Average Error: 0.0 → 0.0

Time: 59.6s

Precision: binary64

Cost: 1536

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

↓

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

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

↓

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

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

↓

(FPCore (x)
 :precision binary64
 (+
  (cbrt (pow (/ x (+ -1.0 (* x x))) 3.0))
  (+ (/ 1.0 (+ -1.0 (* x x))) (/ x (+ x 1.0)))))

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

↓

double code(double x) {
	return cbrt(pow((x / (-1.0 + (x * x))), 3.0)) + ((1.0 / (-1.0 + (x * x))) + (x / (x + 1.0)));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Alternative 1
Accuracy	0.0
Cost	2432

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

Alternative 2
Accuracy	0.0
Cost	1216

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

Alternative 3
Accuracy	0.0
Cost	704

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

Alternative 4
Accuracy	0.0
Cost	1664

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

Derivation

Initial program 0.0
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
Using strategy rm
Applied *-un-lft-identity_binary64_7600.0
\[\leadsto \frac{1}{x - 1} + \frac{\color{blue}{1 \cdot x}}{x + 1}\]
Applied associate-/l*_binary64_7050.0
\[\leadsto \frac{1}{x - 1} + \color{blue}{\frac{1}{\frac{x + 1}{x}}}\]
Simplified0.0
\[\leadsto \frac{1}{x - 1} + \frac{1}{\color{blue}{\frac{1 + x}{x}}}\]
Using strategy rm
Applied flip--_binary64_7350.0
\[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}} + \frac{1}{\frac{1 + x}{x}}\]
Applied associate-/r/_binary64_7060.0
\[\leadsto \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)} + \frac{1}{\frac{1 + x}{x}}\]
Simplified0.0
\[\leadsto \color{blue}{\frac{1}{-1 + x \cdot x}} \cdot \left(x + 1\right) + \frac{1}{\frac{1 + x}{x}}\]
Using strategy rm
Applied distribute-rgt-in_binary64_7100.0
\[\leadsto \color{blue}{\left(x \cdot \frac{1}{-1 + x \cdot x} + 1 \cdot \frac{1}{-1 + x \cdot x}\right)} + \frac{1}{\frac{1 + x}{x}}\]
Applied associate-+l+_binary64_6930.0
\[\leadsto \color{blue}{x \cdot \frac{1}{-1 + x \cdot x} + \left(1 \cdot \frac{1}{-1 + x \cdot x} + \frac{1}{\frac{1 + x}{x}}\right)}\]
Simplified0.0
\[\leadsto x \cdot \frac{1}{-1 + x \cdot x} + \color{blue}{\left(\frac{1}{-1 + x \cdot x} + \frac{x}{1 + x}\right)}\]
Using strategy rm
Applied add-cbrt-cube_binary64_7960.0
\[\leadsto \color{blue}{\sqrt[3]{\left(\left(x \cdot \frac{1}{-1 + x \cdot x}\right) \cdot \left(x \cdot \frac{1}{-1 + x \cdot x}\right)\right) \cdot \left(x \cdot \frac{1}{-1 + x \cdot x}\right)}} + \left(\frac{1}{-1 + x \cdot x} + \frac{x}{1 + x}\right)\]
Simplified0.0
\[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x}{-1 + x \cdot x}\right)}^{3}}} + \left(\frac{1}{-1 + x \cdot x} + \frac{x}{1 + x}\right)\]
Final simplification0.0
\[\leadsto \sqrt[3]{{\left(\frac{x}{-1 + x \cdot x}\right)}^{3}} + \left(\frac{1}{-1 + x \cdot x} + \frac{x}{x + 1}\right)\]

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))