x / (x^2 + 1)

Average Error: 14.9 → 0.1

Time: 1.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{x} + \left(\frac{1}{{x}^{5}} + \left(\frac{-1}{{x}^{3}} + \frac{-1}{{x}^{7}}\right)\right)\\ t_1 := \sqrt{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-0.5}}\\ \mathbf{if}\;x \leq -220.46626152611887:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.03147942073716495:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot \left(t_1 \cdot \left(x \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          (/ 1.0 x)
          (+
           (/ 1.0 (pow x 5.0))
           (+ (/ -1.0 (pow x 3.0)) (/ -1.0 (pow x 7.0))))))
        (t_1 (sqrt (pow (fma x x 1.0) -0.5))))
   (if (<= x -220.46626152611887)
     t_0
     (if (<= x 0.03147942073716495)
       (* (/ 1.0 (sqrt (fma x x 1.0))) (* t_1 (* x t_1)))
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = (1.0 / x) + ((1.0 / pow(x, 5.0)) + ((-1.0 / pow(x, 3.0)) + (-1.0 / pow(x, 7.0))));
	double t_1 = sqrt(pow(fma(x, x, 1.0), -0.5));
	double tmp;
	if (x <= -220.46626152611887) {
		tmp = t_0;
	} else if (x <= 0.03147942073716495) {
		tmp = (1.0 / sqrt(fma(x, x, 1.0))) * (t_1 * (x * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = Float64(Float64(1.0 / x) + Float64(Float64(1.0 / (x ^ 5.0)) + Float64(Float64(-1.0 / (x ^ 3.0)) + Float64(-1.0 / (x ^ 7.0)))))
	t_1 = sqrt((fma(x, x, 1.0) ^ -0.5))
	tmp = 0.0
	if (x <= -220.46626152611887)
		tmp = t_0;
	elseif (x <= 0.03147942073716495)
		tmp = Float64(Float64(1.0 / sqrt(fma(x, x, 1.0))) * Float64(t_1 * Float64(x * t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] + N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Power[N[(x * x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -220.46626152611887], t$95$0, If[LessEqual[x, 0.03147942073716495], N[(N[(1.0 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Target

Original	14.9
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -220.466261526118871 or 0.031479420737164948 < x

   1. Initial program 29.5

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

   2. Simplified 29.5

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

   3. Taylor expanded in x around inf 0.3

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{7}}\right)} \]

   4. Simplified 0.3

      \[\leadsto \color{blue}{\frac{1}{x} + \left(\frac{1}{{x}^{5}} + \left(\frac{-1}{{x}^{3}} + \frac{-1}{{x}^{7}}\right)\right)} \]

   if -220.466261526118871 < x < 0.031479420737164948

   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

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

   4. Applied egg-rr 0.0

      \[\leadsto \frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot \color{blue}{\left(\left(x \cdot \sqrt{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-0.5}}\right) \cdot \sqrt{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-0.5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -220.46626152611887:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} + \left(\frac{-1}{{x}^{3}} + \frac{-1}{{x}^{7}}\right)\right)\\ \mathbf{elif}\;x \leq 0.03147942073716495:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot \left(\sqrt{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-0.5}} \cdot \left(x \cdot \sqrt{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-0.5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} + \left(\frac{-1}{{x}^{3}} + \frac{-1}{{x}^{7}}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022213 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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