Asymptote A

Average Error: 14.6 → 0.1

Time: 4.8s

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 14.6

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

2. Applied egg-rr 14.0

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

3. Simplified 14.0

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

   [Start] 14.0	\[ \frac{x + \left(-2 - x\right)}{1 + x} \cdot \frac{1}{x + -1} \]
   associate-*l/ [=>] 14.0	\[ \color{blue}{\frac{\left(x + \left(-2 - x\right)\right) \cdot \frac{1}{x + -1}}{1 + x}} \]
   associate-*r/ [=>] 14.0	\[ \frac{\color{blue}{\frac{\left(x + \left(-2 - x\right)\right) \cdot 1}{x + -1}}}{1 + x} \]
   *-rgt-identity [=>] 14.0	\[ \frac{\frac{\color{blue}{x + \left(-2 - x\right)}}{x + -1}}{1 + x} \]
   +-commutative [=>] 14.0	\[ \frac{\frac{x + \left(-2 - x\right)}{x + -1}}{\color{blue}{x + 1}} \]

4. Applied egg-rr 14.6

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

5. Simplified 0.1

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

   [Start] 14.6	\[ \frac{\frac{-1}{1 - x} \cdot x + \frac{-1}{1 - x} \cdot \left(-2 - x\right)}{x + 1} \]
   distribute-lft-out [=>] 14.0	\[ \frac{\color{blue}{\frac{-1}{1 - x} \cdot \left(x + \left(-2 - x\right)\right)}}{x + 1} \]
   associate--l+ [<=] 14.0	\[ \frac{\frac{-1}{1 - x} \cdot \color{blue}{\left(\left(x + -2\right) - x\right)}}{x + 1} \]
   +-commutative [=>] 14.0	\[ \frac{\frac{-1}{1 - x} \cdot \left(\color{blue}{\left(-2 + x\right)} - x\right)}{x + 1} \]
   associate-+r- [<=] 0.1	\[ \frac{\frac{-1}{1 - x} \cdot \color{blue}{\left(-2 + \left(x - x\right)\right)}}{x + 1} \]
   associate-*l/ [=>] 0.1	\[ \frac{\color{blue}{\frac{-1 \cdot \left(-2 + \left(x - x\right)\right)}{1 - x}}}{x + 1} \]
   +-inverses [=>] 0.1	\[ \frac{\frac{-1 \cdot \left(-2 + \color{blue}{0}\right)}{1 - x}}{x + 1} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{-1 \cdot \color{blue}{-2}}{1 - x}}{x + 1} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\color{blue}{2}}{1 - x}}{x + 1} \]

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 2
Error	0.9
Cost	584

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

Alternative 3
Error	0.4
Cost	448

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

Alternative 4
Error	31.4
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023027 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))