sqrt A (should all be same)

Average Error: 30.7 → 0.4

Time: 7.5s

Precision: binary64

Cost: 13316

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{x \cdot -2} \cdot \sqrt{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + x} \cdot \sqrt{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (sqrt (* x -2.0)) (sqrt (- x)))
   (* (sqrt (+ x x)) (sqrt x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = sqrt((x * -2.0)) * sqrt(-x);
	} else {
		tmp = sqrt((x + x)) * sqrt(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 <= (-2d-310)) then
        tmp = sqrt((x * (-2.0d0))) * sqrt(-x)
    else
        tmp = sqrt((x + x)) * sqrt(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 <= -2e-310) {
		tmp = Math.sqrt((x * -2.0)) * Math.sqrt(-x);
	} else {
		tmp = Math.sqrt((x + x)) * Math.sqrt(x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = math.sqrt((x * -2.0)) * math.sqrt(-x)
	else:
		tmp = math.sqrt((x + x)) * math.sqrt(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 <= -2e-310)
		tmp = Float64(sqrt(Float64(x * -2.0)) * sqrt(Float64(-x)));
	else
		tmp = Float64(sqrt(Float64(x + x)) * sqrt(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 <= -2e-310)
		tmp = sqrt((x * -2.0)) * sqrt(-x);
	else
		tmp = sqrt((x + x)) * sqrt(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, -2e-310], N[(N[Sqrt[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 31.0

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

   2. Simplified 31.0

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

      [Start] 31.0	\[ \sqrt{x \cdot x + x \cdot x} \]
      rational_best-simplify-63 [=>] 31.0	\[ \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]

   3. Applied egg-rr 0.4

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

   if -1.999999999999994e-310 < x

   1. Initial program 30.3

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

   2. Simplified 30.3

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

      [Start] 30.3	\[ \sqrt{x \cdot x + x \cdot x} \]
      rational_best-simplify-63 [=>] 30.3	\[ \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]

   3. Applied egg-rr 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{x \cdot -2} \cdot \sqrt{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + x} \cdot \sqrt{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13252

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

Alternative 2
Error	0.4
Cost	6788

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

Alternative 3
Error	31.6
Cost	6592

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

Reproduce

herbie shell --seed 2023100 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))