3frac (problem 3.3.3)

Average Error: 9.8 → 0.2

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4144.9360232154404:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\\ \mathbf{elif}\;x \le 169090.902769654087:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x \cdot 1 - 2 \cdot \left(x + 1\right)\right) + 1 \cdot \left(x \cdot \left(x + 1\right)\right)}{{x}^{3} - x \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} - \frac{2}{{x}^{3}}\right)\right)}{\left(x - 1\right) \cdot \left(\frac{1}{x + 1} + \frac{2}{x}\right)}\\ \end{array}\]

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 <= -4144.93602321544)) {
		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 {
		double VAR_1;
		if ((x <= 169090.9027696541)) {
			VAR_1 = ((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) (((double) pow(x, 3.0)) - ((double) (x * 1.0))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (1.0 / ((double) (x + 1.0)))) + ((double) (2.0 / x)))) * ((double) (((double) (2.0 / ((double) (x * x)))) + ((double) (((double) (2.0 / ((double) pow(x, 4.0)))) - ((double) (2.0 / ((double) pow(x, 3.0)))))))))) / ((double) (((double) (x - 1.0)) * ((double) (((double) (1.0 / ((double) (x + 1.0)))) + ((double) (2.0 / x))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	9.8
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 < -4144.9360232154404

   1. Initial program 19.6

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

   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

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

   if -4144.9360232154404 < x < 169090.902769654087

   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

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

   7. Taylor expanded around 0 0.0

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

   8. Simplified 0.0

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

   if 169090.902769654087 < x

   1. Initial program 20.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   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 54.1

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

   5. Simplified 25.5

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

   6. Simplified 25.5

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

   7. Taylor expanded around inf 0.2

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

   8. Simplified 0.2

      \[\leadsto \frac{\left(\frac{1}{1 + x} + \frac{2}{x}\right) \cdot \color{blue}{\left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} - \frac{2}{{x}^{3}}\right)\right)}}{\left(x - 1\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4144.9360232154404:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\\ \mathbf{elif}\;x \le 169090.902769654087:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x \cdot 1 - 2 \cdot \left(x + 1\right)\right) + 1 \cdot \left(x \cdot \left(x + 1\right)\right)}{{x}^{3} - x \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} - \frac{2}{{x}^{3}}\right)\right)}{\left(x - 1\right) \cdot \left(\frac{1}{x + 1} + \frac{2}{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))))