Asymptote A

Average Error: 14.8 → 0.4

Time: 5.7s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ 2.0 (- 1.0 (* x 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 * 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 * 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 * 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 * 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(1.0 - Float64(x * 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 * 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[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.8

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

2. Applied egg-rr 29.8

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

3. Applied egg-rr 0.4

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

4. Simplified 0.4

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

   [Start] 0.4	\[ \frac{2 + \left(x - x\right)}{1 - x \cdot x} \]
   +-commutative [=>] 0.4	\[ \frac{\color{blue}{\left(x - x\right) + 2}}{1 - x \cdot x} \]
   +-inverses [=>] 0.4	\[ \frac{\color{blue}{0} + 2}{1 - x \cdot x} \]
   metadata-eval [=>] 0.4	\[ \frac{\color{blue}{2}}{1 - x \cdot x} \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.0
Cost	585

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

Alternative 2
Error	31.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))