sqrt B (should all be same)

Average Error: 30.3 → 0.4

Time: 2.2s

Precision: binary64

Cost: 19780

Math

\[\sqrt{\left(2 \cdot x\right) \cdot x} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;{2}^{0.25} \cdot \left(x \cdot \left(-\sqrt{\sqrt{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (* (pow 2.0 0.25) (* x (- (sqrt (sqrt 2.0)))))
   (* (sqrt (* x 2.0)) (sqrt x))))

C

double code(double x) {
	return sqrt(((2.0 * x) * x));
}

↓

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = pow(2.0, 0.25) * (x * -sqrt(sqrt(2.0)));
	} else {
		tmp = sqrt((x * 2.0)) * sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((2.0d0 * x) * x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4d-310)) then
        tmp = (2.0d0 ** 0.25d0) * (x * -sqrt(sqrt(2.0d0)))
    else
        tmp = sqrt((x * 2.0d0)) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return Math.sqrt(((2.0 * x) * x));
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = Math.pow(2.0, 0.25) * (x * -Math.sqrt(Math.sqrt(2.0)));
	} else {
		tmp = Math.sqrt((x * 2.0)) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	return math.sqrt(((2.0 * x) * x))

↓

def code(x):
	tmp = 0
	if x <= -4e-310:
		tmp = math.pow(2.0, 0.25) * (x * -math.sqrt(math.sqrt(2.0)))
	else:
		tmp = math.sqrt((x * 2.0)) * math.sqrt(x)
	return tmp

Julia

function code(x)
	return sqrt(Float64(Float64(2.0 * x) * x))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64((2.0 ^ 0.25) * Float64(x * Float64(-sqrt(sqrt(2.0)))));
	else
		tmp = Float64(sqrt(Float64(x * 2.0)) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((2.0 * x) * x));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = (2.0 ^ 0.25) * (x * -sqrt(sqrt(2.0)));
	else
		tmp = sqrt((x * 2.0)) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -4e-310], N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(x * (-N[Sqrt[N[Sqrt[2.0], $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 30.7

      \[\sqrt{\left(2 \cdot x\right) \cdot x} \]

   2. Applied egg-rr 64.0

      \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{2} \cdot x}\right)}^{2}} \]

   3. Applied egg-rr 62.6

      \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]

   4. Applied egg-rr 30.9

      \[\leadsto {2}^{0.25} \cdot \color{blue}{\sqrt{x \cdot \left(x \cdot \sqrt{2}\right)}} \]

   5. Simplified 30.8

      \[\leadsto {2}^{0.25} \cdot \color{blue}{\sqrt{\sqrt{2} \cdot \left(x \cdot x\right)}} \]
      Proof
      (*.f64 (pow.f64 2 1/4) (sqrt.f64 (*.f64 (sqrt.f64 2) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 2 1/4) (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 x x) (sqrt.f64 2))))): 3 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 2 1/4) (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 x (*.f64 x (sqrt.f64 2)))))): 3 points increase in error, 0 points decrease in error

   6. Taylor expanded in x around -inf 0.4

      \[\leadsto {2}^{0.25} \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\sqrt{2}} \cdot x\right)\right)} \]

   7. Simplified 0.4

      \[\leadsto {2}^{0.25} \cdot \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left(-x\right)\right)} \]
      Proof
      (*.f64 (pow.f64 2 1/4) (*.f64 (sqrt.f64 (sqrt.f64 2)) (neg.f64 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 2 1/4) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (sqrt.f64 2)) x)))): 3 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 2 1/4) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (sqrt.f64 (sqrt.f64 2)) x)))): 0 points increase in error, 3 points decrease in error

   if -3.999999999999988e-310 < x

   1. Initial program 29.9

      \[\sqrt{\left(2 \cdot x\right) \cdot x} \]

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;{2}^{0.25} \cdot \left(x \cdot \left(-\sqrt{\sqrt{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

Alternative 3
Error	31.7
Cost	6592

\[x \cdot \sqrt{2} \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))