3frac (problem 3.3.3)

Average Error: 9.6 → 0.6

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -5.4337115324627003 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 0.0\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(2 \cdot \left(\left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}\right)\right)}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))) <= -5.4337115324627) || !((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))) <= 0.0))) {
		VAR = (((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0)));
	} else {
		VAR = ((((1.0 / (x + 1.0)) + (2.0 / x)) * (2.0 * ((pow(x, -2.0) + (1.0 / pow(x, 4.0))) - (1.0 / pow(x, 3.0))))) / (((1.0 / (x + 1.0)) + (2.0 / x)) * (x - 1.0)));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	9.6
Target	0.3
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < -5.4337115324627 or 0.0 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))

   1. Initial program 0.7

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

   if -5.4337115324627 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < 0.0

   1. Initial program 18.9

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

   3. Applied flip-- 53.1

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

   4. Applied frac-add 54.0

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

   5. Simplified 25.0

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

   6. Taylor expanded around inf 0.4

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

   7. Simplified 0.4

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

   9. Applied pow-flip 0.4

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -5.4337115324627003 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 0.0\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(2 \cdot \left(\left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}\right)\right)}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))