Asymptote C

Average Error: 29.1 → 0.0

Time: 24.4s

Precision: binary64

Cost: 20808

Math

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

↓

\[\begin{array}{l} t_0 := -\left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + \frac{3}{x}\right)\right)\right)\\ t_1 := \frac{x + 1}{x + -1}\\ \mathbf{if}\;x \leq -1900:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2000:\\ \;\;\;\;\left(\frac{x}{x + 1} + t_1\right) + \left(-t_1\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 4.0))
           (+ (/ 1.0 (pow x 2.0)) (+ (* 3.0 (/ 1.0 (pow x 3.0))) (/ 3.0 x))))))
        (t_1 (/ (+ x 1.0) (+ x -1.0))))
   (if (<= x -1900.0)
     t_0
     (if (<= x 2000.0) (+ (+ (/ x (+ x 1.0)) t_1) (* (- t_1) 2.0)) 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, 4.0)) + ((1.0 / pow(x, 2.0)) + ((3.0 * (1.0 / pow(x, 3.0))) + (3.0 / x))));
	double t_1 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (x <= -1900.0) {
		tmp = t_0;
	} else if (x <= 2000.0) {
		tmp = ((x / (x + 1.0)) + t_1) + (-t_1 * 2.0);
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -((1.0d0 / (x ** 4.0d0)) + ((1.0d0 / (x ** 2.0d0)) + ((3.0d0 * (1.0d0 / (x ** 3.0d0))) + (3.0d0 / x))))
    t_1 = (x + 1.0d0) / (x + (-1.0d0))
    if (x <= (-1900.0d0)) then
        tmp = t_0
    else if (x <= 2000.0d0) then
        tmp = ((x / (x + 1.0d0)) + t_1) + (-t_1 * 2.0d0)
    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, 4.0)) + ((1.0 / Math.pow(x, 2.0)) + ((3.0 * (1.0 / Math.pow(x, 3.0))) + (3.0 / x))));
	double t_1 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (x <= -1900.0) {
		tmp = t_0;
	} else if (x <= 2000.0) {
		tmp = ((x / (x + 1.0)) + t_1) + (-t_1 * 2.0);
	} 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, 4.0)) + ((1.0 / math.pow(x, 2.0)) + ((3.0 * (1.0 / math.pow(x, 3.0))) + (3.0 / x))))
	t_1 = (x + 1.0) / (x + -1.0)
	tmp = 0
	if x <= -1900.0:
		tmp = t_0
	elif x <= 2000.0:
		tmp = ((x / (x + 1.0)) + t_1) + (-t_1 * 2.0)
	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 ^ 4.0)) + Float64(Float64(1.0 / (x ^ 2.0)) + Float64(Float64(3.0 * Float64(1.0 / (x ^ 3.0))) + Float64(3.0 / x)))))
	t_1 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	tmp = 0.0
	if (x <= -1900.0)
		tmp = t_0;
	elseif (x <= 2000.0)
		tmp = Float64(Float64(Float64(x / Float64(x + 1.0)) + t_1) + Float64(Float64(-t_1) * 2.0));
	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 ^ 4.0)) + ((1.0 / (x ^ 2.0)) + ((3.0 * (1.0 / (x ^ 3.0))) + (3.0 / x))));
	t_1 = (x + 1.0) / (x + -1.0);
	tmp = 0.0;
	if (x <= -1900.0)
		tmp = t_0;
	elseif (x <= 2000.0)
		tmp = ((x / (x + 1.0)) + t_1) + (-t_1 * 2.0);
	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, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 * N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1900.0], t$95$0, If[LessEqual[x, 2000.0], N[(N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[((-t$95$1) * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1900 or 2e3 < x

   1. Initial program 59.4

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

   2. Taylor expanded in x around inf 0.3

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

   3. Taylor expanded in x around 0 0.0

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

   if -1900 < x < 2e3

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1900:\\ \;\;\;\;-\left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + \frac{3}{x}\right)\right)\right)\\ \mathbf{elif}\;x \leq 2000:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{x + 1}{x + -1}\right) + \left(-\frac{x + 1}{x + -1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + \frac{3}{x}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	8776

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := -\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\\ t_2 := \frac{x + 1}{x + -1}\\ t_3 := t_2 - t_0\\ t_4 := t_3 \cdot \frac{1}{t_3}\\ t_5 := t_0 - t_2\\ \mathbf{if}\;x \leq -460000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 290000:\\ \;\;\;\;\left(\left(\left(t_3 \cdot t_4\right) \cdot t_4\right) \cdot \left(t_3 \cdot \frac{-1}{t_3}\right)\right) \cdot \left(t_5 \cdot \frac{1}{t_5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7240

\[\begin{array}{l} t_0 := \frac{x + 1}{x + -1}\\ t_1 := -\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\\ t_2 := \frac{x}{x + 1}\\ t_3 := t_0 - t_2\\ t_4 := t_2 - t_0\\ \mathbf{if}\;x \leq -460000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 290000:\\ \;\;\;\;\left(\left(t_3 \cdot \left(t_3 \cdot \frac{1}{t_3}\right)\right) \cdot \left(t_3 \cdot \frac{-1}{t_3}\right)\right) \cdot \left(t_4 \cdot \frac{1}{t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.3
Cost	6856

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{x + 1}{x + -1}\\ t_2 := t_0 - t_1\\ t_3 := t_1 - t_0\\ \mathbf{if}\;x \leq -120000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 170000000:\\ \;\;\;\;\left(\left(t_3 \cdot \left(t_3 \cdot \frac{1}{t_3}\right)\right) \cdot \left(t_3 \cdot \frac{-1}{t_3}\right)\right) \cdot \left(t_2 \cdot \frac{1}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	4936

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{x + 1}{x + -1}\\ t_2 := t_0 - t_1\\ t_3 := t_1 - t_0\\ \mathbf{if}\;x \leq -120000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 170000000:\\ \;\;\;\;\left(t_3 \cdot \left(t_3 \cdot \frac{-1}{t_3}\right)\right) \cdot \left(t_2 \cdot \frac{1}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	3016

\[\begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;x \leq -120000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 170000000:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot \frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 6
Error	0.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -120000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 170000000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 7
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 8
Error	1.4
Cost	456

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

Alternative 9
Error	31.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023092 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))