x / (x^2 + 1)

Average Accuracy: 76.7% → 100.0%

Time: 2.9s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	76.7%
Target	99.8%
Herbie	100.0%

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

Derivation

1. Initial program 76.7%

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

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] 76.7	\[ \frac{x}{x \cdot x + 1} \]
   *-un-lft-identity [=>] 76.7	\[ \frac{\color{blue}{1 \cdot x}}{x \cdot x + 1} \]
   add-sqr-sqrt [=>] 76.7	\[ \frac{1 \cdot x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}} \]
   times-frac [=>] 76.8	\[ \color{blue}{\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}} \]
   +-commutative [=>] 76.8	\[ \frac{1}{\sqrt{\color{blue}{1 + x \cdot x}}} \cdot \frac{x}{\sqrt{x \cdot x + 1}} \]
   hypot-1-def [=>] 76.8	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{x}{\sqrt{x \cdot x + 1}} \]
   +-commutative [=>] 76.8	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\sqrt{\color{blue}{1 + x \cdot x}}} \]
   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. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right)} \]
   associate-*l/ [=>] 100.0	\[ \color{blue}{\frac{1 \cdot \frac{x}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}} \]
   *-un-lft-identity [<=] 100.0	\[ \frac{\color{blue}{\frac{x}{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+16} \lor \neg \left(x \leq 400000000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 400000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

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	51.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023131 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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