Asymptote A

Average Error: 14.9 → 0.1

Time: 4.9s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = ((1.0d0 / (1.0d0 - x)) / ((-1.0d0) - x)) * (-2.0d0)
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 ((1.0 / (1.0 - x)) / (-1.0 - x)) * -2.0;
}

Python

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

↓

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

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(Float64(1.0 / Float64(1.0 - x)) / Float64(-1.0 - x)) * -2.0)
end

MATLAB

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

↓

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

Derivation

1. Initial program 14.9

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

2. Applied egg-rr 14.9

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

3. Simplified 0.4

   \[\leadsto \color{blue}{\frac{-2}{\left(-1 - x\right) \cdot \left(1 - x\right)}} \]
   Proof
   (/.f64 -2 (*.f64 (-.f64 -1 x) (-.f64 1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 -1 (neg.f64 x))) (-.f64 1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (*.f64 (+.f64 -1 (neg.f64 x)) (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 x))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x))))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (*.f64 (+.f64 -1 (neg.f64 x)) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 -2 (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 x)) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= metadata-eval (-.f64 -2 0)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 -2 (Rewrite<= +-inverses_binary64 (-.f64 x x))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 -2 x) x)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))): 54 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (-.f64 (Rewrite<= metadata-eval (-.f64 -1 1)) x) x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= associate--r+_binary64 (-.f64 -1 (+.f64 1 x))) x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))): 2 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (-.f64 -1 (+.f64 1 x)))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 x -1) (+.f64 1 x))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 1 points decrease in error
   (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (+.f64 x -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (+.f64 1 x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 5 points increase in error, 1 points decrease in error
   (-.f64 (/.f64 (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 x))) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (+.f64 1 x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (+.f64 (neg.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x))) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (+.f64 1 x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 -1 x))) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (+.f64 1 x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (+.f64 (*.f64 -1 (*.f64 -1 x)) (Rewrite<= metadata-eval (*.f64 -1 1))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (+.f64 1 x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 -1 (+.f64 (*.f64 -1 x) 1))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (+.f64 1 x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (*.f64 -1 (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 x)) 1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (+.f64 1 x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (*.f64 -1 (+.f64 (neg.f64 x) 1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (+.f64 (Rewrite<= metadata-eval (-.f64 0 -1)) x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (*.f64 -1 (+.f64 (neg.f64 x) 1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 -1 x))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (*.f64 -1 (+.f64 (neg.f64 x) 1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (-.f64 0 (Rewrite<= unsub-neg_binary64 (+.f64 -1 (neg.f64 x)))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (*.f64 -1 (+.f64 (neg.f64 x) 1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (+.f64 -1 (neg.f64 x)))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (*.f64 -1 (+.f64 (neg.f64 x) 1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (+.f64 -1 (neg.f64 x)))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (*.f64 -1 (+.f64 (neg.f64 x) 1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 -1 (neg.f64 x)) -1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 (*.f64 -1 (+.f64 (neg.f64 x) 1)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 (+.f64 -1 (neg.f64 x))) (/.f64 (+.f64 (neg.f64 x) 1) (+.f64 (neg.f64 x) 1)))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 80 points increase in error, 5 points decrease in error
   (+.f64 (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 1 -1)) (+.f64 -1 (neg.f64 x))) (/.f64 (+.f64 (neg.f64 x) 1) (+.f64 (neg.f64 x) 1))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 -1 (+.f64 -1 (neg.f64 x))))) (/.f64 (+.f64 (neg.f64 x) 1) (+.f64 (neg.f64 x) 1))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (/.f64 1 (Rewrite<= neg-mul-1_binary64 (neg.f64 (+.f64 -1 (neg.f64 x))))) (/.f64 (+.f64 (neg.f64 x) 1) (+.f64 (neg.f64 x) 1))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (/.f64 1 (Rewrite=> neg-sub0_binary64 (-.f64 0 (+.f64 -1 (neg.f64 x))))) (/.f64 (+.f64 (neg.f64 x) 1) (+.f64 (neg.f64 x) 1))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (/.f64 1 (-.f64 0 (Rewrite=> unsub-neg_binary64 (-.f64 -1 x)))) (/.f64 (+.f64 (neg.f64 x) 1) (+.f64 (neg.f64 x) 1))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (/.f64 1 (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 0 -1) x))) (/.f64 (+.f64 (neg.f64 x) 1) (+.f64 (neg.f64 x) 1))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (/.f64 1 (+.f64 (Rewrite=> metadata-eval 1) x)) (/.f64 (+.f64 (neg.f64 x) 1) (+.f64 (neg.f64 x) 1))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (/.f64 1 (+.f64 1 x)) (Rewrite=> *-inverses_binary64 1)) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 1 (+.f64 1 x))) (neg.f64 (/.f64 (*.f64 (+.f64 -1 (neg.f64 x)) -1) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 -1 (+.f64 -1 (neg.f64 x)))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (+.f64 -1 (neg.f64 x)))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (+.f64 -1 (neg.f64 x)))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (-.f64 0 (Rewrite=> unsub-neg_binary64 (-.f64 -1 x))) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 0 -1) x)) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (+.f64 (Rewrite=> metadata-eval 1) x) (*.f64 (+.f64 -1 (neg.f64 x)) (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (+.f64 1 x) (+.f64 -1 (neg.f64 x))) (+.f64 (neg.f64 x) 1))))): 4 points increase in error, 79 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 (/.f64 (+.f64 1 x) (+.f64 -1 (neg.f64 x)))) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (/.