Asymptote C

Average Error: 29.7 → 0.8

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 1.8052226380405045 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(-\frac{3}{x}\right) - \frac{3 + x}{{x}^{3}}\right) - \sqrt[3]{{x}^{-12}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3 + x, 1\right)\\ \end{array} \]

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))) 1.8052226380405045e-13)
   (- (- (- (/ 3.0 x)) (/ (+ 3.0 x) (pow x 3.0))) (cbrt (pow x -12.0)))
   (fma x (+ 3.0 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))) <= 1.8052226380405045e-13) {
		tmp = (-(3.0 / x) - ((3.0 + x) / pow(x, 3.0))) - cbrt(pow(x, -12.0));
	} else {
		tmp = fma(x, (3.0 + 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))) <= 1.8052226380405045e-13)
		tmp = Float64(Float64(Float64(-Float64(3.0 / x)) - Float64(Float64(3.0 + x) / (x ^ 3.0))) - cbrt((x ^ -12.0)));
	else
		tmp = fma(x, Float64(3.0 + 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], 1.8052226380405045e-13], N[(N[((-N[(3.0 / x), $MachinePrecision]) - N[(N[(3.0 + x), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[Power[x, -12.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(x * N[(3.0 + x), $MachinePrecision] + 1.0), $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))) < 1.80522e-13

   1. Initial program 59.4

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

   2. Taylor expanded in x around inf 0.5

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

   3. Simplified 0.2

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

   4. Applied egg-rr 0.2

      \[\leadsto \left(\left(-\frac{3}{x}\right) - \frac{3 + x}{{x}^{3}}\right) - \color{blue}{\sqrt[3]{{x}^{-12}}} \]

   if 1.80522e-13 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 0.4

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

   2. Taylor expanded in x around 0 1.5

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

   3. Simplified 1.5

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 1.8052226380405045 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(-\frac{3}{x}\right) - \frac{3 + x}{{x}^{3}}\right) - \sqrt[3]{{x}^{-12}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3 + x, 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))