Asymptote B

Average Error: 0.0 → 0.0

Time: 2.8s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+ (/ 1.0 (+ x -1.0)) (+ 1.0 (+ (/ x (+ 1.0 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)) + (1.0 + ((x / (1.0 + 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))) + (1.0d0 + ((x / (1.0d0 + 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)) + (1.0 + ((x / (1.0 + 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)) + (1.0 + ((x / (1.0 + 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(1.0 / Float64(x + -1.0)) + Float64(1.0 + Float64(Float64(x / Float64(1.0 + 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)) + (1.0 + ((x / (1.0 + 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[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Simplified 0.0

   \[\leadsto \frac{1}{x - 1} + \color{blue}{\left(1 + \left(\frac{x}{x + 1} - 1\right)\right)} \]
   Proof
   (+.f64 1 (-.f64 (/.f64 x (+.f64 x 1)) 1)): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (-.f64 (/.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 x))) 1)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1 (/.f64 x (+.f64 1 x))) 1)): 3 points increase in error, 3 points decrease in error

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;1 + \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{1 + x} + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	0.7
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;t_0 + \frac{1}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_0 + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	0.0
Cost	704

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

Alternative 4
Error	0.7
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 5
Error	0.7
Cost	584

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

Alternative 6
Error	0.7
Cost	328

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

Alternative 7
Error	31.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022331 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))