Asymptote C

Average Error: 29.0 → 0.1

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{x + 1}{x + -1}\\ \mathbf{if}\;x \leq -10731.07214304758:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{{x}^{3}}\right)\\ \mathbf{elif}\;x \leq 443381.0077497065:\\ \;\;\;\;\frac{1}{\frac{t_0 + t_1}{{t_0}^{2} - {t_1}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (/ (+ x 1.0) (+ x -1.0))))
   (if (<= x -10731.07214304758)
     (+ (/ -3.0 x) (+ (/ -1.0 (* x x)) (/ -3.0 (pow x 3.0))))
     (if (<= x 443381.0077497065)
       (/ 1.0 (/ (+ t_0 t_1) (- (pow t_0 2.0) (pow t_1 2.0))))
       (+ (/ -3.0 x) (/ (/ -1.0 x) x))))))

C

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

↓

double code(double x) {
	double t_0 = x / (x + 1.0);
	double t_1 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (x <= -10731.07214304758) {
		tmp = (-3.0 / x) + ((-1.0 / (x * x)) + (-3.0 / pow(x, 3.0)));
	} else if (x <= 443381.0077497065) {
		tmp = 1.0 / ((t_0 + t_1) / (pow(t_0, 2.0) - pow(t_1, 2.0)));
	} else {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    t_1 = (x + 1.0d0) / (x + (-1.0d0))
    if (x <= (-10731.07214304758d0)) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / (x * x)) + ((-3.0d0) / (x ** 3.0d0)))
    else if (x <= 443381.0077497065d0) then
        tmp = 1.0d0 / ((t_0 + t_1) / ((t_0 ** 2.0d0) - (t_1 ** 2.0d0)))
    else
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    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 = x / (x + 1.0);
	double t_1 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (x <= -10731.07214304758) {
		tmp = (-3.0 / x) + ((-1.0 / (x * x)) + (-3.0 / Math.pow(x, 3.0)));
	} else if (x <= 443381.0077497065) {
		tmp = 1.0 / ((t_0 + t_1) / (Math.pow(t_0, 2.0) - Math.pow(t_1, 2.0)));
	} else {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = x / (x + 1.0)
	t_1 = (x + 1.0) / (x + -1.0)
	tmp = 0
	if x <= -10731.07214304758:
		tmp = (-3.0 / x) + ((-1.0 / (x * x)) + (-3.0 / math.pow(x, 3.0)))
	elif x <= 443381.0077497065:
		tmp = 1.0 / ((t_0 + t_1) / (math.pow(t_0, 2.0) - math.pow(t_1, 2.0)))
	else:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	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(x / Float64(x + 1.0))
	t_1 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	tmp = 0.0
	if (x <= -10731.07214304758)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / Float64(x * x)) + Float64(-3.0 / (x ^ 3.0))));
	elseif (x <= 443381.0077497065)
		tmp = Float64(1.0 / Float64(Float64(t_0 + t_1) / Float64((t_0 ^ 2.0) - (t_1 ^ 2.0))));
	else
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	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 = x / (x + 1.0);
	t_1 = (x + 1.0) / (x + -1.0);
	tmp = 0.0;
	if (x <= -10731.07214304758)
		tmp = (-3.0 / x) + ((-1.0 / (x * x)) + (-3.0 / (x ^ 3.0)));
	elseif (x <= 443381.0077497065)
		tmp = 1.0 / ((t_0 + t_1) / ((t_0 ^ 2.0) - (t_1 ^ 2.0)));
	else
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	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[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -10731.07214304758], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 443381.0077497065], N[(1.0 / N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -10731.0721430475805

   1. Initial program 59.2

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

   2. Taylor expanded in x around inf 0.4

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

   3. Simplified 0.0

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

   if -10731.0721430475805 < x < 443381.007749707

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   if 443381.007749707 < x

   1. Initial program 59.5

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

   2. Applied egg-rr 59.5

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{x}{x + 1}\right)\right)} - \frac{x + 1}{x - 1} \]

   3. Taylor expanded in x around inf 0.4

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{\frac{1}{x}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10731.07214304758:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{{x}^{3}}\right)\\ \mathbf{elif}\;x \leq 443381.0077497065:\\ \;\;\;\;\frac{1}{\frac{\frac{x}{x + 1} + \frac{x + 1}{x + -1}}{{\left(\frac{x}{x + 1}\right)}^{2} - {\left(\frac{x + 1}{x + -1}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))