Asymptote C

Average Error: 29.0 → 0.3

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -23307.635529317067:\\ \;\;\;\;\frac{\frac{-12}{x \cdot x} - \left(\frac{9}{x} + \frac{48}{{x}^{3}}\right)}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\ \mathbf{elif}\;x \leq 13029.469918336774:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}\right)}{\frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1} + {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + 1}}\right)}^{4} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + 1}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + 1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} \cdot \left(\frac{1}{x} + 3\right) - \frac{3}{{x}^{3}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -23307.635529317067)
   (/
    (- (/ (- 12.0) (* x x)) (+ (/ 9.0 x) (/ 48.0 (pow x 3.0))))
    (+
     (* (/ x (+ x 1.0)) (/ x (+ x 1.0)))
     (/ (+ x (* (+ x 1.0) (/ (+ x 1.0) (- x 1.0)))) (- x 1.0))))
   (if (<= x 13029.469918336774)
     (/
      (log
       (exp (- (pow (/ x (+ x 1.0)) 3.0) (pow (/ (+ x 1.0) (- x 1.0)) 3.0))))
      (+
       (/ (+ x (* (+ x 1.0) (/ (+ x 1.0) (- x 1.0)))) (- x 1.0))
       (*
        (pow (/ (cbrt x) (cbrt (+ x 1.0))) 4.0)
        (* (/ (cbrt x) (cbrt (+ x 1.0))) (/ (cbrt x) (cbrt (+ x 1.0)))))))
     (- (* (/ -1.0 x) (+ (/ 1.0 x) 3.0)) (/ 3.0 (pow x 3.0))))))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -23307.635529317067)) {
		tmp = (((double) ((((double) -(12.0)) / ((double) (x * x))) - ((double) ((9.0 / x) + (48.0 / ((double) pow(x, 3.0))))))) / ((double) (((double) ((x / ((double) (x + 1.0))) * (x / ((double) (x + 1.0))))) + (((double) (x + ((double) (((double) (x + 1.0)) * (((double) (x + 1.0)) / ((double) (x - 1.0))))))) / ((double) (x - 1.0))))));
	} else {
		double tmp_1;
		if ((x <= 13029.469918336774)) {
			tmp_1 = (((double) log(((double) exp(((double) (((double) pow((x / ((double) (x + 1.0))), 3.0)) - ((double) pow((((double) (x + 1.0)) / ((double) (x - 1.0))), 3.0)))))))) / ((double) ((((double) (x + ((double) (((double) (x + 1.0)) * (((double) (x + 1.0)) / ((double) (x - 1.0))))))) / ((double) (x - 1.0))) + ((double) (((double) pow((((double) cbrt(x)) / ((double) cbrt(((double) (x + 1.0))))), 4.0)) * ((double) ((((double) cbrt(x)) / ((double) cbrt(((double) (x + 1.0))))) * (((double) cbrt(x)) / ((double) cbrt(((double) (x + 1.0))))))))))));
		} else {
			tmp_1 = ((double) (((double) ((-1.0 / x) * ((double) ((1.0 / x) + 3.0)))) - (3.0 / ((double) pow(x, 3.0)))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -23307.6355293170673

   1. Initial program Error: 59.4 bits

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

   3. Applied flip3-- Error: 59.4 bits

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

   4. Simplified Error: 59.4 bits

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

   5. Taylor expanded around inf Error: 0.5 bits

      \[\leadsto \frac{\color{blue}{-\left(12 \cdot \frac{1}{{x}^{2}} + \left(9 \cdot \frac{1}{x} + 48 \cdot \frac{1}{{x}^{3}}\right)\right)}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\]

   6. Simplified Error: 0.4 bits

      \[\leadsto \frac{\color{blue}{\frac{-12}{x \cdot x} - \left(\frac{9}{x} + \frac{48}{{x}^{3}}\right)}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\]

   if -23307.6355293170673 < x < 13029.469918336774

   1. Initial program Error: 0.1 bits

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

   3. Applied flip3-- Error: 0.1 bits

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

   4. Simplified Error: 0.1 bits

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

   6. Applied add-cube-cbrt Error: 0.1 bits

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

   7. Applied add-cube-cbrt Error: 0.1 bits

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

   8. Applied times-frac Error: 0.1 bits

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

   9. Applied add-cube-cbrt Error: 0.1 bits

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

   10. Applied add-cube-cbrt Error: 0.1 bits

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

   11. Applied times-frac Error: 0.1 bits

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

   12. Applied swap-sqr Error: 0.1 bits

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

   13. Simplified Error: 0.1 bits

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

   15. Applied add-log-exp Error: 0.1 bits

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

   16. Applied add-log-exp Error: 0.2 bits

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

   17. Applied diff-log Error: 0.2 bits

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

   18. Simplified Error: 0.2 bits

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

   if 13029.469918336774 < x

   1. Initial program Error: 59.3 bits

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

   2. Taylor expanded around inf Error: 0.3 bits

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

   3. Simplified Error: 0.3 bits

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

4. Final simplification Error: 0.3 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -23307.635529317067:\\ \;\;\;\;\frac{\frac{-12}{x \cdot x} - \left(\frac{9}{x} + \frac{48}{{x}^{3}}\right)}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\ \mathbf{elif}\;x \leq 13029.469918336774:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}\right)}{\frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1} + {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + 1}}\right)}^{4} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + 1}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + 1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} \cdot \left(\frac{1}{x} + 3\right) - \frac{3}{{x}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))