Asymptote C

Average Error: 29.3 → 0.4

Time: 3.8s

Precision: binary64

Cost: 7876

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))))
   (if (<= t_0 0.0) (- (+ (/ 1.0 (pow x 2.0)) (/ 3.0 x))) 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 / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -((1.0 / pow(x, 2.0)) + (3.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x - 1.0d0))
    if (t_0 <= 0.0d0) then
        tmp = -((1.0d0 / (x ** 2.0d0)) + (3.0d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

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)) - ((x + 1.0) / (x - 1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -((1.0 / Math.pow(x, 2.0)) + (3.0 / x));
	} else {
		tmp = t_0;
	}
	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)) - ((x + 1.0) / (x - 1.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = -((1.0 / math.pow(x, 2.0)) + (3.0 / x))
	else:
		tmp = 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 / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-Float64(Float64(1.0 / (x ^ 2.0)) + Float64(3.0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
end

↓

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -((1.0 / (x ^ 2.0)) + (3.0 / x));
	else
		tmp = t_0;
	end
	tmp_2 = 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 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], (-N[(N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(3.0 / x), $MachinePrecision]), $MachinePrecision]), t$95$0]]

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 59.6

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

   2. Simplified 59.6

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

      [Start] 59.6	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      rational.json-simplify-50 [=>] 59.6	\[ \frac{x}{x + 1} - \color{blue}{\frac{-\left(x + 1\right)}{1 - x}} \]
      rational.json-simplify-12 [=>] 59.6	\[ \frac{x}{x + 1} - \frac{\color{blue}{0 - \left(x + 1\right)}}{1 - x} \]
      rational.json-simplify-1 [=>] 59.6	\[ \frac{x}{x + 1} - \frac{0 - \color{blue}{\left(1 + x\right)}}{1 - x} \]
      rational.json-simplify-17 [=>] 59.6	\[ \frac{x}{x + 1} - \frac{0 - \color{blue}{\left(x - -1\right)}}{1 - x} \]
      rational.json-simplify-45 [=>] 59.6	\[ \frac{x}{x + 1} - \frac{\color{blue}{-1 - \left(x - 0\right)}}{1 - x} \]
      rational.json-simplify-5 [=>] 59.6	\[ \frac{x}{x + 1} - \frac{-1 - \color{blue}{x}}{1 - x} \]

   3. Taylor expanded in x around inf 0.6

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

   4. Taylor expanded in x around 0 0.3

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

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

   1. Initial program 0.5

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0:\\ \;\;\;\;-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -210000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 118000000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	1032

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;3 \cdot x + 1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{1 + 0.3333333333333333 \cdot \frac{1}{x}}{x}}{0.3333333333333333}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;3 \cdot x + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 4
Error	1.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 5
Error	31.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023064 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))