sqrt E (should all be same)

Average Error: 30.7 → 0.4

Time: 1.9s

Precision: binary64

Cost: 6788

Math

\[\sqrt{{x}^{2} + {x}^{2}} \]

↓

\[\begin{array}{l} t_0 := \sqrt{2} \cdot x\\ \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) x))) (if (<= x -1e-310) (- t_0) t_0)))

C

double code(double x) {
	return sqrt((pow(x, 2.0) + pow(x, 2.0)));
}

↓

double code(double x) {
	double t_0 = sqrt(2.0) * x;
	double tmp;
	if (x <= -1e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * x
    if (x <= (-1d-310)) then
        tmp = -t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(x, 2.0) + Math.pow(x, 2.0)));
}

↓

public static double code(double x) {
	double t_0 = Math.sqrt(2.0) * x;
	double tmp;
	if (x <= -1e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	return math.sqrt((math.pow(x, 2.0) + math.pow(x, 2.0)))

↓

def code(x):
	t_0 = math.sqrt(2.0) * x
	tmp = 0
	if x <= -1e-310:
		tmp = -t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	return sqrt(Float64((x ^ 2.0) + (x ^ 2.0)))
end

↓

function code(x)
	t_0 = Float64(sqrt(2.0) * x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(-t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((x ^ 2.0) + (x ^ 2.0)));
end

↓

function tmp_2 = code(x)
	t_0 = sqrt(2.0) * x;
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = -t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1e-310], (-t$95$0), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 30.8

      \[\sqrt{{x}^{2} + {x}^{2}} \]

   2. Taylor expanded in x around -inf 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ -1 \cdot \left(\sqrt{2} \cdot x\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ \color{blue}{\left(\sqrt{2} \cdot x\right) \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.4	\[ \color{blue}{-\sqrt{2} \cdot x} \]

   if -9.999999999999969e-311 < x

   1. Initial program 30.5

      \[\sqrt{{x}^{2} + {x}^{2}} \]

   2. Taylor expanded in x around 0 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	31.8
Cost	6592

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

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))