x / (x^2 + 1)

Average Error: 15.1 → 0.0

Time: 3.3s

Precision: binary64

Cost: 7432

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -10000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{x}{\frac{1 - {x}^{4}}{1 - x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -10000000000.0)
   (/ 1.0 x)
   (if (<= x 100000000.0)
     (/ x (/ (- 1.0 (pow x 4.0)) (- 1.0 (* x x))))
     (/ 1.0 x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -10000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 100000000.0) {
		tmp = x / ((1.0 - pow(x, 4.0)) / (1.0 - (x * x)));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((x * x) + 1.0d0)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-10000000000.0d0)) then
        tmp = 1.0d0 / x
    else if (x <= 100000000.0d0) then
        tmp = x / ((1.0d0 - (x ** 4.0d0)) / (1.0d0 - (x * x)))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -10000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 100000000.0) {
		tmp = x / ((1.0 - Math.pow(x, 4.0)) / (1.0 - (x * x)));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -10000000000.0:
		tmp = 1.0 / x
	elif x <= 100000000.0:
		tmp = x / ((1.0 - math.pow(x, 4.0)) / (1.0 - (x * x)))
	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 <= -10000000000.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 100000000.0)
		tmp = Float64(x / Float64(Float64(1.0 - (x ^ 4.0)) / Float64(1.0 - Float64(x * x))));
	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 <= -10000000000.0)
		tmp = 1.0 / x;
	elseif (x <= 100000000.0)
		tmp = x / ((1.0 - (x ^ 4.0)) / (1.0 - (x * x)));
	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, -10000000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 100000000.0], N[(x / N[(N[(1.0 - N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Target

Original	15.1
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -1e10 or 1e8 < x

   1. Initial program 30.9

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

   2. Taylor expanded in x around inf 0.0

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

   if -1e10 < x < 1e8

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

      \[\leadsto \frac{x}{\color{blue}{\frac{1 - {x}^{4}}{1 - x \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{x}{\frac{1 - {x}^{4}}{1 - x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -50000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 75000000:\\ \;\;\;\;x \cdot \frac{-1}{-1 - x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	712

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

Alternative 3
Error	0.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -10000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 40000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4
Error	0.6
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 5
Error	31.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023046 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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