Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Precision: binary64

Cost: 1216

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 1}{x}\\ \frac{\frac{x + \left(-1 + t_0\right)}{x - 1}}{t_0} \end{array} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ x 1.0) x))) (/ (/ (+ x (+ -1.0 t_0)) (- x 1.0)) t_0)))

C

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

↓

double code(double x) {
	double t_0 = (x + 1.0) / x;
	return ((x + (-1.0 + t_0)) / (x - 1.0)) / t_0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x - 1.0d0)) + (x / (x + 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (x + 1.0d0) / x
    code = ((x + ((-1.0d0) + t_0)) / (x - 1.0d0)) / t_0
end function

Java

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

↓

public static double code(double x) {
	double t_0 = (x + 1.0) / x;
	return ((x + (-1.0 + t_0)) / (x - 1.0)) / t_0;
}

Python

def code(x):
	return (1.0 / (x - 1.0)) + (x / (x + 1.0))

↓

def code(x):
	t_0 = (x + 1.0) / x
	return ((x + (-1.0 + t_0)) / (x - 1.0)) / t_0

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x - 1.0)) + Float64(x / Float64(x + 1.0)))
end

↓

function code(x)
	t_0 = Float64(Float64(x + 1.0) / x)
	return Float64(Float64(Float64(x + Float64(-1.0 + t_0)) / Float64(x - 1.0)) / t_0)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x - 1.0)) + (x / (x + 1.0));
end

↓

function tmp = code(x)
	t_0 = (x + 1.0) / x;
	tmp = ((x + (-1.0 + t_0)) / (x - 1.0)) / t_0;
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]}, N[(N[(N[(x + N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

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

   [Start] 100.0%	\[ \frac{1}{x - 1} + \frac{x}{x + 1} \]
   clear-num [=>] 100.0%	\[ \frac{1}{x - 1} + \color{blue}{\frac{1}{\frac{x + 1}{x}}} \]
   frac-add [=>] 100.0%	\[ \color{blue}{\frac{1 \cdot \frac{x + 1}{x} + \left(x - 1\right) \cdot 1}{\left(x - 1\right) \cdot \frac{x + 1}{x}}} \]
   *-un-lft-identity [<=] 100.0%	\[ \frac{\color{blue}{\frac{x + 1}{x}} + \left(x - 1\right) \cdot 1}{\left(x - 1\right) \cdot \frac{x + 1}{x}} \]
   +-commutative [=>] 100.0%	\[ \frac{\frac{\color{blue}{1 + x}}{x} + \left(x - 1\right) \cdot 1}{\left(x - 1\right) \cdot \frac{x + 1}{x}} \]
   *-commutative [<=] 100.0%	\[ \frac{\frac{1 + x}{x} + \color{blue}{1 \cdot \left(x - 1\right)}}{\left(x - 1\right) \cdot \frac{x + 1}{x}} \]
   *-un-lft-identity [<=] 100.0%	\[ \frac{\frac{1 + x}{x} + \color{blue}{\left(x - 1\right)}}{\left(x - 1\right) \cdot \frac{x + 1}{x}} \]
   sub-neg [=>] 100.0%	\[ \frac{\frac{1 + x}{x} + \color{blue}{\left(x + \left(-1\right)\right)}}{\left(x - 1\right) \cdot \frac{x + 1}{x}} \]
   metadata-eval [=>] 100.0%	\[ \frac{\frac{1 + x}{x} + \left(x + \color{blue}{-1}\right)}{\left(x - 1\right) \cdot \frac{x + 1}{x}} \]
   sub-neg [=>] 100.0%	\[ \frac{\frac{1 + x}{x} + \left(x + -1\right)}{\color{blue}{\left(x + \left(-1\right)\right)} \cdot \frac{x + 1}{x}} \]
   metadata-eval [=>] 100.0%	\[ \frac{\frac{1 + x}{x} + \left(x + -1\right)}{\left(x + \color{blue}{-1}\right) \cdot \frac{x + 1}{x}} \]
   +-commutative [=>] 100.0%	\[ \frac{\frac{1 + x}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{\color{blue}{1 + x}}{x}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{-1 + x}}{\frac{x + 1}{x}}} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \frac{\frac{1 + x}{x} + \left(x + -1\right)}{\left(x + -1\right) \cdot \frac{1 + x}{x}} \]
   associate-/r* [=>] 100.0%	\[ \color{blue}{\frac{\frac{\frac{1 + x}{x} + \left(x + -1\right)}{x + -1}}{\frac{1 + x}{x}}} \]
   /-rgt-identity [<=] 100.0%	\[ \frac{\frac{\frac{1 + x}{x} + \left(x + -1\right)}{x + -1}}{\color{blue}{\frac{\frac{1 + x}{x}}{1}}} \]
   +-commutative [=>] 100.0%	\[ \frac{\frac{\color{blue}{\left(x + -1\right) + \frac{1 + x}{x}}}{x + -1}}{\frac{\frac{1 + x}{x}}{1}} \]
   associate-+l+ [=>] 100.0%	\[ \frac{\frac{\color{blue}{x + \left(-1 + \frac{1 + x}{x}\right)}}{x + -1}}{\frac{\frac{1 + x}{x}}{1}} \]
   +-commutative [=>] 100.0%	\[ \frac{\frac{x + \left(-1 + \frac{\color{blue}{x + 1}}{x}\right)}{x + -1}}{\frac{\frac{1 + x}{x}}{1}} \]
   +-commutative [=>] 100.0%	\[ \frac{\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\color{blue}{-1 + x}}}{\frac{\frac{1 + x}{x}}{1}} \]
   /-rgt-identity [=>] 100.0%	\[ \frac{\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{-1 + x}}{\color{blue}{\frac{1 + x}{x}}} \]
   +-commutative [=>] 100.0%	\[ \frac{\frac{x + \left(-1 + \frac{x + 1}{x}\right)}{-1 + x}}{\frac{\color{blue}{x + 1}}{x}} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	1216

\[\begin{array}{l} t_0 := \frac{x + 1}{x}\\ \frac{\frac{x + \left(-1 + t_0\right)}{x - 1}}{t_0} \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + \frac{\frac{2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	704

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

Alternative 4
Accuracy	98.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Accuracy	50.3%
Cost	192

\[x - 1 \]

Alternative 6
Accuracy	50.0%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))