Asymptote C

Average Error: 29.0 → 0.2

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + 1}{x + -1}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;t_1 + \frac{-1 - x}{x + -1} \leq 0.0004:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-3}{{x}^{3}} + \left(\frac{\frac{-1}{x}}{x} + \frac{-1}{{x}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({t_1}^{3} - {t_0}^{3}\right)}}{\mathsf{fma}\left(t_0, t_1 + t_0, {t_1}^{2}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ x 1.0) (+ x -1.0))) (t_1 (/ x (+ x 1.0))))
   (if (<= (+ t_1 (/ (- -1.0 x) (+ x -1.0))) 0.0004)
     (+
      (/ -3.0 x)
      (+ (/ -3.0 (pow x 3.0)) (+ (/ (/ -1.0 x) x) (/ -1.0 (pow x 4.0)))))
     (/
      (exp (log (- (pow t_1 3.0) (pow t_0 3.0))))
      (fma t_0 (+ t_1 t_0) (pow t_1 2.0))))))

C

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

↓

double code(double x) {
	double t_0 = (x + 1.0) / (x + -1.0);
	double t_1 = x / (x + 1.0);
	double tmp;
	if ((t_1 + ((-1.0 - x) / (x + -1.0))) <= 0.0004) {
		tmp = (-3.0 / x) + ((-3.0 / pow(x, 3.0)) + (((-1.0 / x) / x) + (-1.0 / pow(x, 4.0))));
	} else {
		tmp = exp(log((pow(t_1, 3.0) - pow(t_0, 3.0)))) / fma(t_0, (t_1 + t_0), pow(t_1, 2.0));
	}
	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 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 0.0004)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-3.0 / (x ^ 3.0)) + Float64(Float64(Float64(-1.0 / x) / x) + Float64(-1.0 / (x ^ 4.0)))));
	else
		tmp = Float64(exp(log(Float64((t_1 ^ 3.0) - (t_0 ^ 3.0)))) / fma(t_0, Float64(t_1 + t_0), (t_1 ^ 2.0)));
	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[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0004], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[Log[N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[(t$95$1 + t$95$0), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $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))) < 4.00000000000000019e-4

   1. Initial program 58.7

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

   2. Taylor expanded in x around inf 0.6

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

   3. Simplified 0.3

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

   if 4.00000000000000019e-4 < (-.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

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

   3. Applied egg-rr 0.0

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0004:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-3}{{x}^{3}} + \left(\frac{\frac{-1}{x}}{x} + \frac{-1}{{x}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x + -1}\right)}^{3}\right)}}{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \frac{x}{x + 1} + \frac{x + 1}{x + -1}, {\left(\frac{x}{x + 1}\right)}^{2}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022178 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))