Asymptote B

Average Error: 0.0 → 0.0

Time: 3.9s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

def code(x):
	return (x + (1.0 / x)) / (x + (-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(x + Float64(1.0 / x)) / Float64(x + 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 = (x + (1.0 / x)) / (x + (-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[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Taylor expanded in x around 0 0.0

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

4. Taylor expanded in x around 0 0.0

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.6
Cost	840

\[\begin{array}{l} t_0 := 1 + \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -117.82726752014982:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.014417641856487702:\\ \;\;\;\;\left(-1 - x\right) + \frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -117.82726752014982:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.8570617936911118:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	0.6
Cost	712

\[\begin{array}{l} t_0 := 1 + \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -117.82726752014982:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.014417641856487702:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -117.82726752014982:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.014417641856487702:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	32.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022294 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))