Asymptote A

Average Error: 14.5 → 0.1

Time: 3.6s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 14.5

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

2. Applied egg-rr 13.8

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

3. Taylor expanded in x around 0 0.1

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

4. Applied egg-rr 0.1

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

5. Simplified 0.1

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{-1}{1 - x}}{x + 1}} \]
   Proof
   (/.f64 (*.f64 -2 (/.f64 -1 (-.f64 1 x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 -2 (/.f64 -1 (-.f64 1 x))) (Rewrite<= +-commutative_binary64 (+.f64 1 x))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 -2 (+.f64 1 x)) (/.f64 -1 (-.f64 1 x)))): 9 points increase in error, 1 points decrease in error

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.7
Cost	840

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

Alternative 2
Error	1.1
Cost	584

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

Alternative 3
Error	0.7
Cost	584

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

Alternative 4
Error	0.1
Cost	576

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

Alternative 5
Error	31.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))