sqrt times

Average Accuracy: 99.2% → 99.5%

Time: 2.3s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x - 1.0d0)) * sqrt(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((-0.5d0) + (((-0.125d0) / x) + ((-0.0625d0) / (x * x))))
end function

Java

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

↓

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

Python

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

↓

def code(x):
	return x + (-0.5 + ((-0.125 / x) + (-0.0625 / (x * x))))

Julia

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

↓

function code(x)
	return Float64(x + Float64(-0.5 + Float64(Float64(-0.125 / x) + Float64(-0.0625 / Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x - 1.0)) * sqrt(x);
end

↓

function tmp = code(x)
	tmp = x + (-0.5 + ((-0.125 / x) + (-0.0625 / (x * x))));
end

Wolfram

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

↓

code[x_] := N[(x + N[(-0.5 + N[(N[(-0.125 / x), $MachinePrecision] + N[(-0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 99.5%

   \[\leadsto \color{blue}{x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)} \]

3. Simplified 99.5%

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

   [Start] 99.5	\[ x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   +-commutative [=>] 99.5	\[ x - \left(0.5 + \color{blue}{\left(0.125 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)}\right) \]
   associate-*r/ [=>] 99.5	\[ x - \left(0.5 + \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]
   metadata-eval [=>] 99.5	\[ x - \left(0.5 + \left(\frac{\color{blue}{0.125}}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]
   associate-*r/ [=>] 99.5	\[ x - \left(0.5 + \left(\frac{0.125}{x} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right)\right) \]
   metadata-eval [=>] 99.5	\[ x - \left(0.5 + \left(\frac{0.125}{x} + \frac{\color{blue}{0.0625}}{{x}^{2}}\right)\right) \]
   unpow2 [=>] 99.5	\[ x - \left(0.5 + \left(\frac{0.125}{x} + \frac{0.0625}{\color{blue}{x \cdot x}}\right)\right) \]

4. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	448

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

Alternative 2
Accuracy	99.1%
Cost	192

\[x + -0.5 \]

Alternative 3
Accuracy	98.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))