Asymptote C

Average Error: 29.5 → 0.1

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-1, {x}^{-2}, \mathsf{fma}\left(-3, {x}^{-3}, \frac{-3}{x}\right)\right)\\ \mathbf{if}\;x \leq -6176794.5564302765:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 209.85361716255164:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -1.0 (pow x -2.0) (fma -3.0 (pow x -3.0) (/ -3.0 x)))))
   (if (<= x -6176794.5564302765)
     t_0
     (if (<= x 209.85361716255164)
       (fma x (/ 1.0 (+ x 1.0)) (/ (fma -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 = fma(-1.0, pow(x, -2.0), fma(-3.0, pow(x, -3.0), (-3.0 / x)));
	double tmp;
	if (x <= -6176794.5564302765) {
		tmp = t_0;
	} else if (x <= 209.85361716255164) {
		tmp = fma(x, (1.0 / (x + 1.0)), (fma(-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 = fma(-1.0, (x ^ -2.0), fma(-3.0, (x ^ -3.0), Float64(-3.0 / x)))
	tmp = 0.0
	if (x <= -6176794.5564302765)
		tmp = t_0;
	elseif (x <= 209.85361716255164)
		tmp = fma(x, Float64(1.0 / Float64(x + 1.0)), Float64(fma(-1.0, x, -1.0) / Float64(x + -1.0)));
	else
		tmp = t_0;
	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[(-1.0 * N[Power[x, -2.0], $MachinePrecision] + N[(-3.0 * N[Power[x, -3.0], $MachinePrecision] + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6176794.5564302765], t$95$0, If[LessEqual[x, 209.85361716255164], N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * x + -1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6176794.5564302765 or 209.85361716255164 < x

   1. Initial program 59.5

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

   2. Taylor expanded in x around inf 0.4

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

   3. Simplified 0.1

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

   4. Applied egg-rr 0.1

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

   if -6176794.5564302765 < x < 209.85361716255164

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6176794.5564302765:\\ \;\;\;\;\mathsf{fma}\left(-1, {x}^{-2}, \mathsf{fma}\left(-3, {x}^{-3}, \frac{-3}{x}\right)\right)\\ \mathbf{elif}\;x \leq 209.85361716255164:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, {x}^{-2}, \mathsf{fma}\left(-3, {x}^{-3}, \frac{-3}{x}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))