3frac (problem 3.3.3)

Average Error: 9.5 → 0.0

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -127880528.286486253 \lor \neg \left(x \le 485.728292556334679\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{{\left(\left(x \cdot x - 1 \cdot 1\right) \cdot x\right)}^{1}}\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if (((x <= -127880528.28648625) || !(x <= 485.7282925563347))) {
		temp = (2.0 * ((1.0 / pow(x, 7.0)) + ((1.0 / pow(x, 5.0)) + pow(x, -3.0))));
	} else {
		temp = (((((1.0 * x) - ((x + 1.0) * 2.0)) * (x - 1.0)) + (((x + 1.0) * x) * 1.0)) / pow((((x * x) - (1.0 * 1.0)) * x), 1.0));
	}
	return temp;
}

Error

Bits error versus x

Target

Original	9.5
Target	0.3
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -127880528.28648625 or 485.7282925563347 < x

   1. Initial program 19.4

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

   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

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

   5. Applied pow-flip 0.0

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

   if -127880528.28648625 < x < 485.7282925563347

   1. Initial program 0.3

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

   3. Applied frac-sub 0.3

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

   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. Using strategy rm

   6. Applied pow1 0.0

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

   7. Applied pow1 0.0

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

   8. Applied pow1 0.0

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

   9. Applied pow-prod-down 0.0

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

   10. Applied pow-prod-down 0.0

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

   11. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -127880528.286486253 \lor \neg \left(x \le 485.728292556334679\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{{\left(\left(x \cdot x - 1 \cdot 1\right) \cdot x\right)}^{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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))))