sqrt times

Average Error: 0.5 → 0.4

Time: 1.4s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return -0.5 + (x + (-0.125 / 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 = (-0.5d0) + (x + ((-0.125d0) / 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 -0.5 + (x + (-0.125 / x));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 0.5

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

2. Taylor expanded in x around inf 0.4

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

3. Simplified 0.4

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

4. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022166 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))