3frac (problem 3.3.3)

Average Error: 9.5 → 0.3

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -838439.330952369724 \lor \neg \left(x \le 511.78487604959889\right):\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x \cdot 1 - 2 \cdot \left(x + 1\right)\right) + 1 \cdot \left(x \cdot \left(x + 1\right)\right)}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}\\ \end{array}\]

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 <= -838439.3309523697) || !(x <= 511.7848760495989))) {
		VAR = ((double) (((double) (2.0 / ((double) pow(x, 7.0)))) + ((double) (((double) (2.0 / ((double) pow(x, 5.0)))) + ((double) (2.0 / ((double) pow(x, 3.0))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (x - 1.0)) * ((double) (((double) (x * 1.0)) - ((double) (2.0 * ((double) (x + 1.0)))))))) + ((double) (1.0 * ((double) (x * ((double) (x + 1.0)))))))) / ((double) (x * ((double) (((double) (x * x)) - ((double) (1.0 * 1.0))))))));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	9.5
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 < -838439.330952369724 or 511.78487604959889 < x

   1. Initial program 18.8

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

   2. Taylor expanded around inf 0.5

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

   3. Simplified 0.5

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

   if -838439.330952369724 < x < 511.78487604959889

   1. Initial program 0.2

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

   3. Applied frac-sub 0.2

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

   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

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

   6. Simplified 0.0

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -838439.330952369724 \lor \neg \left(x \le 511.78487604959889\right):\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x \cdot 1 - 2 \cdot \left(x + 1\right)\right) + 1 \cdot \left(x \cdot \left(x + 1\right)\right)}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))))