x / (x^2 + 1)

Average Error: 14.8 → 0.4

Time: 3.3s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -352929180640312.3:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.346374982514411 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -352929180640312.3)
   (/ 1.0 x)
   (if (<= x 9.346374982514411e-7) (* x (/ 1.0 (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 <= -352929180640312.3) {
		tmp = 1.0 / x;
	} else if (x <= 9.346374982514411e-7) {
		tmp = x * (1.0 / 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 <= -352929180640312.3)
		tmp = Float64(1.0 / x);
	elseif (x <= 9.346374982514411e-7)
		tmp = Float64(x * Float64(1.0 / 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, -352929180640312.3], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 9.346374982514411e-7], N[(x * N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Target

Original	14.8
Target	0.1
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if x < -352929180640312.312 or 9.3463749825144108e-7 < x

   1. Initial program 30.4

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

   2. Taylor expanded in x around inf 0.9

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

   if -352929180640312.312 < x < 9.3463749825144108e-7

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -352929180640312.3:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.346374982514411 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	13376

\[\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right)} \]

Alternative 2
Error	0.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1429382063088882:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.346374982514411 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3957232401.0070257:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.346374982514411 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1429382063088882:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.346374982514411 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 5
Error	30.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022294 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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