Time: 15.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.9941623063372853 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 158565.28261171645:\\ \;\;\;\;\frac{-\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)}{\mathsf{fma}\left(x \cdot x, x \cdot x, x\right) \cdot \left(x + \mathsf{fma}\left(x, x, -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + \frac{\frac{1}{x}}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r1873780 = 1.0;
        double r1873781 = x;
        double r1873782 = r1873781 + r1873780;
        double r1873783 = r1873780 / r1873782;
        double r1873784 = r1873780 / r1873781;
        double r1873785 = r1873783 - r1873784;
        return r1873785;
}

↓

double f(double x) {
        double r1873786 = x;
        double r1873787 = -1.9941623063372853e+44;
        bool r1873788 = r1873786 <= r1873787;
        double r1873789 = 1.0;
        double r1873790 = r1873786 * r1873786;
        double r1873791 = r1873790 * r1873786;
        double r1873792 = r1873789 / r1873791;
        double r1873793 = r1873790 * r1873790;
        double r1873794 = r1873789 / r1873793;
        double r1873795 = r1873789 / r1873786;
        double r1873796 = r1873795 / r1873786;
        double r1873797 = r1873794 + r1873796;
        double r1873798 = r1873792 - r1873797;
        double r1873799 = 158565.28261171645;
        bool r1873800 = r1873786 <= r1873799;
        double r1873801 = r1873789 - r1873786;
        double r1873802 = r1873801 * r1873801;
        double r1873803 = r1873793 - r1873802;
        double r1873804 = -r1873803;
        double r1873805 = fma(r1873790, r1873790, r1873786);
        double r1873806 = -1.0;
        double r1873807 = fma(r1873786, r1873786, r1873806);
        double r1873808 = r1873786 + r1873807;
        double r1873809 = r1873805 * r1873808;
        double r1873810 = r1873804 / r1873809;
        double r1873811 = r1873800 ? r1873810 : r1873798;
        double r1873812 = r1873788 ? r1873798 : r1873811;
        return r1873812;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1.9941623063372853e+44 or 158565.28261171645 < x

   1. Initial program 27.7

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

   3. Applied frac-sub 27.1

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

   4. Simplified 27.1

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

   6. Applied flip3-+ 60.7

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

   7. Applied associate-*l/ 60.7

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

   8. Applied associate-/r/ 60.7

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

   9. Simplified 50.3

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

   10. Taylor expanded around inf 0.7

       \[\leadsto \color{blue}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)}\]

   11. Simplified 0.1

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

   if -1.9941623063372853e+44 < x < 158565.28261171645

   1. Initial program 3.9

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

   3. Applied frac-sub 3.2

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

   4. Simplified 3.2

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

   6. Applied flip3-+ 3.2

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

   7. Applied associate-*l/ 3.2

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

   8. Applied associate-/r/ 3.2

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

   9. Simplified 0.1

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

   11. Applied flip-+ 0.1

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

   12. Applied frac-times 0.1

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

   13. Simplified 0.1

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

   14. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.9941623063372853 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 158565.28261171645:\\ \;\;\;\;\frac{-\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 - x\right) \cdot \left(1 - x\right)\right)}{\mathsf{fma}\left(x \cdot x, x \cdot x, x\right) \cdot \left(x + \mathsf{fma}\left(x, x, -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + \frac{\frac{1}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))