Average Error: 0.5 → 0.3

Time: 3.6s

Precision: binary64

Cost: 832

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

↓

\[x - \left(\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)\right)\]

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

↓

x - \left(\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)\right)

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

↓

(FPCore (x)
 :precision binary64
 (- x (+ (/ 0.125 x) (+ 0.5 (/ 0.0625 (* x x))))))

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

↓

double code(double x) {
	return x - ((0.125 / x) + (0.5 + (0.0625 / (x * x))));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	1.3
Cost	58688

\[\sqrt[3]{\sqrt{x - 1} \cdot \sqrt{x}} \cdot \left(\sqrt[3]{\sqrt{x - 1} \cdot \sqrt{x}} \cdot \sqrt[3]{\sqrt{x - 1} \cdot \sqrt{x}}\right)\]

Alternative 2
Error	1.3
Cost	45632

\[\left(\sqrt[3]{\sqrt{x - 1}} \cdot \sqrt[3]{\sqrt{x - 1}}\right) \cdot \left(\sqrt{x} \cdot \sqrt[3]{\sqrt{x - 1}}\right)\]

Alternative 3
Error	1.3
Cost	45376

\[\sqrt[3]{\sqrt{x}} \cdot \left(\sqrt{x - 1} \cdot \left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right)\right)\]

Alternative 4
Error	0.9
Cost	39232

\[\sqrt{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt[3]{x - 1}}\right)\]

Alternative 5
Error	0.7
Cost	32576

\[\sqrt{\sqrt{x - 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt{x - 1}}\right)\]

Alternative 6
Error	0.9
Cost	32448

\[\left(\sqrt{x - 1} \cdot \left|\sqrt[3]{x}\right|\right) \cdot \sqrt{\sqrt[3]{x}}\]

Alternative 7
Error	0.7
Cost	32448

\[\sqrt{\sqrt{x}} \cdot \left(\sqrt{x - 1} \cdot \sqrt{\sqrt{x}}\right)\]

Alternative 8
Error	5.8
Cost	25920

\[e^{\log \left(\sqrt{x - 1} \cdot \sqrt{x}\right)}\]

Alternative 9
Error	0.5
Cost	13120

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

Alternative 10
Error	32.1
Cost	6784

\[{\left(x \cdot \left(x - 1\right)\right)}^{0.5}\]

Alternative 11
Error	32.1
Cost	6720

\[\sqrt{x \cdot \left(x - 1\right)}\]

Alternative 12
Error	0.4
Cost	448

\[x - \left(0.5 + \frac{0.125}{x}\right)\]

Alternative 13
Error	0.6
Cost	192

\[x - 0.5\]

Alternative 14
Error	1.3
Cost	64

\[x\]

Alternative 15
Error	60.6
Cost	64

\[1\]

Alternative 16
Error	62.6
Cost	64

\[0\]

Alternative 17
Error	63.3
Cost	64

\[-1\]

Error

Derivation

Initial program 0.5
\[\sqrt{x - 1} \cdot \sqrt{x}\]
Taylor expanded around inf 0.3
\[\leadsto \color{blue}{x - \left(0.125 \cdot \frac{1}{x} + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.5\right)\right)}\]
Simplified0.3
\[\leadsto \color{blue}{x - \left(\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)\right)}\]
Simplified0.3
\[\leadsto \color{blue}{x - \left(\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)\right)}\]
Final simplification0.3
\[\leadsto x - \left(\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)\right)\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))