Asymptote C

Average Error: 29.2 → 0.8

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + e^{\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 0.0)
     (+ (+ (/ -3.0 x) (/ (/ -1.0 x) x)) (/ -3.0 (pow x 3.0)))
     (+ t_0 (exp (- (log1p x) (log1p (- x))))))))

C

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

↓

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 0.0) {
		tmp = ((-3.0 / x) + ((-1.0 / x) / x)) + (-3.0 / pow(x, 3.0));
	} else {
		tmp = t_0 + exp((log1p(x) - log1p(-x)));
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 0.0) {
		tmp = ((-3.0 / x) + ((-1.0 / x) / x)) + (-3.0 / Math.pow(x, 3.0));
	} else {
		tmp = t_0 + Math.exp((Math.log1p(x) - Math.log1p(-x)));
	}
	return tmp;
}

Python

def code(x):
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))

↓

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 + ((-1.0 - x) / (x + -1.0))) <= 0.0:
		tmp = ((-3.0 / x) + ((-1.0 / x) / x)) + (-3.0 / math.pow(x, 3.0))
	else:
		tmp = t_0 + math.exp((math.log1p(x) - math.log1p(-x)))
	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(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 0.0)
		tmp = Float64(Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x)) + Float64(-3.0 / (x ^ 3.0)));
	else
		tmp = Float64(t_0 + exp(Float64(log1p(x) - log1p(Float64(-x)))));
	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[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Exp[N[(N[Log[1 + x], $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 0.0

   1. Initial program 59.4

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

   2. Simplified 59.4

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

   3. Applied egg-rr 59.4

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

   4. Taylor expanded in x around inf 0.6

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

   5. Simplified 0.3

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 0.5

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

   2. Simplified 0.5

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

   3. Applied egg-rr 1.3

      \[\leadsto \frac{x}{x + 1} + \color{blue}{e^{\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + e^{\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))