Average Error: 29.2 → 0.0
Time: 9.1s
Precision: binary64
Cost: 960
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\frac{\frac{3 + \frac{1}{x}}{1 + \frac{1}{x}}}{1 - x} \]
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (/ (/ (+ 3.0 (/ 1.0 x)) (+ 1.0 (/ 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 ((3.0 + (1.0 / x)) / (1.0 + (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 = ((3.0d0 + (1.0d0 / x)) / (1.0d0 + (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 ((3.0 + (1.0 / x)) / (1.0 + (1.0 / x))) / (1.0 - x);
}

def code(x):
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))
↓
def code(x):
	return ((3.0 + (1.0 / x)) / (1.0 + (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(Float64(3.0 + Float64(1.0 / x)) / Float64(1.0 + Float64(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 = ((3.0 + (1.0 / x)) / (1.0 + (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[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\frac{\frac{3 + \frac{1}{x}}{1 + \frac{1}{x}}}{1 - x}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.2

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

  2. Simplified29.2

    \[\leadsto \color{blue}{\frac{-1 - x}{x + -1} - \frac{x}{-1 - x}} \]
    Proof
    (-.f64 (/.f64 (-.f64 -1 x) (+.f64 x -1)) (/.f64 x (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 (-.f64 (Rewrite<= metadata-eval (neg.f64 1)) x) (+.f64 x -1)) (/.f64 x (-.f64 -1 x))): 0 points increase in error, 31 points decrease in error
    (-.f64 (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 1) (neg.f64 x))) (+.f64 x -1)) (/.f64 x (-.f64 -1 x))): 0 points increase in error, 31 points decrease in error
    (-.f64 (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (neg.f64 1))) (+.f64 x -1)) (/.f64 x (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x 1))) (+.f64 x -1)) (/.f64 x (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 (neg.f64 (+.f64 x 1)) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 x (-.f64 -1 x))): 31 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 (neg.f64 (+.f64 x 1)) (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (/.f64 x (-.f64 -1 x))): 0 points increase in error, 31 points decrease in error
    (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)))) (/.f64 x (-.f64 -1 x))): 31 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (/.f64 (+.f64 x 1) (-.f64 x 1)))) (/.f64 x (-.f64 -1 x))): 0 points increase in error, 31 points decrease in error
    (-.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 x (-.f64 -1 x))): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 x (-.f64 (Rewrite<= metadata-eval (neg.f64 1)) x))): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 1) (neg.f64 x))))): 0 points increase in error, 31 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 x (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (neg.f64 1))))): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 x (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x 1))))): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 x (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (+.f64 x 1))))): 0 points increase in error, 31 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 x (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) (+.f64 x 1)))): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 x (+.f64 x 1)) (neg.f64 1)))): 0 points increase in error, 31 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 (/.f64 x (+.f64 x 1)) (Rewrite=> metadata-eval -1))): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 (/.f64 x (+.f64 x 1)) (Rewrite<= metadata-eval (/.f64 1 -1)))): 0 points increase in error, 31 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 (/.f64 x (+.f64 x 1)) (/.f64 1 (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 31 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 x (+.f64 x 1)) (neg.f64 1)) 1))): 0 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 1) (/.f64 x (+.f64 x 1)))) 1)): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 x (+.f64 x 1))) 1)): 0 points increase in error, 31 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 x (+.f64 x 1)))) 1)): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (neg.f64 1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (Rewrite=> /-rgt-identity_binary64 (neg.f64 (/.f64 x (+.f64 x 1))))): 31 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (+.f64 x 1) (-.f64 x 1))) (neg.f64 (/.f64 x (+.f64 x 1)))): 0 points increase in error, 31 points decrease in error
    (-.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)))) (neg.f64 (/.f64 x (+.f64 x 1)))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> sub-neg_binary64 (+.f64 (neg.f64 (/.f64 (+.f64 x 1) (-.f64 x 1))) (neg.f64 (neg.f64 (/.f64 x (+.f64 x 1)))))): 31 points increase in error, 0 points decrease in error
    (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (neg.f64 (/.f64 x (+.f64 x 1)))) (neg.f64 (/.f64 (+.f64 x 1) (-.f64 x 1))))): 0 points increase in error, 31 points decrease in error
    (+.f64 (Rewrite=> remove-double-neg_binary64 (/.f64 x (+.f64 x 1))) (neg.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)))): 0 points increase in error, 31 points decrease in error
    (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))): 0 points increase in error, 31 points decrease in error

