x / (x^2 + 1)

Average Accuracy: 76.3% → 99.8%

Time: 2.3s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\frac{{x}^{3} - x}{-1 + {x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e+52)
   (/ 1.0 x)
   (if (<= x 50.0) (/ (- (pow x 3.0) x) (+ -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 <= -2e+52) {
		tmp = 1.0 / x;
	} else if (x <= 50.0) {
		tmp = (pow(x, 3.0) - x) / (-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 <= (-2d+52)) then
        tmp = 1.0d0 / x
    else if (x <= 50.0d0) then
        tmp = ((x ** 3.0d0) - x) / ((-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 <= -2e+52) {
		tmp = 1.0 / x;
	} else if (x <= 50.0) {
		tmp = (Math.pow(x, 3.0) - x) / (-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 <= -2e+52:
		tmp = 1.0 / x
	elif x <= 50.0:
		tmp = (math.pow(x, 3.0) - x) / (-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 <= -2e+52)
		tmp = Float64(1.0 / x);
	elseif (x <= 50.0)
		tmp = Float64(Float64((x ^ 3.0) - x) / 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 <= -2e+52)
		tmp = 1.0 / x;
	elseif (x <= 50.0)
		tmp = ((x ^ 3.0) - x) / (-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, -2e+52], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 50.0], N[(N[(N[Power[x, 3.0], $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 + N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Target

Original	76.3%
Target	99.9%
Herbie	99.8%

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

Derivation

1. Split input into 2 regimes

2. if x < -2e52 or 50 < x

   1. Initial program 48.1%

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

   2. Taylor expanded in x around inf 99.7%

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

   if -2e52 < x < 50

   1. Initial program 100.0%

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

   2. Applied egg-rr 99.9%

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

   3. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{x}{{x}^{4} + -1} \cdot \mathsf{fma}\left(x, x, -1\right) \]
      associate-*l/ [=>] 99.9	\[ \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, x, -1\right)}{{x}^{4} + -1}} \]
      fma-udef [=>] 99.9	\[ \frac{x \cdot \color{blue}{\left(x \cdot x + -1\right)}}{{x}^{4} + -1} \]
      distribute-rgt-in [=>] 99.9	\[ \frac{\color{blue}{\left(x \cdot x\right) \cdot x + -1 \cdot x}}{{x}^{4} + -1} \]
      neg-mul-1 [<=] 99.9	\[ \frac{\left(x \cdot x\right) \cdot x + \color{blue}{\left(-x\right)}}{{x}^{4} + -1} \]
      unpow3 [<=] 99.9	\[ \frac{\color{blue}{{x}^{3}} + \left(-x\right)}{{x}^{4} + -1} \]
      unsub-neg [=>] 99.9	\[ \frac{\color{blue}{{x}^{3} - x}}{{x}^{4} + -1} \]
      +-commutative [=>] 99.9	\[ \frac{{x}^{3} - x}{\color{blue}{-1 + {x}^{4}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\frac{{x}^{3} - x}{-1 + {x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
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
Accuracy	51.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023141 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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