Asymptote A

Average Error: 14.6 → 0.1

Time: 6.3s

Precision: binary64

Cost: 576

Math

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

↓

\[\frac{\frac{-2}{1 + x}}{x + -1} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (/ -2.0 (+ 1.0 x)) (+ x -1.0)))

C

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

↓

double code(double x) {
	return (-2.0 / (1.0 + x)) / (x + -1.0);
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-2.0d0) / (1.0d0 + x)) / (x + (-1.0d0))
end function

Java

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

↓

public static double code(double x) {
	return (-2.0 / (1.0 + x)) / (x + -1.0);
}

Python

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

↓

def code(x):
	return (-2.0 / (1.0 + x)) / (x + -1.0)

Julia

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

↓

function code(x)
	return Float64(Float64(-2.0 / Float64(1.0 + x)) / Float64(x + -1.0))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (-2.0 / (1.0 + x)) / (x + -1.0);
end

Wolfram

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

↓

code[x_] := N[(N[(-2.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.6

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

2. Applied egg-rr 13.9

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

3. Taylor expanded in x around 0 0.1

   \[\leadsto \frac{\frac{\color{blue}{-2}}{1 + x}}{x + -1} \]

4. Final simplification 0.1

   \[\leadsto \frac{\frac{-2}{1 + x}}{x + -1} \]

Alternatives

Alternative 1
Error	1.0
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x}\\ \mathbf{if}\;x \leq -12637.300802874537:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.9920167451963254 \cdot 10^{-7}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.4
Cost	448

\[\frac{2}{1 - x \cdot x} \]

Alternative 3
Error	31.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022298 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))