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

Average Accuracy: 100.0% → 100.0%

Time: 2.9s

Precision: binary64

Cost: 13056

Math

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

↓

\[\sqrt{-\mathsf{fma}\left(x, x, -1\right)} \]

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

function code(x)
	return sqrt(Float64(-fma(x, x, -1.0)))
end

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{fma}\left(x, x, -1\right)}} \]
   Proof

   [Start] 100.0	\[ \sqrt{1 - x \cdot x} \]
   sub-neg [=>] 100.0	\[ \sqrt{\color{blue}{1 + \left(-x \cdot x\right)}} \]
   +-commutative [=>] 100.0	\[ \sqrt{\color{blue}{\left(-x \cdot x\right) + 1}} \]
   neg-sub0 [=>] 100.0	\[ \sqrt{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]
   associate-+l- [=>] 100.0	\[ \sqrt{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]
   sub0-neg [=>] 100.0	\[ \sqrt{\color{blue}{-\left(x \cdot x - 1\right)}} \]
   fma-neg [=>] 100.0	\[ \sqrt{-\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]
   metadata-eval [=>] 100.0	\[ \sqrt{-\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

3. Final simplification 100.0%

   \[\leadsto \sqrt{-\mathsf{fma}\left(x, x, -1\right)} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6720

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

Alternative 2
Accuracy	99.5%
Cost	448

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

Alternative 3
Accuracy	1.6%
Cost	64

\[-1 \]

Alternative 4
Accuracy	99.0%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))