3frac (problem 3.3.3)

Average Error: 9.7 → 0.2

Time: 13.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -102.94433550507199:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x \cdot x}}{x}\right)\right)\\ \mathbf{elif}\;x \le 99.938672500306552:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{1}{x}, \left(-2\right) + 2, \left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) + \frac{-2}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\ \end{array}\]

C

double code(double x) {
	return ((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 ((x <= -102.94433550507199)) {
		VAR = ((double) fma(2.0, ((double) (1.0 / ((double) pow(x, 7.0)))), ((double) fma(2.0, ((double) (1.0 / ((double) pow(x, 5.0)))), ((double) (((double) (2.0 / ((double) (x * x)))) / x))))));
	} else {
		double VAR_1;
		if ((x <= 99.93867250030655)) {
			VAR_1 = ((double) (1.0 * ((double) fma(((double) (1.0 / x)), ((double) (((double) -(2.0)) + 2.0)), ((double) (((double) (((double) (1.0 / ((double) (x - 1.0)))) + ((double) (1.0 / ((double) (x + 1.0)))))) + ((double) (((double) -(2.0)) / x))))))));
		} else {
			VAR_1 = ((double) fma(2.0, ((double) (1.0 / ((double) pow(x, 7.0)))), ((double) fma(2.0, ((double) (1.0 / ((double) pow(x, 5.0)))), ((double) (2.0 / ((double) pow(x, 3.0))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	9.7
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 < -102.94433550507199

   1. Initial program 20.1

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

   3. Applied div-inv 20.1

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

   4. Applied *-un-lft-identity 20.1

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

   5. Applied prod-diff 20.1

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

   6. Applied associate-+l+ 20.1

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

   7. Taylor expanded around inf 0.5

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

   8. Simplified 0.5

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

   10. Applied unpow3 0.6

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

   11. Applied associate-/r* 0.1

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

   if -102.94433550507199 < x < 99.93867250030655

   1. Initial program 0.0

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

   3. Applied div-inv 0.0

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

   4. Applied *-un-lft-identity 0.0

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

   5. Applied prod-diff 0.0

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

   6. Applied associate-+l+ 0.0

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

   8. Applied *-un-lft-identity 0.0

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

   9. Applied *-un-lft-identity 0.0

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

   10. Applied distribute-lft-out 0.0

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

   11. Simplified 0.0

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

   if 99.93867250030655 < x

   1. Initial program 19.8

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

   3. Applied div-inv 19.8

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

   4. Applied *-un-lft-identity 19.8

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

   5. Applied prod-diff 19.8

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

   6. Applied associate-+l+ 19.8

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

   7. Taylor expanded around inf 0.6

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

   8. Simplified 0.6

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -102.94433550507199:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x \cdot x}}{x}\right)\right)\\ \mathbf{elif}\;x \le 99.938672500306552:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{1}{x}, \left(-2\right) + 2, \left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) + \frac{-2}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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))))