Asymptote B

Average Error: 0.0 → 0.0

Time: 5.7s

Precision: binary64

Cost: 1216

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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
    code = (((1.0d0 + x) / x) + (x + (-1.0d0))) / ((x + (-1.0d0)) * (1.0d0 + (1.0d0 / x)))
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) {
	return (((1.0 + x) / x) + (x + -1.0)) / ((x + -1.0) * (1.0 + (1.0 / x)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x)
	tmp = (((1.0 + x) / x) + (x + -1.0)) / ((x + -1.0) * (1.0 + (1.0 / x)));
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_] := N[(N[(N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Taylor expanded in x around 0 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.6
Cost	840

\[\begin{array}{l} t_0 := 1 + \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -5.732353929192063:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.027639669428479896:\\ \;\;\;\;\left(-1 - x\right) + \frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5.732353929192063:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.027639669428479896:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	0.6
Cost	712

\[\begin{array}{l} t_0 := 1 + \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -5.732353929192063:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.027639669428479896:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.0
Cost	704

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

Alternative 5
Error	0.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -5.732353929192063:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.027639669428479896:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	32.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))