x / (x^2 + 1)

Average Error: 14.7 → 0.0

Time: 1.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1e+22)
   (/ 1.0 x)
   (if (<= x 100000000.0) (/ x (fma x x 1.0)) (/ 1.0 x))))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

↓

double code(double x) {
	double tmp;
	if (x <= -1e+22) {
		tmp = 1.0 / x;
	} else if (x <= 100000000.0) {
		tmp = x / fma(x, x, 1.0);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -1e+22)
		tmp = Float64(1.0 / x);
	elseif (x <= 100000000.0)
		tmp = Float64(x / fma(x, x, 1.0));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -1e+22], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 100000000.0], N[(x / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Error

Bits error versus x

Target

Original	14.7
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -1e22 or 1e8 < x

   1. Initial program 31.1

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

   2. Simplified 31.1

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

   3. Taylor expanded in x around inf 0.0

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

   if -1e22 < x < 1e8

   1. Initial program 0.0

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

   2. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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