Asymptote C

Average Error: 29.3 → 0.0

Time: 15.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \mathbf{if}\;x \leq -3606.908350404843:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2601.5280327281093:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{x}{x + 1}\\ t_2 := \frac{x + 1}{x + -1}\\ \frac{{t_1}^{3} - {t_2}^{3}}{{t_1}^{2} + \frac{\mathsf{fma}\left(x + 1, t_2, x\right)}{x + -1}} \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (- (/ -3.0 x) (+ (/ 1.0 (* x x)) (/ 3.0 (pow x 3.0))))
          (/ 1.0 (pow x 4.0)))))
   (if (<= x -3606.908350404843)
     t_0
     (if (<= x 2601.5280327281093)
       (let* ((t_1 (/ x (+ x 1.0))) (t_2 (/ (+ x 1.0) (+ x -1.0))))
         (/
          (- (pow t_1 3.0) (pow t_2 3.0))
          (+ (pow t_1 2.0) (/ (fma (+ x 1.0) t_2 x) (+ x -1.0)))))
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = ((-3.0 / x) - ((1.0 / (x * x)) + (3.0 / pow(x, 3.0)))) - (1.0 / pow(x, 4.0));
	double tmp;
	if (x <= -3606.908350404843) {
		tmp = t_0;
	} else if (x <= 2601.5280327281093) {
		double t_1 = x / (x + 1.0);
		double t_2 = (x + 1.0) / (x + -1.0);
		tmp = (pow(t_1, 3.0) - pow(t_2, 3.0)) / (pow(t_1, 2.0) + (fma((x + 1.0), t_2, x) / (x + -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -3606.9083504048431 or 2601.52803272810934 < x

   1. Initial program 59.3

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

   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 -3606.9083504048431 < x < 2601.52803272810934

   1. Initial program 0.1

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

   2. Applied div-inv_binary64 0.1

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

   3. Applied flip3--_binary64 0.1

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

   4. Simplified 0.1

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

   5. Simplified 0.1

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

   6. Applied *-un-lft-identity_binary64 0.1

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

   7. Applied *-un-lft-identity_binary64 0.1

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

   8. Applied times-frac_binary64 0.1

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

   9. Simplified 0.1

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

   10. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3606.908350404843:\\ \;\;\;\;\left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \mathbf{elif}\;x \leq 2601.5280327281093:\\ \;\;\;\;\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x + -1}\right)}^{3}}{{\left(\frac{x}{x + 1}\right)}^{2} + \frac{\mathsf{fma}\left(x + 1, \frac{x + 1}{x + -1}, x\right)}{x + -1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))