Asymptote C

Average Error: 29.4 → 0.1

Time: 17.8s

Precision: binary64

Cost: 7240

Math

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

↓

\[\begin{array}{l} t_0 := -\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\\ \mathbf{if}\;x \leq -360000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 400000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (+ (/ 1.0 (pow x 2.0)) (/ 3.0 x)))))
   (if (<= x -360000.0)
     t_0
     (if (<= x 400000.0) (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) t_0))))

C

double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}

↓

double code(double x) {
	double t_0 = -((1.0 / pow(x, 2.0)) + (3.0 / x));
	double tmp;
	if (x <= -360000.0) {
		tmp = t_0;
	} else if (x <= 400000.0) {
		tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -((1.0d0 / (x ** 2.0d0)) + (3.0d0 / x))
    if (x <= (-360000.0d0)) then
        tmp = t_0
    else if (x <= 400000.0d0) then
        tmp = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x - 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}

↓

public static double code(double x) {
	double t_0 = -((1.0 / Math.pow(x, 2.0)) + (3.0 / x));
	double tmp;
	if (x <= -360000.0) {
		tmp = t_0;
	} else if (x <= 400000.0) {
		tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = -((1.0 / math.pow(x, 2.0)) + (3.0 / x))
	tmp = 0
	if x <= -360000.0:
		tmp = t_0
	elif x <= 400000.0:
		tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	return Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0)))
end

↓

function code(x)
	t_0 = Float64(-Float64(Float64(1.0 / (x ^ 2.0)) + Float64(3.0 / x)))
	tmp = 0.0
	if (x <= -360000.0)
		tmp = t_0;
	elseif (x <= 400000.0)
		tmp = Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
end

↓

function tmp_2 = code(x)
	t_0 = -((1.0 / (x ^ 2.0)) + (3.0 / x));
	tmp = 0.0;
	if (x <= -360000.0)
		tmp = t_0;
	elseif (x <= 400000.0)
		tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = (-N[(N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(3.0 / x), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -360000.0], t$95$0, If[LessEqual[x, 400000.0], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e5 or 4e5 < x

   1. Initial program 59.6

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

   2. Taylor expanded in x around inf 0.4

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

   3. Taylor expanded in x around 0 0.1

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

   if -3.6e5 < x < 4e5

   1. Initial program 0.2

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -360000:\\ \;\;\;\;-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \leq 400000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -155000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 170000000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;3 \cdot x + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 3
Error	1.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 4
Error	31.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))