x / (x^2 + 1)

Average Accuracy: 76.9% → 100.0%

Time: 4.3s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

public static double code(double x) {
	return (1.0 / Math.hypot(1.0, x)) * (x / Math.hypot(1.0, x));
}

Python

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

↓

def code(x):
	return (1.0 / math.hypot(1.0, x)) * (x / math.hypot(1.0, x))

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

↓

function code(x)
	return Float64(Float64(1.0 / hypot(1.0, x)) * Float64(x / hypot(1.0, x)))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

↓

function tmp = code(x)
	tmp = (1.0 / hypot(1.0, x)) * (x / hypot(1.0, x));
end

Wolfram

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

↓

code[x_] := N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	76.9%
Target	99.9%
Herbie	100.0%

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

Derivation

1. Initial program 76.9%

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

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] 76.9	\[ \frac{x}{x \cdot x + 1} \]
   *-un-lft-identity [=>] 76.9	\[ \frac{\color{blue}{1 \cdot x}}{x \cdot x + 1} \]
   add-sqr-sqrt [=>] 76.9	\[ \frac{1 \cdot x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}} \]
   times-frac [=>] 77.0	\[ \color{blue}{\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}} \]
   +-commutative [=>] 77.0	\[ \frac{1}{\sqrt{\color{blue}{1 + x \cdot x}}} \cdot \frac{x}{\sqrt{x \cdot x + 1}} \]
   hypot-1-def [=>] 77.0	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{x}{\sqrt{x \cdot x + 1}} \]
   +-commutative [=>] 77.0	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\sqrt{\color{blue}{1 + x \cdot x}}} \]
   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. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.86:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
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 3
Accuracy	99.9%
Cost	448

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

Alternative 4
Accuracy	51.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023144 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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