Asymptote C

Average Error: 29.2 → 0.0

Time: 5.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -15104.003133029944:\\ \;\;\;\;\left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{x + 1}{x + -1}\\ \mathbf{if}\;x \leq 3580589034911187:\\ \;\;\;\;\frac{\frac{-1 - \left(5 \cdot {x}^{2} + \left(x \cdot 4 + {x}^{3} \cdot 6\right)\right)}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-6}{x}}{t_0}\\ \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -15104.003133029944)
   (-
    (- (/ -3.0 x) (+ (/ 1.0 (* x x)) (/ 3.0 (pow x 3.0))))
    (/ 1.0 (pow x 4.0)))
   (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0)))))
     (if (<= x 3580589034911187.0)
       (/
        (/
         (- -1.0 (+ (* 5.0 (pow x 2.0)) (+ (* x 4.0) (* (pow x 3.0) 6.0))))
         (* (fma x x -1.0) (fma x x -1.0)))
        t_0)
       (/ (/ -6.0 x) t_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 <= -15104.003133029944) {
		tmp = ((-3.0 / x) - ((1.0 / (x * x)) + (3.0 / pow(x, 3.0)))) - (1.0 / pow(x, 4.0));
	} else {
		double t_0 = (x / (x + 1.0)) + ((x + 1.0) / (x + -1.0));
		double tmp_1;
		if (x <= 3580589034911187.0) {
			tmp_1 = ((-1.0 - ((5.0 * pow(x, 2.0)) + ((x * 4.0) + (pow(x, 3.0) * 6.0)))) / (fma(x, x, -1.0) * fma(x, x, -1.0))) / t_0;
		} else {
			tmp_1 = (-6.0 / x) / t_0;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -15104.0031330299444

   1. Initial program 59.4

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

   2. Taylor expanded in x around inf 0.3

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{4}} + \left(3 \cdot \frac{1}{x} + \left(3 \cdot \frac{1}{{x}^{3}} + \frac{1}{{x}^{2}}\right)\right)\right)} \]

   3. Simplified 0.0

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

   if -15104.0031330299444 < x < 3580589034911187

   1. Initial program 0.6

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

   2. Applied flip--_binary64 0.6

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}} \]

   3. Applied frac-times_binary64 0.6

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

   4. Applied frac-times_binary64 0.6

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

   5. Applied frac-sub_binary64 0.6

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

   6. Simplified 0.6

      \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x + -1\right) \cdot \left(x + -1\right)\right) - {\left(1 + x\right)}^{4}}}{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}} \]

   7. Simplified 0.6

      \[\leadsto \frac{\frac{\left(x \cdot x\right) \cdot \left(\left(x + -1\right) \cdot \left(x + -1\right)\right) - {\left(1 + x\right)}^{4}}{\color{blue}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}} \]

   8. Taylor expanded in x around 0 0.1

      \[\leadsto \frac{\frac{\color{blue}{-\left(1 + \left(5 \cdot {x}^{2} + \left(4 \cdot x + 6 \cdot {x}^{3}\right)\right)\right)}}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}} \]

   if 3580589034911187 < x

   1. Initial program 60.5

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

   2. Applied flip--_binary64 60.5

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}} \]

   3. Taylor expanded in x around inf 0

      \[\leadsto \frac{\color{blue}{\frac{-6}{x}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15104.003133029944:\\ \;\;\;\;\left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \mathbf{elif}\;x \leq 3580589034911187:\\ \;\;\;\;\frac{\frac{-1 - \left(5 \cdot {x}^{2} + \left(x \cdot 4 + {x}^{3} \cdot 6\right)\right)}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-6}{x}}{\frac{x}{x + 1} + \frac{x + 1}{x + -1}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021225 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))