x / (x^2 + 1)

Average Error: 14.9 → 0.0

Time: 1.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7957506658810687 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 127757795.41906317:\\ \;\;\;\;x \cdot {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.7957506658810687e+42)
   (/ 1.0 x)
   (if (<= x 127757795.41906317) (* x (pow (hypot 1.0 x) -2.0)) (/ 1.0 x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1.7957506658810687e+42) {
		tmp = 1.0 / x;
	} else if (x <= 127757795.41906317) {
		tmp = x * pow(hypot(1.0, x), -2.0);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -1.7957506658810687e+42) {
		tmp = 1.0 / x;
	} else if (x <= 127757795.41906317) {
		tmp = x * Math.pow(Math.hypot(1.0, x), -2.0);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	return x / ((x * x) + 1.0)

↓

def code(x):
	tmp = 0
	if x <= -1.7957506658810687e+42:
		tmp = 1.0 / x
	elif x <= 127757795.41906317:
		tmp = x * math.pow(math.hypot(1.0, x), -2.0)
	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 <= -1.7957506658810687e+42)
		tmp = Float64(1.0 / x);
	elseif (x <= 127757795.41906317)
		tmp = Float64(x * (hypot(1.0, x) ^ -2.0));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.7957506658810687e+42)
		tmp = 1.0 / x;
	elseif (x <= 127757795.41906317)
		tmp = x * (hypot(1.0, x) ^ -2.0);
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -1.7957506658810687e+42], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 127757795.41906317], N[(x * N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Error

Bits error versus x

Target

Original	14.9
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -1.79575066588106868e42 or 127757795.419063166 < x

   1. Initial program 32.8

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

   2. Simplified 32.8

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

   3. Taylor expanded in x around inf 0

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

   if -1.79575066588106868e42 < x < 127757795.419063166

   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. Applied egg-rr 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7957506658810687 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 127757795.41906317:\\ \;\;\;\;x \cdot {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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