Average Error: 9.6 → 0.3
Time: 6.1s
Precision: binary64
Cost: 704
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
↓
\[\frac{2}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ 2.0 (* (+ 1.0 x) (- (* x x) x))))
double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}
↓
double code(double x) {
	return 2.0 / ((1.0 + x) * ((x * x) - x));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((1.0d0 + x) * ((x * x) - x))
end function

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}
↓
public static double code(double x) {
	return 2.0 / ((1.0 + x) * ((x * x) - x));
}

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

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

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end
↓
function tmp = code(x)
	tmp = 2.0 / ((1.0 + x) * ((x * x) - x));
end

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(2.0 / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

  1. Initial program 9.6

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

  2. Simplified9.6

    \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
    Proof
    (+.f64 (/.f64 1 (+.f64 1 x)) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 x))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (/.f64 2 x)) (/.f64 1 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 1 (+.f64 x 1)) (neg.f64 (/.f64 2 x))) (/.f64 1 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x))) (/.f64 1 (-.f64 x 1))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr26.0

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

  4. Applied egg-rr30.1

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

  5. Applied egg-rr25.1

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

  6. Taylor expanded in x around 0 0.3

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

  7. Final simplification0.3

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

Alternatives

Alternative 1
Error15.4
Cost584
\[\begin{array}{l} t_0 := \frac{1}{x \cdot x}\\ \mathbf{if}\;x \leq -4068.093615561638:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.832892512719917 \cdot 10^{-13}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error10.4
Cost448
\[1 + \left(-1 + \frac{-2}{x}\right) \]

Error

Reproduce

herbie shell --seed 2022289 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))