x / (x^2 + 1)

Average Accuracy: 76.2% → 100.0%

Time: 2.4s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1e+19)
   (/ 1.0 x)
   (if (<= x 200000000.0) (* 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 <= -1e+19) {
		tmp = 1.0 / x;
	} else if (x <= 200000000.0) {
		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 <= -1e+19)
		tmp = Float64(1.0 / x);
	elseif (x <= 200000000.0)
		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, -1e+19], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 200000000.0], N[(x * N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Target

Original	76.2%
Target	99.9%
Herbie	100.0%

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

Derivation

1. Split input into 2 regimes

2. if x < -1e19 or 2e8 < x

   1. Initial program 50.2%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -1e19 < x < 2e8

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \frac{x}{x \cdot x + 1} \]
      clear-num [=>] 99.7	\[ \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]
      associate-/r/ [=>] 100.0	\[ \color{blue}{\frac{1}{x \cdot x + 1} \cdot x} \]
      fma-def [=>] 100.0	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} \cdot x \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.86:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Accuracy	98.9%
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 4
Accuracy	51.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023153 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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