Asymptote A

Average Error: 22.39% → 1.13%

Time: 4.1s

Precision: binary64

Cost: 585

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;2 + x \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ (/ -2.0 x) x) (+ 2.0 (* x x))))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-2.0 / x) / x;
	} else {
		tmp = 2.0 + (x * x);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = ((-2.0d0) / x) / x
    else
        tmp = 2.0d0 + (x * x)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-2.0 / x) / x;
	} else {
		tmp = 2.0 + (x * x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-2.0 / x) / x
	else:
		tmp = 2.0 + (x * x)
	return tmp

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(-2.0 / x) / x);
	else
		tmp = Float64(2.0 + Float64(x * x));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = (-2.0 / x) / x;
	else
		tmp = 2.0 + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 44.52

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

   2. Taylor expanded in x around inf 2.05

      \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]

   3. Simplified 2.05

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
      Proof

      [Start] 2.05	\[ \frac{-2}{{x}^{2}} \]
      unpow2 [=>] 2.05	\[ \frac{-2}{\color{blue}{x \cdot x}} \]

   4. Applied egg-rr 1.41

      \[\leadsto \color{blue}{\frac{-2}{x} \cdot \frac{1}{x}} \]

   5. Applied egg-rr 1.23

      \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]

   if -1 < x < 1

   1. Initial program 0.01

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

   2. Taylor expanded in x around 0 1.02

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

   3. Simplified 1.02

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

      [Start] 1.02	\[ \frac{1}{x + 1} - \left(-1 \cdot x - 1\right) \]
      sub-neg [=>] 1.02	\[ \frac{1}{x + 1} - \color{blue}{\left(-1 \cdot x + \left(-1\right)\right)} \]
      metadata-eval [=>] 1.02	\[ \frac{1}{x + 1} - \left(-1 \cdot x + \color{blue}{-1}\right) \]
      +-commutative [<=] 1.02	\[ \frac{1}{x + 1} - \color{blue}{\left(-1 + -1 \cdot x\right)} \]
      mul-1-neg [=>] 1.02	\[ \frac{1}{x + 1} - \left(-1 + \color{blue}{\left(-x\right)}\right) \]
      sub-neg [<=] 1.02	\[ \frac{1}{x + 1} - \color{blue}{\left(-1 - x\right)} \]

   4. Taylor expanded in x around 0 1.02

      \[\leadsto \color{blue}{2 + {x}^{2}} \]

   5. Simplified 1.02

      \[\leadsto \color{blue}{2 + x \cdot x} \]
      Proof

      [Start] 1.02	\[ 2 + {x}^{2} \]
      unpow2 [=>] 1.02	\[ 2 + \color{blue}{x \cdot x} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.13

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

Alternatives

Alternative 1
Error	1.54%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 + x \cdot x\\ \end{array} \]

Alternative 2
Error	0.52%
Cost	576

\[\frac{2}{\left(-1 - x\right) \cdot \left(-1 + x\right)} \]

Alternative 3
Error	49.46%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))