sqrt A (should all be same)

Average Error: 30.8 → 0.4

Time: 7.2s

Precision: binary64

Cost: 7044

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot -0.25}{\frac{0.25}{-x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (sqrt 2.0) (- x))
   (/ (* (sqrt 2.0) -0.25) (/ 0.25 (- x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = sqrt(2.0) * -x;
	} else {
		tmp = (sqrt(2.0) * -0.25) / (0.25 / -x);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = sqrt(2.0d0) * -x
    else
        tmp = (sqrt(2.0d0) * (-0.25d0)) / (0.25d0 / -x)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = (Math.sqrt(2.0) * -0.25) / (0.25 / -x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = (math.sqrt(2.0) * -0.25) / (0.25 / -x)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	else
		tmp = Float64(Float64(sqrt(2.0) * -0.25) / Float64(0.25 / Float64(-x)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = (sqrt(2.0) * -0.25) / (0.25 / -x);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -5e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.25), $MachinePrecision] / N[(0.25 / (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 30.6

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

   2. Simplified 30.6

      \[\leadsto \color{blue}{\sqrt{x \cdot \left(x + x\right)}} \]
      Proof

      [Start] 30.6	\[ \sqrt{x \cdot x + x \cdot x} \]
      rational.json-simplify-44 [=>] 30.6	\[ \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]

   3. Taylor expanded in x around -inf 0.4

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

   4. Simplified 0.4

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

      [Start] 0.4	\[ -1 \cdot \left(\sqrt{2} \cdot x\right) \]
      rational.json-simplify-24 [=>] 0.4	\[ \color{blue}{\sqrt{2} \cdot \left(-1 \cdot x\right)} \]
      rational.json-simplify-50 [=>] 0.4	\[ \sqrt{2} \cdot \color{blue}{\left(x \cdot -1\right)} \]
      rational.json-simplify-38 [=>] 0.4	\[ \sqrt{2} \cdot \color{blue}{\left(-x\right)} \]

   if -4.999999999999985e-310 < x

   1. Initial program 30.9

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

   2. Simplified 30.9

      \[\leadsto \color{blue}{\sqrt{x \cdot \left(x + x\right)}} \]
      Proof

      [Start] 30.9	\[ \sqrt{x \cdot x + x \cdot x} \]
      rational.json-simplify-44 [=>] 30.9	\[ \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]

   3. Taylor expanded in x around 0 0.4

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

   4. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{1}{x}}} \]

   5. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-2} \cdot \left(x - x \cdot 3\right)} \]

   6. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot -0.25}{\frac{0.25}{-x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot -0.25}{\frac{0.25}{-x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{1}{x}}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \]

Alternative 3
Error	31.8
Cost	6592

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

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))