Asymptote B

Average Error: 0.02% → 0.02%

Time: 2.7s

Precision: binary64

Cost: 1216

Math

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

↓

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

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 + -1.0) + t_0) / Float64(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 + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.02

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

2. Applied egg-rr 0.02

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

3. Final simplification 0.02

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

Alternatives

Alternative 1
Error	0.88%
Cost	713

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

Alternative 2
Error	0.88%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \]

Alternative 3
Error	0.02%
Cost	704

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

Alternative 4
Error	48.52%
Cost	320

\[x + \left(-1 - x\right) \]

Alternative 5
Error	49%
Cost	192

\[x + -1 \]

Alternative 6
Error	49.25%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))