Average Error: 14.2 → 0.4
Time: 3.3s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x} \]
↓
\[\frac{-1}{\mathsf{fma}\left(x, x, x\right)} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))
↓
(FPCore (x) :precision binary64 (/ -1.0 (fma x x x)))
double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / x);
}
↓
double code(double x) {
	return -1.0 / fma(x, x, x);
}

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

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

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

Error

Bits error versus x

Derivation

  1. Initial program 14.2

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

  2. Applied frac-sub_binary6413.6

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

  3. Simplified0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))