Asymptote C

Average Error: 29.1 → 0.1

Time: 25.7s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\frac{1 - \frac{x \cdot \left(\left(x + -2\right) - x\right)}{1 + x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \frac{-3}{x}}{x}}{x} - \frac{3}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e+16)
   (/ -3.0 x)
   (if (<= x 500.0)
     (/ (- 1.0 (/ (* x (- (+ x -2.0) x)) (+ 1.0 x))) (- 1.0 x))
     (- (/ (/ (+ -1.0 (/ -3.0 x)) x) x) (/ 3.0 x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -5e+16) {
		tmp = -3.0 / x;
	} else if (x <= 500.0) {
		tmp = (1.0 - ((x * ((x + -2.0) - x)) / (1.0 + x))) / (1.0 - x);
	} else {
		tmp = (((-1.0 + (-3.0 / x)) / x) / x) - (3.0 / x);
	}
	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) :: tmp
    if (x <= (-5d+16)) then
        tmp = (-3.0d0) / x
    else if (x <= 500.0d0) then
        tmp = (1.0d0 - ((x * ((x + (-2.0d0)) - x)) / (1.0d0 + x))) / (1.0d0 - x)
    else
        tmp = ((((-1.0d0) + ((-3.0d0) / x)) / x) / x) - (3.0d0 / x)
    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 tmp;
	if (x <= -5e+16) {
		tmp = -3.0 / x;
	} else if (x <= 500.0) {
		tmp = (1.0 - ((x * ((x + -2.0) - x)) / (1.0 + x))) / (1.0 - x);
	} else {
		tmp = (((-1.0 + (-3.0 / x)) / x) / x) - (3.0 / x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -5e+16:
		tmp = -3.0 / x
	elif x <= 500.0:
		tmp = (1.0 - ((x * ((x + -2.0) - x)) / (1.0 + x))) / (1.0 - x)
	else:
		tmp = (((-1.0 + (-3.0 / x)) / x) / x) - (3.0 / 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)
	tmp = 0.0
	if (x <= -5e+16)
		tmp = Float64(-3.0 / x);
	elseif (x <= 500.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * Float64(Float64(x + -2.0) - x)) / Float64(1.0 + x))) / Float64(1.0 - x));
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 + Float64(-3.0 / x)) / x) / x) - Float64(3.0 / x));
	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)
	tmp = 0.0;
	if (x <= -5e+16)
		tmp = -3.0 / x;
	elseif (x <= 500.0)
		tmp = (1.0 - ((x * ((x + -2.0) - x)) / (1.0 + x))) / (1.0 - x);
	else
		tmp = (((-1.0 + (-3.0 / x)) / x) / x) - (3.0 / x);
	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_] := If[LessEqual[x, -5e+16], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 500.0], N[(N[(1.0 - N[(N[(x * N[(N[(x + -2.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] - N[(3.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e16

   1. Initial program 60.5

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

   2. Taylor expanded in x around inf 0

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -5e16 < x < 500

   1. Initial program 0.7

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

   2. Applied egg-rr 0.7

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

   3. Applied egg-rr 0.7

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

   4. Simplified 0.7

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

   5. Applied egg-rr 0.1

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

   6. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, 0, \frac{1 - \frac{x \cdot \left(\left(x + -2\right) - x\right)}{1 + x}}{1 - x}\right)} \]

   7. Simplified 0.1

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

   if 500 < x

   1. Initial program 58.9

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

   2. Taylor expanded in x around inf 0.4

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

   3. Simplified 0.1

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

   4. Applied egg-rr 0.1

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

   5. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{-1}{x} + \frac{-3}{x \cdot x}}{x}} - \frac{3}{x} \]
      Proof

   6. Applied egg-rr 0.1

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

Alternatives

Alternative 1
Error	0.0
Cost	1352

\[\begin{array}{l} t_0 := \frac{-1 + \frac{-3}{x}}{x}\\ \mathbf{if}\;x \leq -14500:\\ \;\;\;\;\frac{t_0 + -3}{x}\\ \mathbf{elif}\;x \leq 10500:\\ \;\;\;\;\frac{1}{1 - x} - \left(\frac{x}{-1 + x} + \frac{x}{-1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x} - \frac{3}{x}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	1096

\[\begin{array}{l} t_0 := \frac{\frac{-1 + \frac{-3}{x}}{x} + -3}{x}\\ \mathbf{if}\;x \leq -13200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 12000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.0
Cost	1096

\[\begin{array}{l} t_0 := \frac{-1 + \frac{-3}{x}}{x}\\ \mathbf{if}\;x \leq -13200:\\ \;\;\;\;\frac{t_0 + -3}{x}\\ \mathbf{elif}\;x \leq 12000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x} - \frac{3}{x}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	968

\[\begin{array}{l} t_0 := \frac{\frac{-1 + \frac{-3}{x}}{x} + -3}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\frac{1}{1 - x} - -2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.6
Cost	840

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

Alternative 6
Error	0.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-1}{x} + -3}{x}\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;\frac{1}{1 - x} - -2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} - \frac{3}{x}\\ \end{array} \]

Alternative 7
Error	0.6
Cost	712

\[\begin{array}{l} t_0 := \frac{\frac{-1}{x} + -3}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;3 \cdot x + 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
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 9
Error	1.3
Cost	456

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

Alternative 10
Error	31.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))