x / (x^2 + 1)

Average Error: 23.3% → 0.03%

Time: 1.5s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -4e+47)
   (/ 1.0 x)
   (if (<= x 50000000.0) (* 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 <= -4e+47) {
		tmp = 1.0 / x;
	} else if (x <= 50000000.0) {
		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 <= -4e+47)
		tmp = Float64(1.0 / x);
	elseif (x <= 50000000.0)
		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, -4e+47], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 50000000.0], N[(x * N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Target

Original	23.3%
Target	0.15%
Herbie	0.03%

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

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000002e47 or 5e7 < x

   1. Initial program 50.55

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

   2. Taylor expanded in x around inf 0

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

   if -4.0000000000000002e47 < x < 5e7

   1. Initial program 0.05

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

   2. Applied egg-rr 0.06

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

4. Final simplification 0.03

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.03%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
Error	0.94%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Error	49.29%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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