Average Error: 29.5 → 25.4
Time: 13.0s
Precision: binary64
Cost: 1472
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\frac{-1 + \left(x + \left(x + 1\right) \cdot \frac{-1 - x}{x}\right)}{\frac{x + 1}{x} \cdot \left(-1 + x\right)} \]
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (/
  (+ -1.0 (+ x (* (+ x 1.0) (/ (- -1.0 x) x))))
  (* (/ (+ x 1.0) x) (+ -1.0 x))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	return (-1.0 + (x + ((x + 1.0) * ((-1.0 - x) / x)))) / (((x + 1.0) / x) * (-1.0 + x));
}

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

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

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

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

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

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

\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\frac{-1 + \left(x + \left(x + 1\right) \cdot \frac{-1 - x}{x}\right)}{\frac{x + 1}{x} \cdot \left(-1 + x\right)}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.5

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

  2. Applied egg-rr29.2

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

  3. Applied egg-rr30.3

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

  4. Simplified28.4

    \[\leadsto \frac{\color{blue}{-1 + \left(x - \frac{{\left(x + 1\right)}^{2}}{x}\right)}}{\frac{x + 1}{x} \cdot \left(x + -1\right)} \]
    Proof
    (+.f64 -1 (-.f64 x (/.f64 (pow.f64 (+.f64 x 1) 2) x))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 -1 x) (/.f64 (pow.f64 (+.f64 x 1) 2) x))): 34 points increase in error, 28 points decrease in error
    (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)) (/.f64 (pow.f64 (+.f64 x 1) 2) x)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate--l+_binary64 (+.f64 x (-.f64 -1 (/.f64 (pow.f64 (+.f64 x 1) 2) x)))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr25.4

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

  6. Final simplification25.4

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

Alternatives

Alternative 1
Error28.4
Cost1472
\[\frac{-1 + \left(x + \frac{\left(x + 1\right) \cdot \left(-1 - x\right)}{x}\right)}{\frac{x + 1}{x} \cdot \left(-1 + x\right)} \]
Alternative 2
Error29.2
Cost1088
\[\frac{\left(-1 - x\right) + \frac{1 - x}{-1 + \frac{-1}{x}}}{-1 + x} \]
Alternative 3
Error29.5
Cost832
\[\frac{x}{x + 1} + \frac{-1 - x}{-1 + x} \]

Error

Reproduce

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