Average Error: 14.7 → 0.1
Time: 1.7s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x} \]
↓
\[\frac{-1}{x} \cdot \frac{1}{x + 1} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))
↓
(FPCore (x) :precision binary64 (* (/ -1.0 x) (/ 1.0 (+ x 1.0))))
double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / x);
}
↓
double code(double x) {
	return (-1.0 / x) * (1.0 / (x + 1.0));
}

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

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

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

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

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

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

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.7

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

  2. Applied egg-rr14.0

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

  3. Taylor expanded in x around 0 0.4

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022170 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))