3frac (problem 3.3.3)

Average Error: 9.6 → 0.5

Time: 9.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -5.4337115324627003 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 0.0\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(1 \cdot x - \left(x + 1\right) \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

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 = (fma(((1.0 * x) - ((x + 1.0) * 2.0)), (x - 1.0), (((x + 1.0) * x) * 1.0)) / (((x + 1.0) * x) * (x - 1.0)));
	} else {
		VAR = fma(2.0, (1.0 / pow(x, 7.0)), fma(2.0, (1.0 / pow(x, 5.0)), ((2.0 / x) / (x * x))));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	9.6
Target	0.3
Herbie	0.5

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

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

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

   3. Applied frac-sub 0.8

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

   4. Applied frac-add 0.5

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

   5. Simplified 0.7

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

   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. Taylor expanded around inf 0.7

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

   3. Simplified 0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)}\]
   4. Using strategy rm

   5. Applied cube-mult 0.7

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

   6. Applied associate-/r* 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -5.4337115324627003 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 0.0\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(1 \cdot x - \left(x + 1\right) \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(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))))