3frac (problem 3.3.3)

Average Error: 9.9 → 0.2

Time: 6.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -89.3311912530264891:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 99.322725633338678:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \frac{1}{\sqrt[3]{x + 1}}, -\frac{2}{x} \cdot 1\right) + \mathsf{fma}\left(\frac{2}{x}, \left(-1\right) + 1, \frac{1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\ \end{array}\]

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 ((x <= -89.33119125302649)) {
		VAR = fma(2.0, (1.0 / pow(x, 7.0)), fma(2.0, (1.0 / pow(x, 5.0)), ((2.0 / x) / (x * x))));
	} else {
		double VAR_1;
		if ((x <= 99.32272563333868)) {
			VAR_1 = (fma((1.0 / (cbrt((x + 1.0)) * cbrt((x + 1.0)))), (1.0 / cbrt((x + 1.0))), -((2.0 / x) * 1.0)) + fma((2.0 / x), (-1.0 + 1.0), (1.0 / (x - 1.0))));
		} else {
			VAR_1 = fma(2.0, (1.0 / pow(x, 7.0)), fma(2.0, (1.0 / pow(x, 5.0)), (2.0 / pow(x, 3.0))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	9.9
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 < -89.33119125302649

   1. Initial program 20.4

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

   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

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

   5. Applied cube-mult 0.6

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

   6. Applied associate-/r* 0.1

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

   if -89.33119125302649 < x < 99.32272563333868

   1. Initial program 0.0

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

   3. Applied *-un-lft-identity 0.0

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

   4. Applied add-cube-cbrt 0.0

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

   5. Applied *-un-lft-identity 0.0

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

   6. Applied times-frac 0.0

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

   7. Applied prod-diff 0.0

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

   8. Applied associate-+l+ 0.0

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

   9. Simplified 0.0

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

   if 99.32272563333868 < x

   1. Initial program 19.3

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

   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

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -89.3311912530264891:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 99.322725633338678:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \frac{1}{\sqrt[3]{x + 1}}, -\frac{2}{x} \cdot 1\right) + \mathsf{fma}\left(\frac{2}{x}, \left(-1\right) + 1, \frac{1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +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))))