x / (x^2 + 1)

Average Accuracy: 77.1% → 100.0%

Time: 2.7s

Precision: binary64

Cost: 7432

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{-0.5}{x \cdot x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e+47)
   (/ 1.0 x)
   (if (<= x 1000.0)
     (/ x (+ 1.0 (* x x)))
     (* (/ 1.0 (hypot 1.0 x)) (+ 1.0 (/ -0.5 (* x x)))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -2e+47) {
		tmp = 1.0 / x;
	} else if (x <= 1000.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = (1.0 / hypot(1.0, x)) * (1.0 + (-0.5 / (x * x)));
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -2e+47) {
		tmp = 1.0 / x;
	} else if (x <= 1000.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = (1.0 / Math.hypot(1.0, x)) * (1.0 + (-0.5 / (x * x)));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -2e+47:
		tmp = 1.0 / x
	elif x <= 1000.0:
		tmp = x / (1.0 + (x * x))
	else:
		tmp = (1.0 / math.hypot(1.0, x)) * (1.0 + (-0.5 / (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 <= -2e+47)
		tmp = Float64(1.0 / x);
	elseif (x <= 1000.0)
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, x)) * Float64(1.0 + Float64(-0.5 / 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 <= -2e+47)
		tmp = 1.0 / x;
	elseif (x <= 1000.0)
		tmp = x / (1.0 + (x * x));
	else
		tmp = (1.0 / hypot(1.0, x)) * (1.0 + (-0.5 / (x * 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+47], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1000.0], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	77.1%
Target	99.8%
Herbie	100.0%

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

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e47

   1. Initial program 44.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -2.0000000000000001e47 < x < 1e3

   1. Initial program 100.0%

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

   if 1e3 < x

   1. Initial program 54.5%

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

   2. Applied egg-rr 100.0%

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

      [Start] 54.5	\[ \frac{x}{x \cdot x + 1} \]
      *-un-lft-identity [=>] 54.5	\[ \frac{\color{blue}{1 \cdot x}}{x \cdot x + 1} \]
      add-sqr-sqrt [=>] 54.5	\[ \frac{1 \cdot x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}} \]
      times-frac [=>] 54.7	\[ \color{blue}{\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}} \]
      +-commutative [=>] 54.7	\[ \frac{1}{\sqrt{\color{blue}{1 + x \cdot x}}} \cdot \frac{x}{\sqrt{x \cdot x + 1}} \]
      hypot-1-def [=>] 54.7	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{x}{\sqrt{x \cdot x + 1}} \]
      +-commutative [=>] 54.7	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\sqrt{\color{blue}{1 + x \cdot x}}} \]
      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%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \color{blue}{\left(1 - 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]

   4. Simplified 100.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \color{blue}{\left(1 - \frac{0.5}{x \cdot x}\right)} \]
      Proof

      [Start] 100.0	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 - 0.5 \cdot \frac{1}{{x}^{2}}\right) \]
      associate-*r/ [=>] 100.0	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 - \frac{\color{blue}{0.5}}{{x}^{2}}\right) \]
      unpow2 [=>] 100.0	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 - \frac{0.5}{\color{blue}{x \cdot x}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{-0.5}{x \cdot x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13376

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

Alternative 2
Accuracy	100.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 500000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Accuracy	98.8%
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
Accuracy	51.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023126 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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