sqrt times

Average Error: 0.5 → 0.2

Time: 3.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 + ((((-0.125d0) + ((-0.0625d0) / x)) / x) + (-0.5d0))
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 + (((-0.125 + (-0.0625 / x)) / x) + -0.5);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.5

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

2. Taylor expanded in x around inf 0.2

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

3. Simplified 0.2

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

   [Start] 0.2	\[ x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   associate--r+ [=>] 0.2	\[ \color{blue}{\left(x - 0.5\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)} \]
   sub-neg [=>] 0.2	\[ \color{blue}{\left(x + \left(-0.5\right)\right)} - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]
   associate--l+ [=>] 0.2	\[ \color{blue}{x + \left(\left(-0.5\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)} \]
   metadata-eval [=>] 0.2	\[ x + \left(\color{blue}{-0.5} - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   metadata-eval [<=] 0.2	\[ x + \left(\color{blue}{0.5 \cdot -1} - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   rem-square-sqrt [<=] 64.0	\[ x + \left(0.5 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   unpow2 [<=] 64.0	\[ x + \left(0.5 \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}} - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   +-commutative [=>] 64.0	\[ x + \left(0.5 \cdot {\left(\sqrt{-1}\right)}^{2} - \color{blue}{\left(0.125 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)}\right) \]
   associate--r+ [=>] 64.0	\[ x + \color{blue}{\left(\left(0.5 \cdot {\left(\sqrt{-1}\right)}^{2} - 0.125 \cdot \frac{1}{x}\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]
   cancel-sign-sub-inv [=>] 64.0	\[ x + \left(\color{blue}{\left(0.5 \cdot {\left(\sqrt{-1}\right)}^{2} + \left(-0.125\right) \cdot \frac{1}{x}\right)} - 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   unpow2 [=>] 64.0	\[ x + \left(\left(0.5 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} + \left(-0.125\right) \cdot \frac{1}{x}\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   rem-square-sqrt [=>] 0.2	\[ x + \left(\left(0.5 \cdot \color{blue}{-1} + \left(-0.125\right) \cdot \frac{1}{x}\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   metadata-eval [=>] 0.2	\[ x + \left(\left(\color{blue}{-0.5} + \left(-0.125\right) \cdot \frac{1}{x}\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   metadata-eval [=>] 0.2	\[ x + \left(\left(-0.5 + \color{blue}{-0.125} \cdot \frac{1}{x}\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   associate-*r/ [=>] 0.2	\[ x + \left(\left(-0.5 + \color{blue}{\frac{-0.125 \cdot 1}{x}}\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   metadata-eval [=>] 0.2	\[ x + \left(\left(-0.5 + \frac{\color{blue}{-0.125}}{x}\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   associate-*r/ [=>] 0.2	\[ x + \left(\left(-0.5 + \frac{-0.125}{x}\right) - \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right) \]
   metadata-eval [=>] 0.2	\[ x + \left(\left(-0.5 + \frac{-0.125}{x}\right) - \frac{\color{blue}{0.0625}}{{x}^{2}}\right) \]
   unpow2 [=>] 0.2	\[ x + \left(\left(-0.5 + \frac{-0.125}{x}\right) - \frac{0.0625}{\color{blue}{x \cdot x}}\right) \]

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	448

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

Alternative 2
Error	0.5
Cost	192

\[x + -0.5 \]

Alternative 3
Error	1.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))