x / (x^2 + 1)

Average Error: 14.9 → 0.0

Time: 2.3s

Precision: binary64

Cost: 7432

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 400000:\\ \;\;\;\;\frac{x \cdot \left(x \cdot x + -1\right)}{-1 + {x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1000000000000.0)
   (/ 1.0 x)
   (if (<= x 400000.0)
     (/ (* x (+ (* x x) -1.0)) (+ -1.0 (pow x 4.0)))
     (/ 1.0 x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 400000.0) {
		tmp = (x * ((x * x) + -1.0)) / (-1.0 + pow(x, 4.0));
	} 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 <= (-1000000000000.0d0)) then
        tmp = 1.0d0 / x
    else if (x <= 400000.0d0) then
        tmp = (x * ((x * x) + (-1.0d0))) / ((-1.0d0) + (x ** 4.0d0))
    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 <= -1000000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 400000.0) {
		tmp = (x * ((x * x) + -1.0)) / (-1.0 + Math.pow(x, 4.0));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -1000000000000.0:
		tmp = 1.0 / x
	elif x <= 400000.0:
		tmp = (x * ((x * x) + -1.0)) / (-1.0 + math.pow(x, 4.0))
	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 <= -1000000000000.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 400000.0)
		tmp = Float64(Float64(x * Float64(Float64(x * x) + -1.0)) / Float64(-1.0 + (x ^ 4.0)));
	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 <= -1000000000000.0)
		tmp = 1.0 / x;
	elseif (x <= 400000.0)
		tmp = (x * ((x * x) + -1.0)) / (-1.0 + (x ^ 4.0));
	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, -1000000000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 400000.0], N[(N[(x * N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Target

Original	14.9
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -1e12 or 4e5 < x

   1. Initial program 30.9

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

   2. Taylor expanded in x around inf 0.1

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

   if -1e12 < x < 4e5

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{x}{{x}^{4} + -1} \cdot \mathsf{fma}\left(x, x, -1\right)} \]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\frac{{x}^{3} - x}{-1 + {x}^{4}}} \]
      Proof

      [Start] 0.0	\[ \frac{x}{{x}^{4} + -1} \cdot \mathsf{fma}\left(x, x, -1\right) \]
      associate-*l/ [=>] 0.0	\[ \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, x, -1\right)}{{x}^{4} + -1}} \]
      fma-udef [=>] 0.0	\[ \frac{x \cdot \color{blue}{\left(x \cdot x + -1\right)}}{{x}^{4} + -1} \]
      distribute-rgt-in [=>] 0.0	\[ \frac{\color{blue}{\left(x \cdot x\right) \cdot x + -1 \cdot x}}{{x}^{4} + -1} \]
      neg-mul-1 [<=] 0.0	\[ \frac{\left(x \cdot x\right) \cdot x + \color{blue}{\left(-x\right)}}{{x}^{4} + -1} \]
      unpow3 [<=] 0.0	\[ \frac{\color{blue}{{x}^{3}} + \left(-x\right)}{{x}^{4} + -1} \]
      unsub-neg [=>] 0.0	\[ \frac{\color{blue}{{x}^{3} - x}}{{x}^{4} + -1} \]
      +-commutative [=>] 0.0	\[ \frac{{x}^{3} - x}{\color{blue}{-1 + {x}^{4}}} \]

   4. Applied egg-rr 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 400000:\\ \;\;\;\;\frac{x \cdot \left(x \cdot x + -1\right)}{-1 + {x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -500000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 400000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Error	0.7
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	30.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023083 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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