Asymptote C

Average Error: 28.9 → 0.2

Time: 5.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.001:\\ \;\;\;\;\left(-\frac{3}{x}\right) - \mathsf{fma}\left(\mathsf{fma}\left({x}^{-3}, x - -3, {x}^{-4}\right), 1, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \left(x + 1\right) \cdot \frac{x + 1}{x + -1}}{x + 1}\\ \end{array} \]

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))) 0.001)
   (- (- (/ 3.0 x)) (fma (fma (pow x -3.0) (- x -3.0) (pow x -4.0)) 1.0 0.0))
   (/ (- x (* (+ x 1.0) (/ (+ 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))) <= 0.001) {
		tmp = -(3.0 / x) - fma(fma(pow(x, -3.0), (x - -3.0), pow(x, -4.0)), 1.0, 0.0);
	} else {
		tmp = (x - ((x + 1.0) * ((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))) <= 0.001)
		tmp = Float64(Float64(-Float64(3.0 / x)) - fma(fma((x ^ -3.0), Float64(x - -3.0), (x ^ -4.0)), 1.0, 0.0));
	else
		tmp = Float64(Float64(x - Float64(Float64(x + 1.0) * Float64(Float64(x + 1.0) / 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], 0.001], N[((-N[(3.0 / x), $MachinePrecision]) - N[(N[(N[Power[x, -3.0], $MachinePrecision] * N[(x - -3.0), $MachinePrecision] + N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision] * 1.0 + 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 1e-3

   1. Initial program 58.8

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

   2. Taylor expanded in x around inf 0.6

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

   3. Simplified 0.3

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

   4. Applied egg-rr 0.8

      \[\leadsto \left(-\frac{3}{x}\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left({x}^{-3}, x - -3, {x}^{-4}\right)\right)} + -1\right)} \]

   5. Applied egg-rr 0.3

      \[\leadsto \left(-\frac{3}{x}\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{-3}, x - -3, {x}^{-4}\right), 1, 0\right)} \]

   if 1e-3 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.001:\\ \;\;\;\;\left(-\frac{3}{x}\right) - \mathsf{fma}\left(\mathsf{fma}\left({x}^{-3}, x - -3, {x}^{-4}\right), 1, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \left(x + 1\right) \cdot \frac{x + 1}{x + -1}}{x + 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))