Asymptote C

Average Error: 29.6 → 0.1

Time: 3.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;x \leq -12339.105848947931:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{{x}^{3}}\right)\\ \mathbf{elif}\;x \leq 16767.2313256428:\\ \;\;\;\;\frac{\frac{-1 - x}{x} + t_0}{t_0 \cdot \frac{x + 1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, {x}^{-3}, \frac{-3}{x} + \frac{2}{x \cdot x}\right)}{\frac{1}{\frac{x}{x + -1}}}\\ \end{array} \]

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))))
   (if (<= x -12339.105848947931)
     (+ (/ -3.0 x) (+ (/ -1.0 (* x x)) (/ -3.0 (pow x 3.0))))
     (if (<= x 16767.2313256428)
       (/ (+ (/ (- -1.0 x) x) t_0) (* t_0 (/ (+ x 1.0) x)))
       (/
        (fma -2.0 (pow x -3.0) (+ (/ -3.0 x) (/ 2.0 (* x x))))
        (/ 1.0 (/ x (+ x -1.0))))))))

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 tmp;
	if (x <= -12339.105848947931) {
		tmp = (-3.0 / x) + ((-1.0 / (x * x)) + (-3.0 / pow(x, 3.0)));
	} else if (x <= 16767.2313256428) {
		tmp = (((-1.0 - x) / x) + t_0) / (t_0 * ((x + 1.0) / x));
	} else {
		tmp = fma(-2.0, pow(x, -3.0), ((-3.0 / x) + (2.0 / (x * x)))) / (1.0 / (x / (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)
	t_0 = Float64(Float64(x + -1.0) / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -12339.105848947931)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / Float64(x * x)) + Float64(-3.0 / (x ^ 3.0))));
	elseif (x <= 16767.2313256428)
		tmp = Float64(Float64(Float64(Float64(-1.0 - x) / x) + t_0) / Float64(t_0 * Float64(Float64(x + 1.0) / x)));
	else
		tmp = Float64(fma(-2.0, (x ^ -3.0), Float64(Float64(-3.0 / x) + Float64(2.0 / Float64(x * x)))) / Float64(1.0 / Float64(x / 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_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -12339.105848947931], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 16767.2313256428], N[(N[(N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(t$95$0 * N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Power[x, -3.0], $MachinePrecision] + N[(N[(-3.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -12339.105848947931

   1. Initial program 59.2

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

   2. Taylor expanded in x around inf 0.4

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

   3. Simplified 0.0

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

   if -12339.105848947931 < x < 16767.2313256427988

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}} \]

   3. Applied egg-rr 0.1

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

   if 16767.2313256427988 < x

   1. Initial program 59.5

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

   2. Applied egg-rr 59.5

      \[\leadsto \color{blue}{\frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}} \]

   3. Applied egg-rr 59.5

      \[\leadsto \frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x} \cdot 1}{\color{blue}{\frac{1}{\frac{x}{x + -1}}}} \]

   4. Taylor expanded in x around inf 0.3

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

   5. Simplified 0.0

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

   6. Applied egg-rr 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12339.105848947931:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{{x}^{3}}\right)\\ \mathbf{elif}\;x \leq 16767.2313256428:\\ \;\;\;\;\frac{\frac{-1 - x}{x} + \frac{x + -1}{x + 1}}{\frac{x + -1}{x + 1} \cdot \frac{x + 1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, {x}^{-3}, \frac{-3}{x} + \frac{2}{x \cdot x}\right)}{\frac{1}{\frac{x}{x + -1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))