3frac (problem 3.3.3)

Average Error: 9.9 → 0.2

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1792.4066915638655:\\ \;\;\;\;1 \cdot \frac{2 \cdot \left(\left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}\right)}{x - 1}\\ \mathbf{elif}\;x \le 142.430799048476587:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\ \end{array}\]

C

double code(double x) {
	return ((double) (((double) (((double) (1.0 / ((double) (x + 1.0)))) - ((double) (2.0 / x)))) + ((double) (1.0 / ((double) (x - 1.0))))));
}

↓

double code(double x) {
	double VAR;
	if ((x <= -1792.4066915638655)) {
		VAR = ((double) (1.0 * ((double) (((double) (2.0 * ((double) (((double) (((double) (1.0 / ((double) pow(x, 2.0)))) + ((double) (1.0 / ((double) pow(x, 4.0)))))) - ((double) (1.0 / ((double) pow(x, 3.0)))))))) / ((double) (x - 1.0))))));
	} else {
		double VAR_1;
		if ((x <= 142.4307990484766)) {
			VAR_1 = ((double) (((double) (((double) (1.0 / ((double) (x + 1.0)))) - ((double) (2.0 / x)))) + ((double) (1.0 / ((double) (x - 1.0))))));
		} else {
			VAR_1 = ((double) (2.0 * ((double) (((double) (1.0 / ((double) pow(x, 7.0)))) + ((double) (((double) (1.0 / ((double) pow(x, 5.0)))) + ((double) (1.0 / ((double) pow(x, 3.0))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	9.9
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x < -1792.4066915638655

   1. Initial program 19.6

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

   3. Applied flip-- 53.2

      \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} \cdot \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{2}{x}}{\frac{1}{x + 1} + \frac{2}{x}}} + \frac{1}{x - 1}\]

   4. Applied frac-add 54.3

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

   5. Simplified 25.4

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

   6. Taylor expanded around inf 0.2

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

   7. Simplified 0.2

      \[\leadsto \frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \color{blue}{\left(2 \cdot \left(\left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}\right)\right)}}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\]
   8. Using strategy rm

   9. Applied times-frac 0.2

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

   10. Simplified 0.2

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

   if -1792.4066915638655 < x < 142.430799048476587

   1. Initial program 0.1

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

   if 142.430799048476587 < x

   1. Initial program 20.4

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

   2. Taylor expanded around inf 0.6

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

   3. Simplified 0.6

      \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1792.4066915638655:\\ \;\;\;\;1 \cdot \frac{2 \cdot \left(\left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}\right)}{x - 1}\\ \mathbf{elif}\;x \le 142.430799048476587:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))