x / (x^2 + 1)

Average Error: 14.5 → 0.0

Time: 2.2s

Precision: binary64

Cost: 6916

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -400000:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{elif}\;x \leq 20000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -400000.0)
   (- (/ 1.0 x) (pow x -3.0))
   (if (<= x 20000000.0) (/ x (+ 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 <= -400000.0) {
		tmp = (1.0 / x) - pow(x, -3.0);
	} else if (x <= 20000000.0) {
		tmp = x / (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 <= (-400000.0d0)) then
        tmp = (1.0d0 / x) - (x ** (-3.0d0))
    else if (x <= 20000000.0d0) then
        tmp = x / (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 <= -400000.0) {
		tmp = (1.0 / x) - Math.pow(x, -3.0);
	} else if (x <= 20000000.0) {
		tmp = x / (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 <= -400000.0:
		tmp = (1.0 / x) - math.pow(x, -3.0)
	elif x <= 20000000.0:
		tmp = x / (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 <= -400000.0)
		tmp = Float64(Float64(1.0 / x) - (x ^ -3.0));
	elseif (x <= 20000000.0)
		tmp = Float64(x / 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 <= -400000.0)
		tmp = (1.0 / x) - (x ^ -3.0);
	elseif (x <= 20000000.0)
		tmp = x / (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, -400000.0], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 20000000.0], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $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 < -4e5

   1. Initial program 30.2

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

   2. Taylor expanded in x around inf 0.0

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

   3. Applied egg-rr 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \frac{1}{x} - \left(\left(1 + {x}^{-3}\right) - 1\right) \]
      +-commutative [=>] 0.0	\[ \frac{1}{x} - \left(\color{blue}{\left({x}^{-3} + 1\right)} - 1\right) \]
      associate--l+ [=>] 0.0	\[ \frac{1}{x} - \color{blue}{\left({x}^{-3} + \left(1 - 1\right)\right)} \]
      metadata-eval [=>] 0.0	\[ \frac{1}{x} - \left({x}^{-3} + \color{blue}{0}\right) \]
      +-rgt-identity [=>] 0.0	\[ \frac{1}{x} - \color{blue}{{x}^{-3}} \]

   if -4e5 < x < 2e7

   1. Initial program 0.0

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

   if 2e7 < x

   1. Initial program 29.8

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

   2. Taylor expanded in x around inf 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -400000:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{elif}\;x \leq 20000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
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
Error	0.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 20000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
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 4
Error	30.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023012 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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