Asymptote A

Average Error: 14.4 → 0.1

Time: 1.5min

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (/ 2.0 (- x -1.0)) (- 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 / (x - -1.0)) / (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 / (x - (-1.0d0))) / (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 / (x - -1.0)) / (1.0 - x);
}

Python

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

↓

def code(x):
	return (2.0 / (x - -1.0)) / (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(x - -1.0)) / 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 / (x - -1.0)) / (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[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.4

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

2. Applied egg-rr 13.8

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

3. Applied egg-rr 0.7

   \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\frac{2}{1 + x}\right)}^{2}} \cdot \sqrt[3]{\frac{2}{1 + x}}}}{1 - x} \]

4. Simplified 0.1

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

Alternatives

Alternative 1
Error	0.7
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	1.0
Cost	584

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

Alternative 3
Error	0.7
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 4
Error	57.2
Cost	64

\[1 \]

Alternative 5
Error	31.8
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))