Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3

Average Accuracy: 100.0% → 100.0%

Time: 2.1s

Precision: binary64

Cost: 13184

Math

\[\sqrt{1 - x \cdot x} \]

↓

\[e^{\mathsf{log1p}\left(-x \cdot x\right) \cdot 0.5} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (- 1.0 (* x x))))

↓

(FPCore (x) :precision binary64 (exp (* (log1p (- (* x x))) 0.5)))

C

double code(double x) {
	return sqrt((1.0 - (x * x)));
}

↓

double code(double x) {
	return exp((log1p(-(x * x)) * 0.5));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 - (x * x)));
}

↓

public static double code(double x) {
	return Math.exp((Math.log1p(-(x * x)) * 0.5));
}

Python

def code(x):
	return math.sqrt((1.0 - (x * x)))

↓

def code(x):
	return math.exp((math.log1p(-(x * x)) * 0.5))

Julia

function code(x)
	return sqrt(Float64(1.0 - Float64(x * x)))
end

↓

function code(x)
	return exp(Float64(log1p(Float64(-Float64(x * x))) * 0.5))
end

Wolfram

code[x_] := N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := N[Exp[N[(N[Log[1 + (-N[(x * x), $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{1 - x \cdot x} \]

2. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(-x\right)\right) \cdot 0.5}} \]
   Proof

   [Start] 100.0	\[ \sqrt{1 - x \cdot x} \]
   pow1/2 [=>] 100.0	\[ \color{blue}{{\left(1 - x \cdot x\right)}^{0.5}} \]
   pow-to-exp [=>] 100.0	\[ \color{blue}{e^{\log \left(1 - x \cdot x\right) \cdot 0.5}} \]
   sub-neg [=>] 100.0	\[ e^{\log \color{blue}{\left(1 + \left(-x \cdot x\right)\right)} \cdot 0.5} \]
   log1p-def [=>] 100.0	\[ e^{\color{blue}{\mathsf{log1p}\left(-x \cdot x\right)} \cdot 0.5} \]
   distribute-rgt-neg-in [=>] 100.0	\[ e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(-x\right)}\right) \cdot 0.5} \]

3. Final simplification 100.0%

   \[\leadsto e^{\mathsf{log1p}\left(-x \cdot x\right) \cdot 0.5} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6720

\[\sqrt{1 - x \cdot x} \]

Alternative 2
Accuracy	99.4%
Cost	448

\[1 + \left(x \cdot x\right) \cdot -0.5 \]

Alternative 3
Accuracy	98.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023136 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))