Asymptote C

Average Error: 29.9 → 0.0

Time: 9.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\\ \mathbf{if}\;x \leq -586015145795.7788:\\ \;\;\;\;\left(\frac{-3}{x} - t_0\right) - \frac{1}{{x}^{4}}\\ \mathbf{elif}\;x \leq 39024.27154499961:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x + -1\right) + \left(-1 - x\right), -1\right) - x}{\left(x + -1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{{x}^{4}} - t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (* x x)) (/ 3.0 (pow x 3.0)))))
   (if (<= x -586015145795.7788)
     (- (- (/ -3.0 x) t_0) (/ 1.0 (pow x 4.0)))
     (if (<= x 39024.27154499961)
       (/
        (- (fma x (+ (+ x -1.0) (- -1.0 x)) -1.0) x)
        (* (+ x -1.0) (+ x 1.0)))
       (+ (/ -3.0 x) (- (/ -1.0 (pow x 4.0)) 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 = (1.0 / (x * x)) + (3.0 / pow(x, 3.0));
	double tmp;
	if (x <= -586015145795.7788) {
		tmp = ((-3.0 / x) - t_0) - (1.0 / pow(x, 4.0));
	} else if (x <= 39024.27154499961) {
		tmp = (fma(x, ((x + -1.0) + (-1.0 - x)), -1.0) - x) / ((x + -1.0) * (x + 1.0));
	} else {
		tmp = (-3.0 / x) + ((-1.0 / pow(x, 4.0)) - t_0);
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -586015145795.77881

   1. Initial program 60.2

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

   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 -586015145795.77881 < x < 39024.2715449996103

   1. Initial program 0.4

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

   2. Applied add-cube-cbrt_binary64 0.4

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

   3. Applied frac-sub_binary64 0.4

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

   4. Applied cbrt-div_binary64 0.4

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

   5. Applied frac-sub_binary64 0.4

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

   6. Applied cbrt-div_binary64 0.4

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

   7. Applied frac-sub_binary64 0.4

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

   8. Applied cbrt-div_binary64 0.4

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

   9. Applied frac-times_binary64 0.4

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

   10. Applied frac-times_binary64 0.4

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

   11. Simplified 0.1

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

   12. Simplified 0.0

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

   if 39024.2715449996103 < x

   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

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

   4. Applied sub-neg_binary64 0.0

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

   5. Applied associate--l+_binary64 0.0

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

   6. Simplified 0.0

      \[\leadsto \frac{-3}{x} + \color{blue}{\left(\frac{-1}{{x}^{4}} - \left(\frac{3}{{x}^{3}} + \frac{1}{x \cdot x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -586015145795.7788:\\ \;\;\;\;\left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \mathbf{elif}\;x \leq 39024.27154499961:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x + -1\right) + \left(-1 - x\right), -1\right) - x}{\left(x + -1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{{x}^{4}} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))