Asymptote A

Average Error: 14.2 → 0.1

Time: 16.9s

Precision: binary64

Cost: 576

Math

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

↓

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

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

Derivation

1. Initial program 14.2

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

2. Applied egg-rr 13.6

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

3. Applied egg-rr 0.1

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

Alternatives

Alternative 1
Error	0.8
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	584

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

Alternative 3
Error	0.4
Cost	576

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

Alternative 4
Error	30.7
Cost	320

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

Alternative 5
Error	57.1
Cost	64

\[1 \]

Alternative 6
Error	31.4
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))