Asymptote A

Average Error: 14.5 → 0.1

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -207.49900421996719 \lor \neg \left(x \le 234.24057061613541\right):\\ \;\;\;\;\left(\frac{-2}{{x}^{6}} - \frac{\frac{2}{x}}{x}\right) - \frac{2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{x}^{3} + {1}^{3}}, x \cdot x + \left(1 \cdot 1 - x \cdot 1\right), \frac{-1}{x - 1}\right)\\ \end{array}\]

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (1.0 / (x - 1.0)));
}

↓

double code(double x) {
	double VAR;
	if (((x <= -207.4990042199672) || !(x <= 234.24057061613541))) {
		VAR = (((-2.0 / pow(x, 6.0)) - ((2.0 / x) / x)) - (2.0 / pow(x, 4.0)));
	} else {
		VAR = fma((1.0 / (pow(x, 3.0) + pow(1.0, 3.0))), ((x * x) + ((1.0 * 1.0) - (x * 1.0))), (-1.0 / (x - 1.0)));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -207.4990042199672 or 234.24057061613541 < x

   1. Initial program 28.9

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

   3. Applied flip3-+ 60.8

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

   4. Applied associate-/r/ 60.8

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

   5. Applied fma-neg 60.7

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

   6. Simplified 60.7

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

   7. Taylor expanded around inf 0.9

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)}\]

   8. Simplified 0.1

      \[\leadsto \color{blue}{\left(\frac{-2}{{x}^{6}} - \frac{\frac{2}{x}}{x}\right) - \frac{2}{{x}^{4}}}\]

   if -207.4990042199672 < x < 234.24057061613541

   1. Initial program 0.0

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

   3. Applied flip3-+ 0.0

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

   4. Applied associate-/r/ 0.0

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

   5. Applied fma-neg 0.0

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

   6. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -207.49900421996719 \lor \neg \left(x \le 234.24057061613541\right):\\ \;\;\;\;\left(\frac{-2}{{x}^{6}} - \frac{\frac{2}{x}}{x}\right) - \frac{2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{x}^{3} + {1}^{3}}, x \cdot x + \left(1 \cdot 1 - x \cdot 1\right), \frac{-1}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))