Asymptote A

Average Error: 14.4 → 0.1

Time: 6.4s

Precision: binary64

Cost: 576

Math

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

↓

\[\frac{\frac{2}{-1 - x}}{-1 + x} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (/ 2.0 (- -1.0 x)) (+ -1.0 x)))

C

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

↓

double code(double x) {
	return (2.0 / (-1.0 - x)) / (-1.0 + x);
}

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
    code = (2.0d0 / ((-1.0d0) - x)) / ((-1.0d0) + x)
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) {
	return (2.0 / (-1.0 - x)) / (-1.0 + x);
}

Python

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

↓

def code(x):
	return (2.0 / (-1.0 - x)) / (-1.0 + x)

Julia

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

↓

function code(x)
	return Float64(Float64(2.0 / Float64(-1.0 - x)) / Float64(-1.0 + x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (2.0 / (-1.0 - x)) / (-1.0 + x);
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_] := N[(N[(2.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.4

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

2. Applied egg-rr 13.9

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

3. Applied egg-rr 13.9

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

4. Simplified 0.1

   \[\leadsto \frac{\color{blue}{\frac{2}{-1 - x}}}{x + -1} \]
   Proof
   (/.f64 2 (-.f64 -1 x)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (-.f64 -1 x)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 -2)) 1) (-.f64 -1 x)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (neg.f64 (Rewrite<= metadata-eval (-.f64 -2 0))) 1) (-.f64 -1 x)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (neg.f64 (-.f64 -2 (Rewrite<= +-inverses_binary64 (-.f64 x x)))) 1) (-.f64 -1 x)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (neg.f64 (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 -2 x) x))) 1) (-.f64 -1 x)): 8 points increase in error, 101 points decrease in error
   (/.f64 (*.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (-.f64 -2 x)))) 1) (-.f64 -1 x)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (neg.f64 (+.f64 x (-.f64 -2 x))) 1) (Rewrite=> sub-neg_binary64 (+.f64 -1 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (neg.f64 (+.f64 x (-.f64 -2 x))) 1) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 x) -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (neg.f64 (+.f64 x (-.f64 -2 x))) 1) (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x)) -1)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (neg.f64 (+.f64 x (-.f64 -2 x))) 1) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 x -1)) -1)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (neg.f64 (+.f64 x (-.f64 -2 x))) 1) (Rewrite<= fma-udef_binary64 (fma.f64 x -1 -1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 (+.f64 x (-.f64 -2 x))) (/.f64 1 (fma.f64 x -1 -1)))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.1

   \[\leadsto \frac{\frac{2}{-1 - x}}{-1 + x} \]

Alternatives

Alternative 1
Error	1.2
Cost	584

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

Alternative 2
Error	0.8
Cost	584

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

Alternative 3
Error	0.4
Cost	576

\[\frac{-2}{\left(-1 - x\right) \cdot \left(1 - x\right)} \]

Alternative 4
Error	0.4
Cost	448

\[\frac{-2}{-1 + x \cdot x} \]

Alternative 5
Error	57.2
Cost	64

\[1 \]

Alternative 6
Error	31.8
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022321 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))