x / (x^2 + 1)

Percentage Accurate: 76.7% → 100.0%

Time: 4.9s

Precision: binary64

Cost: 7049

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2000000 \lor \neg \left(x \leq 1000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -2000000.0) (not (<= x 1000.0)))
   (- (/ 1.0 x) (pow x -3.0))
   (/ x (+ 1.0 (* x x)))))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -2000000.0) || !(x <= 1000.0)) {
		tmp = (1.0 / x) - pow(x, -3.0);
	} else {
		tmp = x / (1.0 + (x * 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 <= (-2000000.0d0)) .or. (.not. (x <= 1000.0d0))) then
        tmp = (1.0d0 / x) - (x ** (-3.0d0))
    else
        tmp = x / (1.0d0 + (x * 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 <= -2000000.0) || !(x <= 1000.0)) {
		tmp = (1.0 / x) - Math.pow(x, -3.0);
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if (x <= -2000000.0) or not (x <= 1000.0):
		tmp = (1.0 / x) - math.pow(x, -3.0)
	else:
		tmp = x / (1.0 + (x * 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 <= -2000000.0) || !(x <= 1000.0))
		tmp = Float64(Float64(1.0 / x) - (x ^ -3.0));
	else
		tmp = Float64(x / Float64(1.0 + Float64(x * 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 <= -2000000.0) || ~((x <= 1000.0)))
		tmp = (1.0 / x) - (x ^ -3.0);
	else
		tmp = x / (1.0 + (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := If[Or[LessEqual[x, -2000000.0], N[Not[LessEqual[x, 1000.0]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	76.7%
Target	99.9%
Herbie	100.0%

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

Derivation

1. Split input into 2 regimes

2. if x < -2e6 or 1e3 < x

   1. Initial program 54.1%

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

   2. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right)}} \]
      Step-by-step derivation

      [Start] 54.1%	\[ \frac{x}{x \cdot x + 1} \]
      *-un-lft-identity [=>] 54.1%	\[ \frac{\color{blue}{1 \cdot x}}{x \cdot x + 1} \]
      add-sqr-sqrt [=>] 54.1%	\[ \frac{1 \cdot x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}} \]
      times-frac [=>] 54.3%	\[ \color{blue}{\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}} \]
      +-commutative [=>] 54.3%	\[ \frac{1}{\sqrt{\color{blue}{1 + x \cdot x}}} \cdot \frac{x}{\sqrt{x \cdot x + 1}} \]
      hypot-1-def [=>] 54.3%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{x}{\sqrt{x \cdot x + 1}} \]
      +-commutative [=>] 54.3%	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\sqrt{\color{blue}{1 + x \cdot x}}} \]
      hypot-1-def [=>] 100.0%	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\color{blue}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Taylor expanded in x around inf 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - {x}^{-3}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \frac{1}{x} - \frac{1}{{x}^{3}} \]
      exp-to-pow [<=] 44.0%	\[ \frac{1}{x} - \frac{1}{\color{blue}{e^{\log x \cdot 3}}} \]
      *-commutative [<=] 44.0%	\[ \frac{1}{x} - \frac{1}{e^{\color{blue}{3 \cdot \log x}}} \]
      exp-neg [<=] 44.0%	\[ \frac{1}{x} - \color{blue}{e^{-3 \cdot \log x}} \]
      distribute-lft-neg-in [=>] 44.0%	\[ \frac{1}{x} - e^{\color{blue}{\left(-3\right) \cdot \log x}} \]
      metadata-eval [=>] 44.0%	\[ \frac{1}{x} - e^{\color{blue}{-3} \cdot \log x} \]
      *-commutative [<=] 44.0%	\[ \frac{1}{x} - e^{\color{blue}{\log x \cdot -3}} \]
      exp-to-pow [=>] 100.0%	\[ \frac{1}{x} - \color{blue}{{x}^{-3}} \]

   if -2e6 < x < 1e3

   1. Initial program 100.0%

      \[\frac{x}{x \cdot x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000000 \lor \neg \left(x \leq 1000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -2000000 \lor \neg \left(x \leq 1000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	13376

\[\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right)} \]

Alternative 3
Accuracy	100.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 120000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4
Accuracy	99.0%
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 5
Accuracy	51.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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