Asymptote A

Average Error: 14.6 → 0.4

Time: 5.2s

Precision: binary64

Cost: 576

Math

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

↓

\[\frac{2}{\left(1 + x\right) \cdot \left(1 - x\right)} \]

FPCore

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

↓

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

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) * (1.0 - x));
}

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) * (1.0d0 - x))
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) * (1.0 - x));
}

Python

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

↓

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

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(2.0 / Float64(Float64(1.0 + x) * Float64(1.0 - x)))
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) * (1.0 - x));
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[(2.0 / N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.6

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

2. Applied egg-rr 14.1

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

3. Taylor expanded in x around 0 0.4

   \[\leadsto \frac{\color{blue}{2}}{\left(1 + x\right) \cdot \left(\left(-x\right) + 1\right)} \]

4. Final simplification 0.4

   \[\leadsto \frac{2}{\left(1 + x\right) \cdot \left(1 - x\right)} \]

Alternatives

Alternative 1
Error	0.7
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x}\\ \mathbf{if}\;x \leq -3.5407440912576877:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.10170565312564367:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	584

\[\begin{array}{l} t_0 := \frac{-2}{x \cdot x}\\ \mathbf{if}\;x \leq -3.5407440912576877:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.10170565312564367:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	576

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

Alternative 4
Error	62.0
Cost	64

\[x \]

Alternative 5
Error	31.4
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022313 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))