x / (x^2 + 1)

Average Error: 22.54% → 0.02%

Time: 5.7s

Precision: binary64

Cost: 7048

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot x + -1\\ \mathbf{if}\;x \leq -40000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 5000:\\ \;\;\;\;\frac{\frac{x \cdot t_0}{1 + x \cdot x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (* x x) -1.0)))
   (if (<= x -40000000000.0)
     (/ 1.0 x)
     (if (<= x 5000.0)
       (/ (/ (* x t_0) (+ 1.0 (* x x))) t_0)
       (- (/ 1.0 x) (pow x -3.0))))))

C

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

↓

double code(double x) {
	double t_0 = (x * x) + -1.0;
	double tmp;
	if (x <= -40000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 5000.0) {
		tmp = ((x * t_0) / (1.0 + (x * x))) / t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x * x) + (-1.0d0)
    if (x <= (-40000000000.0d0)) then
        tmp = 1.0d0 / x
    else if (x <= 5000.0d0) then
        tmp = ((x * t_0) / (1.0d0 + (x * x))) / t_0
    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 t_0 = (x * x) + -1.0;
	double tmp;
	if (x <= -40000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 5000.0) {
		tmp = ((x * t_0) / (1.0 + (x * x))) / t_0;
	} 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):
	t_0 = (x * x) + -1.0
	tmp = 0
	if x <= -40000000000.0:
		tmp = 1.0 / x
	elif x <= 5000.0:
		tmp = ((x * t_0) / (1.0 + (x * x))) / t_0
	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)
	t_0 = Float64(Float64(x * x) + -1.0)
	tmp = 0.0
	if (x <= -40000000000.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 5000.0)
		tmp = Float64(Float64(Float64(x * t_0) / Float64(1.0 + Float64(x * x))) / t_0);
	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)
	t_0 = (x * x) + -1.0;
	tmp = 0.0;
	if (x <= -40000000000.0)
		tmp = 1.0 / x;
	elseif (x <= 5000.0)
		tmp = ((x * t_0) / (1.0 + (x * x))) / t_0;
	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_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -40000000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 5000.0], N[(N[(N[(x * t$95$0), $MachinePrecision] / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]]]]

Target

Original	22.54%
Target	0.17%
Herbie	0.02%

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

Derivation

1. Split input into 3 regimes

2. if x < -4e10

   1. Initial program 47.77

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

   2. Taylor expanded in x around inf 0

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

   if -4e10 < x < 5e3

   1. Initial program 0.02

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

   2. Applied egg-rr 0.03

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

   3. Applied egg-rr 0.03

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

   4. Applied egg-rr 0.03

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

   if 5e3 < x

   1. Initial program 45.59

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

   2. Taylor expanded in x around inf 0.01

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

   3. Applied egg-rr 0.22

      \[\leadsto \frac{1}{x} - \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-3}\right)} - 1\right)} \]

   4. Simplified 0.01

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

      [Start] 0.22	\[ \frac{1}{x} - \left(e^{\mathsf{log1p}\left({x}^{-3}\right)} - 1\right) \]
      expm1-def [=>] 0.01	\[ \frac{1}{x} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-3}\right)\right)} \]
      expm1-log1p [=>] 0.01	\[ \frac{1}{x} - \color{blue}{{x}^{-3}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.02

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 5000:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot x + -1\right)}{1 + x \cdot x}}{x \cdot x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.03%
Cost	1480

\[\begin{array}{l} t_0 := x \cdot x + -1\\ \mathbf{if}\;x \leq -40000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{\frac{x \cdot t_0}{1 + x \cdot x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
Error	0.04%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Error	1.04%
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	48.14%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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