x / (x^2 + 1)

Average Error: 14.5 → 0.0

Time: 4.6s

Precision: binary64

Cost: 7176

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{{x}^{3}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e+45)
   (/ 1.0 x)
   (if (<= x 10000.0)
     (/ x (+ (* x x) 1.0))
     (- (/ 1.0 x) (/ 1.0 (pow x 3.0))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -2e+45) {
		tmp = 1.0 / x;
	} else if (x <= 10000.0) {
		tmp = x / ((x * x) + 1.0);
	} else {
		tmp = (1.0 / x) - (1.0 / 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 <= (-2d+45)) then
        tmp = 1.0d0 / x
    else if (x <= 10000.0d0) then
        tmp = x / ((x * x) + 1.0d0)
    else
        tmp = (1.0d0 / x) - (1.0d0 / (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 <= -2e+45) {
		tmp = 1.0 / x;
	} else if (x <= 10000.0) {
		tmp = x / ((x * x) + 1.0);
	} else {
		tmp = (1.0 / x) - (1.0 / Math.pow(x, 3.0));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -2e+45:
		tmp = 1.0 / x
	elif x <= 10000.0:
		tmp = x / ((x * x) + 1.0)
	else:
		tmp = (1.0 / x) - (1.0 / 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 <= -2e+45)
		tmp = Float64(1.0 / x);
	elseif (x <= 10000.0)
		tmp = Float64(x / Float64(Float64(x * x) + 1.0));
	else
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 / (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 <= -2e+45)
		tmp = 1.0 / x;
	elseif (x <= 10000.0)
		tmp = x / ((x * x) + 1.0);
	else
		tmp = (1.0 / x) - (1.0 / (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, -2e+45], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 10000.0], N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	14.5
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999999e45

   1. Initial program 34.1

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

   2. Taylor expanded in x around inf 0

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

   if -1.9999999999999999e45 < x < 1e4

   1. Initial program 0.0

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

   if 1e4 < x

   1. Initial program 29.9

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

   2. Taylor expanded in x around inf 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{{x}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 40000000:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
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 3
Error	30.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023098 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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