Asymptote C

Average Error: 29.0 → 0.2

Time: 5.5s

Precision: binary64

Math

\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5.7469619081373935 \cdot 10^{-06}:\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(x + 1\right)}^{3} \cdot \frac{1}{{\left(x - 1\right)}^{3}}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\ \end{array}\]

FPCore

(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))

↓

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 5.7469619081373935e-06)
   (- (- (/ -3.0 x) (pow x -2.0)) (/ 3.0 (pow x 3.0)))
   (/
    (-
     (pow (/ x (+ x 1.0)) 3.0)
     (* (pow (+ x 1.0) 3.0) (/ 1.0 (pow (- x 1.0) 3.0))))
    (+
     (* (/ x (+ x 1.0)) (/ x (+ x 1.0)))
     (/ (+ x (* (+ x 1.0) (/ (+ x 1.0) (- x 1.0)))) (- x 1.0))))))

C

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

↓

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 5.7469619081373935e-06) {
		tmp = ((-3.0 / x) - pow(x, -2.0)) - (3.0 / pow(x, 3.0));
	} else {
		tmp = (pow((x / (x + 1.0)), 3.0) - (pow((x + 1.0), 3.0) * (1.0 / pow((x - 1.0), 3.0)))) / (((x / (x + 1.0)) * (x / (x + 1.0))) + ((x + ((x + 1.0) * ((x + 1.0) / (x - 1.0)))) / (x - 1.0)));
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.74696190814e-6

   1. Initial program 59.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

   2. Taylor expanded around inf 0.5

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

   3. Simplified 0.2

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

   5. Applied pow2_binary64 0.2

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

   6. Applied pow-flip_binary64 0.2

      \[\leadsto \left(\frac{-3}{x} - \color{blue}{{x}^{\left(-2\right)}}\right) - \frac{3}{{x}^{3}}\]

   if 5.74696190814e-6 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip3--_binary64 0.1

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

   4. Simplified 0.1

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

   5. Simplified 0.1

      \[\leadsto \frac{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x + -1}\right)}^{3}}{\color{blue}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \frac{x + \left(1 + x\right) \cdot \frac{1 + x}{x + -1}}{x + -1}}}\]
   6. Using strategy rm

   7. Applied div-inv_binary64 0.1

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

   8. Applied unpow-prod-down_binary64 0.1

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

   9. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5.7469619081373935 \cdot 10^{-06}:\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(x + 1\right)}^{3} \cdot \frac{1}{{\left(x - 1\right)}^{3}}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021149 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))