Average Error: 14.7 → 0.0
Time: 2.7s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\begin{array}{l} t_0 := \left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) + \frac{-1}{{x}^{3}}\\ \mathbf{if}\;x \leq -1.711991797943903 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1732693.7619002569:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (+ (/ 1.0 (pow x 5.0)) (/ 1.0 x)) (/ -1.0 (pow x 3.0)))))
   (if (<= x -1.711991797943903e+38)
     t_0
     (if (<= x 1732693.7619002569) (* x (/ 1.0 (fma x x 1.0))) t_0))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double t_0 = ((1.0 / pow(x, 5.0)) + (1.0 / x)) + (-1.0 / pow(x, 3.0));
	double tmp;
	if (x <= -1.711991797943903e+38) {
		tmp = t_0;
	} else if (x <= 1732693.7619002569) {
		tmp = x * (1.0 / fma(x, x, 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end
↓
function code(x)
	t_0 = Float64(Float64(Float64(1.0 / (x ^ 5.0)) + Float64(1.0 / x)) + Float64(-1.0 / (x ^ 3.0)))
	tmp = 0.0
	if (x <= -1.711991797943903e+38)
		tmp = t_0;
	elseif (x <= 1732693.7619002569)
		tmp = Float64(x * Float64(1.0 / fma(x, x, 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.711991797943903e+38], t$95$0, If[LessEqual[x, 1732693.7619002569], N[(x * N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
t_0 := \left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) + \frac{-1}{{x}^{3}}\\
\mathbf{if}\;x \leq -1.711991797943903 \cdot 10^{+38}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 1732693.7619002569:\\
\;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}} \]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.711991797943903e38 or 1732693.7619002569 < x

    1. Initial program 32.0

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

    2. Simplified32.0

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

    3. Applied egg-rr31.9

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

    4. Taylor expanded in x around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}} \]

    if -1.711991797943903e38 < x < 1732693.7619002569

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.711991797943903 \cdot 10^{+38}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) + \frac{-1}{{x}^{3}}\\ \mathbf{elif}\;x \leq 1732693.7619002569:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) + \frac{-1}{{x}^{3}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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