Asymptote C

Average Error: 29.1 → 0.1

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := \frac{1}{x \cdot x}\\ t_2 := {\left(x + 1\right)}^{2}\\ \mathbf{if}\;x \leq -380000:\\ \;\;\;\;\frac{-3}{x} - t_1\\ \mathbf{elif}\;x \leq 12000:\\ \;\;\;\;\frac{\frac{{t_0}^{3} + {\left(-t_2\right)}^{3}}{{t_0}^{2} + \left(t_2 \cdot t_2 + t_0 \cdot t_2\right)}}{\mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-3}{{x}^{3}} - t_1\right)\\ \end{array} \]

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 (/ 1.0 (* x x))) (t_2 (pow (+ x 1.0) 2.0)))
   (if (<= x -380000.0)
     (- (/ -3.0 x) t_1)
     (if (<= x 12000.0)
       (/
        (/
         (+ (pow t_0 3.0) (pow (- t_2) 3.0))
         (+ (pow t_0 2.0) (+ (* t_2 t_2) (* t_0 t_2))))
        (fma x x -1.0))
       (+ (/ -3.0 x) (- (/ -3.0 (pow x 3.0)) t_1))))))

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 = 1.0 / (x * x);
	double t_2 = pow((x + 1.0), 2.0);
	double tmp;
	if (x <= -380000.0) {
		tmp = (-3.0 / x) - t_1;
	} else if (x <= 12000.0) {
		tmp = ((pow(t_0, 3.0) + pow(-t_2, 3.0)) / (pow(t_0, 2.0) + ((t_2 * t_2) + (t_0 * t_2)))) / fma(x, x, -1.0);
	} else {
		tmp = (-3.0 / x) + ((-3.0 / pow(x, 3.0)) - t_1);
	}
	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(1.0 / Float64(x * x))
	t_2 = Float64(x + 1.0) ^ 2.0
	tmp = 0.0
	if (x <= -380000.0)
		tmp = Float64(Float64(-3.0 / x) - t_1);
	elseif (x <= 12000.0)
		tmp = Float64(Float64(Float64((t_0 ^ 3.0) + (Float64(-t_2) ^ 3.0)) / Float64((t_0 ^ 2.0) + Float64(Float64(t_2 * t_2) + Float64(t_0 * t_2)))) / fma(x, x, -1.0));
	else
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-3.0 / (x ^ 3.0)) - t_1));
	end
	return 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[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -380000.0], N[(N[(-3.0 / x), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[x, 12000.0], N[(N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[(-t$95$2), 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -3.8e5

   1. Initial program 59.4

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

   2. Applied egg-rr 61.8

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

   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

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1}{x \cdot x}} \]

   if -3.8e5 < x < 12000

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   3. Applied egg-rr 0.1

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

   if 12000 < x

   1. Initial program 59.0

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

   2. Taylor expanded in x around inf 0.3

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

   3. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -380000:\\ \;\;\;\;\frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \leq 12000:\\ \;\;\;\;\frac{\frac{{\left(x \cdot \left(x + -1\right)\right)}^{3} + {\left(-{\left(x + 1\right)}^{2}\right)}^{3}}{{\left(x \cdot \left(x + -1\right)\right)}^{2} + \left({\left(x + 1\right)}^{2} \cdot {\left(x + 1\right)}^{2} + \left(x \cdot \left(x + -1\right)\right) \cdot {\left(x + 1\right)}^{2}\right)}}{\mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-3}{{x}^{3}} - \frac{1}{x \cdot x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))