Asymptote C

Average Error: 29.2 → 0.0

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;x \leq -2900:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{{x}^{3}} + \left(\frac{-3}{x} + \frac{-1}{{x}^{4}}\right)\right)\\ \mathbf{elif}\;x \leq 3200:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t_0, \mathsf{fma}\left(-1, x, -1\right)\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \left(\frac{2}{x} + \mathsf{fma}\left(-2, {x}^{-2}, 2 \cdot {x}^{-3}\right)\right)}{\left(x + 1\right) \cdot t_0}\\ \end{array} \]

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 -2900.0)
     (+
      (/ -1.0 (* x x))
      (+ (/ -3.0 (pow x 3.0)) (+ (/ -3.0 x) (/ -1.0 (pow x 4.0)))))
     (if (<= x 3200.0)
       (/ (fma x t_0 (fma -1.0 x -1.0)) (+ x -1.0))
       (/
        (+ -3.0 (+ (/ 2.0 x) (fma -2.0 (pow x -2.0) (* 2.0 (pow x -3.0)))))
        (* (+ x 1.0) t_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 <= -2900.0) {
		tmp = (-1.0 / (x * x)) + ((-3.0 / pow(x, 3.0)) + ((-3.0 / x) + (-1.0 / pow(x, 4.0))));
	} else if (x <= 3200.0) {
		tmp = fma(x, t_0, fma(-1.0, x, -1.0)) / (x + -1.0);
	} else {
		tmp = (-3.0 + ((2.0 / x) + fma(-2.0, pow(x, -2.0), (2.0 * pow(x, -3.0))))) / ((x + 1.0) * t_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 <= -2900.0)
		tmp = Float64(Float64(-1.0 / Float64(x * x)) + Float64(Float64(-3.0 / (x ^ 3.0)) + Float64(Float64(-3.0 / x) + Float64(-1.0 / (x ^ 4.0)))));
	elseif (x <= 3200.0)
		tmp = Float64(fma(x, t_0, fma(-1.0, x, -1.0)) / Float64(x + -1.0));
	else
		tmp = Float64(Float64(-3.0 + Float64(Float64(2.0 / x) + fma(-2.0, (x ^ -2.0), Float64(2.0 * (x ^ -3.0))))) / Float64(Float64(x + 1.0) * t_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, -2900.0], N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 / x), $MachinePrecision] + N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3200.0], N[(N[(x * t$95$0 + N[(-1.0 * x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 + N[(N[(2.0 / x), $MachinePrecision] + N[(-2.0 * N[Power[x, -2.0], $MachinePrecision] + N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -2900

   1. Initial program 58.8

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

   2. Taylor expanded in x around inf 0.3

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

   3. Simplified 0.0

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

   if -2900 < x < 3200

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   3. Taylor expanded in x around 0 0.1

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

   4. Simplified 0.1

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

   if 3200 < x

   1. Initial program 59.4

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

   2. Applied egg-rr 59.3

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

   3. Taylor expanded in x around inf 0.0

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

   4. Simplified 0.0

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

   5. Applied egg-rr 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2900:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{{x}^{3}} + \left(\frac{-3}{x} + \frac{-1}{{x}^{4}}\right)\right)\\ \mathbf{elif}\;x \leq 3200:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, -1\right)\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \left(\frac{2}{x} + \mathsf{fma}\left(-2, {x}^{-2}, 2 \cdot {x}^{-3}\right)\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))