Asymptote C

Average Error: 29.2 → 0.3

Time: 8.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 1.64651002377 \cdot 10^{-6}:\\ \;\;\;\;\left(-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\right) + \frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \left(\left(-\frac{\sqrt[3]{x + 1}}{\sqrt[3]{x - 1}}\right) + \frac{\sqrt[3]{x + 1}}{\sqrt[3]{x - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 1.6465100237672559e-06)) {
		VAR = (-((1.0 * (1.0 / pow(x, 2.0))) + ((3.0 * (1.0 / x)) + (3.0 * (1.0 / pow(x, 3.0))))) + (((cbrt((x + 1.0)) * cbrt((x + 1.0))) / (cbrt((x - 1.0)) * cbrt((x - 1.0)))) * (-(cbrt((x + 1.0)) / cbrt((x - 1.0))) + (cbrt((x + 1.0)) / cbrt((x - 1.0))))));
	} else {
		VAR = log(exp(((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0)))));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) < 1.6465100237672559e-06

   1. Initial program 59.0

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

   3. Applied add-cube-cbrt 59.9

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

   4. Applied add-cube-cbrt 59.1

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

   5. Applied times-frac 59.1

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

   6. Applied add-cube-cbrt 59.1

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

   7. Applied prod-diff 59.1

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

   8. Simplified 59.1

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

   9. Simplified 59.1

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

   10. Taylor expanded around inf 0.5

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

   if 1.6465100237672559e-06 < (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Applied add-log-exp 0.1

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

   5. Applied diff-log 0.1

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

   6. Simplified 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 1.64651002377 \cdot 10^{-6}:\\ \;\;\;\;\left(-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\right) + \frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \left(\left(-\frac{\sqrt[3]{x + 1}}{\sqrt[3]{x - 1}}\right) + \frac{\sqrt[3]{x + 1}}{\sqrt[3]{x - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))