Average Error: 14.5 → 0.1
Time: 7.4s
Precision: binary64
Cost: 576
\[\frac{1}{x + 1} - \frac{1}{x - 1} \]
↓
\[\frac{\frac{2}{1 - x}}{1 + x} \]
(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)))
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);
}

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

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);
}

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)

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

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

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]

\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{\frac{2}{1 - x}}{1 + x}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.5

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

  2. Applied egg-rr13.9

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

  3. Applied egg-rr13.9

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

  4. Simplified0.1

    \[\leadsto \color{blue}{\frac{-2}{x + 1} \cdot \frac{-1}{1 - x}} \]
    Proof
    (*.f64 (/.f64 -2 (+.f64 x 1)) (/.f64 -1 (-.f64 1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (Rewrite<= metadata-eval (-.f64 -2 0)) (+.f64 x 1)) (/.f64 -1 (-.f64 1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (-.f64 -2 (Rewrite<= +-inverses_binary64 (-.f64 x x))) (+.f64 x 1)) (/.f64 -1 (-.f64 1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 -2 x) x)) (+.f64 x 1)) (/.f64 -1 (-.f64 1 x))): 71 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (-.f64 -2 x))) (+.f64 x 1)) (/.f64 -1 (-.f64 1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 x (-.f64 -2 x)) 1)) (+.f64 x 1)) (/.f64 -1 (-.f64 1 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 x (-.f64 -2 x)) (/.f64 1 (+.f64 x 1)))) (/.f64 -1 (-.f64 1 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*r*_binary64 (*.f64 (+.f64 x (-.f64 -2 x)) (*.f64 (/.f64 1 (+.f64 x 1)) (/.f64 -1 (-.f64 1 x))))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr0.1

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

  6. Final simplification0.1

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

Alternatives

Alternative 1
Error0.1
Cost576
\[\frac{\frac{2}{-1 - x}}{x + -1} \]
Alternative 2
Error0.4
Cost448
\[\frac{2}{1 - x \cdot x} \]

Error

Reproduce

herbie shell --seed 2022334 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))