x / (x^2 + 1)

Average Error: 14.8 → 0.4

Time: 2.1s

Precision: binary64

Cost: 7048

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -7.17214385910263 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 7.357464671497419 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -7.17214385910263e+18)
   (/ 1.0 x)
   (if (<= x 7.357464671497419e-7)
     (/ x (+ 1.0 (* x x)))
     (- (/ 1.0 x) (pow x -3.0)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -7.17214385910263e+18) {
		tmp = 1.0 / x;
	} else if (x <= 7.357464671497419e-7) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = (1.0 / x) - pow(x, -3.0);
	}
	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 <= (-7.17214385910263d+18)) then
        tmp = 1.0d0 / x
    else if (x <= 7.357464671497419d-7) then
        tmp = x / (1.0d0 + (x * x))
    else
        tmp = (1.0d0 / x) - (x ** (-3.0d0))
    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 <= -7.17214385910263e+18) {
		tmp = 1.0 / x;
	} else if (x <= 7.357464671497419e-7) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = (1.0 / x) - Math.pow(x, -3.0);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -7.17214385910263e+18:
		tmp = 1.0 / x
	elif x <= 7.357464671497419e-7:
		tmp = x / (1.0 + (x * x))
	else:
		tmp = (1.0 / x) - math.pow(x, -3.0)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -7.17214385910263e+18)
		tmp = Float64(1.0 / x);
	elseif (x <= 7.357464671497419e-7)
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / x) - (x ^ -3.0));
	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 <= -7.17214385910263e+18)
		tmp = 1.0 / x;
	elseif (x <= 7.357464671497419e-7)
		tmp = x / (1.0 + (x * x));
	else
		tmp = (1.0 / x) - (x ^ -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -7.17214385910263e+18], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 7.357464671497419e-7], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]]]

Target

Original	14.8
Target	0.1
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if x < -7172143859102629900

   1. Initial program 31.3

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

   2. Taylor expanded in x around inf 0

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

   if -7172143859102629900 < x < 7.35746467149741875e-7

   1. Initial program 0.0

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

   if 7.35746467149741875e-7 < x

   1. Initial program 30.1

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

   2. Taylor expanded in x around inf 1.7

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}} \]

   3. Applied egg-rr 1.7

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.17214385910263 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 7.357464671497419 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -216712653.40025404:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 7.357464671497419 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -7.17214385910263 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 747582.6933135068:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Error	1.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -216712653.40025404:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 7.357464671497419 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4
Error	30.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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