3frac (problem 3.3.3)

Average Error: 10.0 → 0.3

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -356.53497037483038:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(1 \cdot x - \left(1 + x\right) \cdot 2\right) + 1 \cdot \left(x \cdot \left(1 + x\right)\right)}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 1.3535536 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{x} \cdot \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]

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 ((((double) (((double) (((double) (1.0 / ((double) (1.0 + x)))) - ((double) (2.0 / x)))) + ((double) (1.0 / ((double) (x - 1.0)))))) <= -356.5349703748304)) {
		VAR = ((double) (((double) (((double) (((double) (x - 1.0)) * ((double) (((double) (1.0 * x)) - ((double) (((double) (1.0 + x)) * 2.0)))))) + ((double) (1.0 * ((double) (x * ((double) (1.0 + x)))))))) / ((double) (x * ((double) (((double) (x * x)) - ((double) (1.0 * 1.0))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (1.0 / ((double) (1.0 + x)))) - ((double) (2.0 / x)))) + ((double) (1.0 / ((double) (x - 1.0)))))) <= 1.3535535675146884e-15)) {
			VAR_1 = ((double) (((double) (2.0 / ((double) pow(x, 7.0)))) + ((double) (((double) (2.0 / ((double) pow(x, 5.0)))) + ((double) (((double) (2.0 / x)) * ((double) (1.0 / ((double) (x * x))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (1.0 / ((double) (1.0 + x)))) - ((double) (2.0 / x)))) + ((double) (1.0 / ((double) (x - 1.0))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	10.0
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < -356.53497037483038

   1. Initial program 0.0

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

   3. Applied frac-sub 0.0

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

   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

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

   if -356.53497037483038 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < 1.3535536e-15

   1. Initial program 19.7

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

   2. Taylor expanded around inf 0.9

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

   3. Simplified 0.9

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

   5. Applied add-cube-cbrt 1.6

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

   6. Applied unpow-prod-down 1.6

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

   7. Applied *-un-lft-identity 1.6

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

   8. Applied times-frac 1.2

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

   9. Simplified 0.8

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

   10. Simplified 0.5

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

   if 1.3535536e-15 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))

   1. Initial program 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -356.53497037483038:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(1 \cdot x - \left(1 + x\right) \cdot 2\right) + 1 \cdot \left(x \cdot \left(1 + x\right)\right)}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 1.3535536 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{x} \cdot \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))))