  3. Applied egg-rr28.9

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

  4. Simplified28.9

    \[\leadsto \color{blue}{\frac{\frac{\left(1 + x\right) \cdot \frac{-1 - x}{x} - \left(1 - x\right)}{1 - x}}{\frac{-1 - x}{x}}} \]
    Proof
    (/.f64 (/.f64 (-.f64 (*.f64 (+.f64 1 x) (/.f64 (-.f64 -1 x) x)) (-.f64 1 x)) (-.f64 1 x)) (/.f64 (-.f64 -1 x) x)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (-.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)) (/.f64 (-.f64 -1 x) x)) (-.f64 1 x)) (-.f64 1 x)) (/.f64 (-.f64 -1 x) x)): 0 points increase in error, 4 points decrease in error
    (/.f64 (/.f64 (-.f64 (*.f64 (+.f64 x 1) (/.f64 (-.f64 -1 x) x)) (Rewrite<= *-rgt-identity_binary64 (*.f64 (-.f64 1 x) 1))) (-.f64 1 x)) (/.f64 (-.f64 -1 x) x)): 0 points increase in error, 4 points decrease in error
    (Rewrite<= associate-/r*_binary64 (/.f64 (-.f64 (*.f64 (+.f64 x 1) (/.f64 (-.f64 -1 x) x)) (*.f64 (-.f64 1 x) 1)) (*.f64 (-.f64 1 x) (/.f64 (-.f64 -1 x) x)))): 0 points increase in error, 0 points decrease in error

  5. Taylor expanded in x around 0 0.0

    \[\leadsto \frac{\frac{\color{blue}{-\left(3 + \frac{1}{x}\right)}}{1 - x}}{\frac{-1 - x}{x}} \]

  6. Simplified0.0

    \[\leadsto \frac{\frac{\color{blue}{-3 - \frac{1}{x}}}{1 - x}}{\frac{-1 - x}{x}} \]
    Proof
    (/.f64 (/.f64 (-.f64 -3 (/.f64 1 x)) (-.f64 1 x)) (/.f64 (-.f64 -1 x) x)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (-.f64 (Rewrite<= metadata-eval (-.f64 0 3)) (/.f64 1 x)) (-.f64 1 x)) (/.f64 (-.f64 -1 x) x)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 0 (+.f64 3 (/.f64 1 x)))) (-.f64 1 x)) (/.f64 (-.f64 -1 x) x)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (+.f64 3 (/.f64 1 x)))) (-.f64 1 x)) (/.f64 (-.f64 -1 x) x)): 0 points increase in error, 0 points decrease in error

  7. Applied egg-rr0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x} + 3}{1 - x} \cdot \frac{1}{1 + \frac{1}{x}}} \]

  8. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{3 + \frac{1}{x}}{1 + \frac{1}{x}}}{1 - x}} \]
    Proof
    (/.f64 (/.f64 (+.f64 3 (/.f64 1 x)) (+.f64 1 (/.f64 1 x))) (-.f64 1 x)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 x) 3)) (+.f64 1 (/.f64 1 x))) (-.f64 1 x)): 0 points increase in error, 0 points decrease in error
    (/.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (/.f64 1 x) 3) 1)) (+.f64 1 (/.f64 1 x))) (-.f64 1 x)): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (/.f64 1 x) 3) (/.f64 1 (+.f64 1 (/.f64 1 x))))) (-.f64 1 x)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 (/.f64 1 x) 3) (-.f64 1 x)) (/.f64 1 (+.f64 1 (/.f64 1 x))))): 2 points increase in error, 0 points decrease in error

  9. Final simplification0.0

    \[\leadsto \frac{\frac{3 + \frac{1}{x}}{1 + \frac{1}{x}}}{1 - x} \]

Alternatives

Alternative 1
Error0.1
Cost1096
\[\begin{array}{l} \mathbf{if}\;x \leq -300000:\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{elif}\;x \leq 420000:\\ \;\;\;\;\frac{x}{1 + x} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \end{array} \]
Alternative 2
Error0.7
Cost841
\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot x\\ \end{array} \]
Alternative 3
Error0.7
Cost841
\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.82\right):\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot x\\ \end{array} \]
Alternative 4
Error0.6
Cost841
\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-1 - x\right) \cdot \left(-1 - x\right)\\ \end{array} \]
Alternative 5
Error1.0
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]
Alternative 6
Error1.4
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]
Alternative 7
Error31.4
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))