Average Error: 14.6 → 0.3
Time: 2.5s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1} \]
↓
\[\frac{-2}{\mathsf{fma}\left(x, x, -1\right)} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ -2.0 (fma x x -1.0)))
double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}
↓
double code(double x) {
	return -2.0 / fma(x, x, -1.0);
}

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(-2.0 / fma(x, x, -1.0))
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[(-2.0 / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]

\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{-2}{\mathsf{fma}\left(x, x, -1\right)}

Error

Bits error versus x

Derivation

  1. Initial program 14.6

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

  2. Applied egg-rr14.0

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

  3. Taylor expanded in x around 0 0.3

    \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]

  4. Final simplification0.3

    \[\leadsto \frac{-2}{\mathsf{fma}\left(x, x, -1\right)} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))