Asymptote B

Average Error: 0.0 → 0.0

Time: 5.5s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 16.2

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

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.4
Cost	713

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

Alternative 2
Error	0.0
Cost	704

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

Alternative 3
Error	0.5
Cost	584

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

Alternative 4
Error	0.5
Cost	328

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

Alternative 5
Error	32.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023045 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))