sqrt times

Average Error: 0.5 → 0.3

Time: 2.3s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (+ x (+ -0.5 (/ (+ -0.125 (/ -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 + (-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) + ((-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 + (-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 + (-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 + Float64(-0.0625 / 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 + (-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 + N[(-0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

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

2. Simplified 0.5

   \[\leadsto \color{blue}{\sqrt{x + -1} \cdot \sqrt{x}} \]

3. Taylor expanded in x around inf 0.3

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

4. Simplified 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	448

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

Alternative 2
Error	0.6
Cost	192

\[x + -0.5 \]

Alternative 3
Error	1.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022225 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))