Average Error: 14.8 → 0.0
Time: 1.6s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -8.27254731064536 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9969059349.580149:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -8.27254731064536e+18)
   (/ 1.0 x)
   (if (<= x 9969059349.580149)
     (expm1 (log1p (/ x (fma x x 1.0))))
     (/ 1.0 x))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if (x <= -8.27254731064536e+18) {
		tmp = 1.0 / x;
	} else if (x <= 9969059349.580149) {
		tmp = expm1(log1p((x / fma(x, x, 1.0))));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end
↓
function code(x)
	tmp = 0.0
	if (x <= -8.27254731064536e+18)
		tmp = Float64(1.0 / x);
	elseif (x <= 9969059349.580149)
		tmp = expm1(log1p(Float64(x / fma(x, x, 1.0))));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := If[LessEqual[x, -8.27254731064536e+18], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 9969059349.580149], N[(Exp[N[Log[1 + N[(x / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -8.27254731064536 \cdot 10^{+18}:\\
\;\;\;\;\frac{1}{x}\\

\mathbf{elif}\;x \leq 9969059349.580149:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x}\\


\end{array}

Error

Bits error versus x

Target

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}} \]

Derivation

  1. Split input into 2 regimes

  2. if x < -8272547310645359620 or 9969059349.5801487 < x

    1. Initial program 31.1

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

    2. Simplified31.1

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

    3. Taylor expanded in x around inf 0

      \[\leadsto \color{blue}{\frac{1}{x}} \]

    if -8272547310645359620 < x < 9969059349.5801487

    1. Initial program 0.0

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

    2. Simplified0.0

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

    3. Applied egg-rr0.0

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.27254731064536 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9969059349.580149:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))