3frac (problem 3.3.3)

Average Error: 9.6 → 0.1

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -114.36312374053232 \lor \neg \left(x \le 105.87813595844015\right):\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \left(\frac{1}{x - 1} - \frac{2}{x}\right)\\ \end{array}\]

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 <= -114.36312374053232) || !(x <= 105.87813595844015))) {
		VAR = ((double) (((double) (2.0 / ((double) pow(x, 7.0)))) + ((double) (((double) (2.0 / ((double) pow(x, 5.0)))) + ((double) (((double) (2.0 / x)) / ((double) (x * x))))))));
	} else {
		VAR = ((double) (((double) (1.0 / ((double) (x + 1.0)))) + ((double) (((double) (1.0 / ((double) (x - 1.0)))) - ((double) (2.0 / x))))));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	9.6
Target	0.3
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -114.363123740532316 or 105.878135958440154 < x

   1. Initial program 19.7

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

   2. Simplified 19.7

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

   3. Taylor expanded around inf 0.5

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

   4. Simplified 0.5

      \[\leadsto \color{blue}{\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)}\]
   5. Using strategy rm

   6. Applied cube-mult 0.5

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

   7. Applied associate-/r* 0.1

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

   if -114.363123740532316 < x < 105.878135958440154

   1. Initial program 0.0

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

   2. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -114.36312374053232 \lor \neg \left(x \le 105.87813595844015\right):\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \left(\frac{1}{x - 1} - \frac{2}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))