Asymptote C

Average Error: 29.0 → 0.2

Time: 1.4min

Precision: binary64

Cost: 832

Math

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

↓

\[\frac{\frac{3 \cdot x + 1}{-1 - x}}{-1 + x} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ (/ (+ (* 3.0 x) 1.0) (- -1.0 x)) (+ -1.0 x)))

C

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

↓

double code(double x) {
	return (((3.0 * x) + 1.0) / (-1.0 - x)) / (-1.0 + x);
}

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
    code = (((3.0d0 * x) + 1.0d0) / ((-1.0d0) - x)) / ((-1.0d0) + x)
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) {
	return (((3.0 * x) + 1.0) / (-1.0 - x)) / (-1.0 + x);
}

Python

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

↓

def code(x):
	return (((3.0 * x) + 1.0) / (-1.0 - x)) / (-1.0 + x)

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)
	return Float64(Float64(Float64(Float64(3.0 * x) + 1.0) / Float64(-1.0 - x)) / Float64(-1.0 + x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (((3.0 * x) + 1.0) / (-1.0 - x)) / (-1.0 + x);
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_] := N[(N[(N[(N[(3.0 * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.0

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

2. Applied egg-rr 30.0

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

3. Taylor expanded in x around 0 0.2

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

Alternatives

Alternative 1
Error	0.1
Cost	1096

\[\begin{array}{l} t_0 := \frac{\frac{2}{x} + -3}{-1 + x}\\ \mathbf{if}\;x \leq -330000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 255000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	840

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

Alternative 3
Error	0.7
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{2}{x} + -3}{-1 + x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;x - \left(-1 - x\right) \cdot \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	712

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

Alternative 5
Error	1.1
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 6
Error	1.4
Cost	456

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

Alternative 7
Error	31.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))