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (+.f64 1 x)) (+.f64 -1 (neg.f64 x)))) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (/.f64 (/.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 1) (neg.f64 x))) (+.f64 -1 (neg.f64 x))) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (/.f64 (/.f64 (+.f64 (Rewrite=> metadata-eval -1) (neg.f64 x)) (+.f64 -1 (neg.f64 x))) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (/.f64 (Rewrite=> *-inverses_binary64 1) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (+.f64 (neg.f64 x) 1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 -1 (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 1 -1)) (+.f64 (neg.f64 x) 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 -1 (+.f64 (neg.f64 x) 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (Rewrite<= metadata-eval (pow.f64 1 3)) (*.f64 -1 (+.f64 (neg.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (pow.f64 1 3) (*.f64 -1 (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x)) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (pow.f64 1 3) (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 -1 (*.f64 -1 x)) (*.f64 -1 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (pow.f64 1 3) (+.f64 (*.f64 -1 (*.f64 -1 x)) (Rewrite=> metadata-eval -1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (pow.f64 1 3) (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 -1 x))) -1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (pow.f64 1 3) (+.f64 (neg.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 x))) -1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (pow.f64 1 3) (+.f64 (Rewrite=> remove-double-neg_binary64 x) -1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (/.f64 (pow.f64 1 3) (Rewrite<= rem-cube-cbrt_binary64 (pow.f64 (cbrt.f64 (+.f64 x -1)) 3))))): 93 points increase in error, 15 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (Rewrite<= cube-div_binary64 (pow.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) 3)))): 46 points increase in error, 31 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (Rewrite=> cube-mult_binary64 (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (/.f64 1 (cbrt.f64 (+.f64 x -1)))))))): 18 points increase in error, 9 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (*.f64 (Rewrite<= unpow-1_binary64 (pow.f64 (cbrt.f64 (+.f64 x -1)) -1)) (/.f64 1 (cbrt.f64 (+.f64 x -1))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (*.f64 (pow.f64 (cbrt.f64 (+.f64 x -1)) -1) (Rewrite<= unpow-1_binary64 (pow.f64 (cbrt.f64 (+.f64 x -1)) -1)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (Rewrite=> pow-sqr_binary64 (pow.f64 (cbrt.f64 (+.f64 x -1)) (*.f64 2 -1)))))): 20 points increase in error, 43 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (neg.f64 (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (pow.f64 (cbrt.f64 (+.f64 x -1)) (Rewrite=> metadata-eval -2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> unsub-neg_binary64 (-.f64 (/.f64 1 (+.f64 1 x)) (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (pow.f64 (cbrt.f64 (+.f64 x -1)) -2)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (pow.f64 (cbrt.f64 (+.f64 x -1)) (Rewrite<= metadata-eval (*.f64 2 -1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (cbrt.f64 (+.f64 x -1)) -1) (pow.f64 (cbrt.f64 (+.f64 x -1)) -1))))): 43 points increase in error, 20 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (*.f64 (Rewrite=> unpow-1_binary64 (/.f64 1 (cbrt.f64 (+.f64 x -1)))) (pow.f64 (cbrt.f64 (+.f64 x -1)) -1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (*.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) (Rewrite=> unpow-1_binary64 (/.f64 1 (cbrt.f64 (+.f64 x -1))))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (Rewrite<= cube-mult_binary64 (pow.f64 (/.f64 1 (cbrt.f64 (+.f64 x -1))) 3))): 9 points increase in error, 18 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (Rewrite=> cube-div_binary64 (/.f64 (pow.f64 1 3) (pow.f64 (cbrt.f64 (+.f64 x -1)) 3)))): 31 points increase in error, 46 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (/.f64 (Rewrite=> metadata-eval 1) (pow.f64 (cbrt.f64 (+.f64 x -1)) 3))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (/.f64 1 (Rewrite=> rem-cube-cbrt_binary64 (+.f64 x -1)))): 15 points increase in error, 93 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 1 (+.f64 x -1)) 1))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (*.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (+.f64 x -1)) 1)): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (+.f64 1 x)) (*.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 -1 (+.f64 x -1)))) 1)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub_binary64 (+.f64 (/.f64 1 (+.f64 1 x)) (*.f64 (/.f64 -1 (+.f64 x -1)) 1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 1 x)) (Rewrite=> *-rgt-identity_binary64 (/.f64 -1 (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\frac{1}{1 - x}}{-1 - x} \cdot -2} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.6
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 2
Error	0.4
Cost	448

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

Alternative 3
Error	31.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022318 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))