Asymptote C

Average Error: 29.2 → 0.1

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;-\left(\frac{3 + \frac{1}{x}}{x} + \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x + x, \frac{0.5}{x + 1}, -\frac{x + 1}{x + -1}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 5e-5)
   (- (+ (/ (+ 3.0 (/ 1.0 x)) x) (+ (/ 3.0 (pow x 3.0)) (/ 1.0 (pow x 4.0)))))
   (fma (+ x x) (/ 0.5 (+ x 1.0)) (- (/ (+ x 1.0) (+ x -1.0))))))

C

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

↓

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 5e-5) {
		tmp = -(((3.0 + (1.0 / x)) / x) + ((3.0 / pow(x, 3.0)) + (1.0 / pow(x, 4.0))));
	} else {
		tmp = fma((x + x), (0.5 / (x + 1.0)), -((x + 1.0) / (x + -1.0)));
	}
	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)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 5e-5)
		tmp = Float64(-Float64(Float64(Float64(3.0 + Float64(1.0 / x)) / x) + Float64(Float64(3.0 / (x ^ 3.0)) + Float64(1.0 / (x ^ 4.0)))));
	else
		tmp = fma(Float64(x + x), Float64(0.5 / Float64(x + 1.0)), Float64(-Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	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_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-5], (-N[(N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(x + x), $MachinePrecision] * N[(0.5 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + (-N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.00000000000000024e-5

   1. Initial program 58.9

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

   2. Taylor expanded in x around inf 0.5

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

   3. Simplified 0.2

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + x, \frac{0.5}{x + 1}, -\frac{x + 1}{x + -1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;-\left(\frac{3 + \frac{1}{x}}{x} + \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x + x, \frac{0.5}{x + 1}, -\frac{x + 1}{x + -1}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))