Asymptote C

Average Error: 29.0 → 0.1

Time: 7.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x} + \left(\frac{-3}{{x}^{3}} + \left(\frac{-3}{x} + \frac{-1}{{x}^{4}}\right)\right)\\ \mathbf{if}\;x \leq -915718.0760814397:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1672.3439861122827:\\ \;\;\;\;{\left(\frac{x + 1}{x}\right)}^{-1} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          (/ -1.0 (* x x))
          (+ (/ -3.0 (pow x 3.0)) (+ (/ -3.0 x) (/ -1.0 (pow x 4.0)))))))
   (if (<= x -915718.0760814397)
     t_0
     (if (<= x 1672.3439861122827)
       (+ (pow (/ (+ x 1.0) x) -1.0) (/ (- -1.0 x) (+ 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 / (x * x)) + ((-3.0 / pow(x, 3.0)) + ((-3.0 / x) + (-1.0 / pow(x, 4.0))));
	double tmp;
	if (x <= -915718.0760814397) {
		tmp = t_0;
	} else if (x <= 1672.3439861122827) {
		tmp = pow(((x + 1.0) / x), -1.0) + ((-1.0 - x) / (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 * x)) + (((-3.0d0) / (x ** 3.0d0)) + (((-3.0d0) / x) + ((-1.0d0) / (x ** 4.0d0))))
    if (x <= (-915718.0760814397d0)) then
        tmp = t_0
    else if (x <= 1672.3439861122827d0) then
        tmp = (((x + 1.0d0) / x) ** (-1.0d0)) + (((-1.0d0) - x) / (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 / (x * x)) + ((-3.0 / Math.pow(x, 3.0)) + ((-3.0 / x) + (-1.0 / Math.pow(x, 4.0))));
	double tmp;
	if (x <= -915718.0760814397) {
		tmp = t_0;
	} else if (x <= 1672.3439861122827) {
		tmp = Math.pow(((x + 1.0) / x), -1.0) + ((-1.0 - x) / (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 / (x * x)) + ((-3.0 / math.pow(x, 3.0)) + ((-3.0 / x) + (-1.0 / math.pow(x, 4.0))))
	tmp = 0
	if x <= -915718.0760814397:
		tmp = t_0
	elif x <= 1672.3439861122827:
		tmp = math.pow(((x + 1.0) / x), -1.0) + ((-1.0 - x) / (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(-1.0 / Float64(x * x)) + Float64(Float64(-3.0 / (x ^ 3.0)) + Float64(Float64(-3.0 / x) + Float64(-1.0 / (x ^ 4.0)))))
	tmp = 0.0
	if (x <= -915718.0760814397)
		tmp = t_0;
	elseif (x <= 1672.3439861122827)
		tmp = Float64((Float64(Float64(x + 1.0) / x) ^ -1.0) + Float64(Float64(-1.0 - x) / 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 * x)) + ((-3.0 / (x ^ 3.0)) + ((-3.0 / x) + (-1.0 / (x ^ 4.0))));
	tmp = 0.0;
	if (x <= -915718.0760814397)
		tmp = t_0;
	elseif (x <= 1672.3439861122827)
		tmp = (((x + 1.0) / x) ^ -1.0) + ((-1.0 - x) / (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[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 / x), $MachinePrecision] + N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -915718.0760814397], t$95$0, If[LessEqual[x, 1672.3439861122827], N[(N[Power[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -915718.076081439736 or 1672.34398611228266 < x

   1. Initial program 59.4

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

   2. Applied egg-rr 59.4

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

   3. Taylor expanded in x around inf 0.3

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)\right)} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x} + \left(\frac{-3}{{x}^{3}} + \left(\frac{-3}{x} + \frac{-1}{{x}^{4}}\right)\right)} \]

   if -915718.076081439736 < x < 1672.34398611228266

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x}\right)}^{-1}} - \frac{x + 1}{x - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -915718.0760814397:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{{x}^{3}} + \left(\frac{-3}{x} + \frac{-1}{{x}^{4}}\right)\right)\\ \mathbf{elif}\;x \leq 1672.3439861122827:\\ \;\;\;\;{\left(\frac{x + 1}{x}\right)}^{-1} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{{x}^{3}} + \left(\frac{-3}{x} + \frac{-1}{{x}^{4}}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))