Asymptote A

Average Error: 14.6 → 0.4

Time: 3.2s

Precision: binary64

Cost: 7428

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 (+ x -1.0)))))
   (if (<= t_0 0.0) (* -2.0 (pow x -2.0)) t_0)))

C

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

↓

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -2.0 * pow(x, -2.0);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / (x + (-1.0d0)))
    if (t_0 <= 0.0d0) then
        tmp = (-2.0d0) * (x ** (-2.0d0))
    else
        tmp = t_0
    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 t_0 = (1.0 / (1.0 + x)) + (-1.0 / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -2.0 * Math.pow(x, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 29.0

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

   2. Taylor expanded in x around inf 1.5

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

   3. Simplified 1.5

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
      Proof
      (/.f64 -2 (*.f64 x x)): 0 points increase in error, 0 points decrease in error
      (/.f64 -2 (Rewrite<= unpow2_binary64 (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.7

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

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

   1. Initial program 0.0

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	1476

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

Alternative 2
Error	2.0
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x}\\ \mathbf{if}\;x \leq -1000338689465.2806:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.609788294294903 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{1 + x} + \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.1
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x}\\ \mathbf{if}\;x \leq -1000338689465.2806:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.609788294294903 \cdot 10^{-22}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	31.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))