sqrt E (should all be same)

Average Error: 30.2 → 0.4

Time: 9.6s

Precision: binary64

Cost: 19716

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\frac{x \cdot \sqrt[3]{\sqrt{2}}}{\sqrt[3]{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (- (/ (* x (cbrt (sqrt 2.0))) (cbrt 0.5)))
   (* (sqrt 2.0) x)))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -((x * cbrt(sqrt(2.0))) / cbrt(0.5));
	} else {
		tmp = sqrt(2.0) * x;
	}
	return tmp;
}

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 tmp;
	if (x <= -5e-310) {
		tmp = -((x * Math.cbrt(Math.sqrt(2.0))) / Math.cbrt(0.5));
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(-Float64(Float64(x * cbrt(sqrt(2.0))) / cbrt(0.5)));
	else
		tmp = Float64(sqrt(2.0) * x);
	end
	return tmp
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -5e-310], (-N[(N[(x * N[Power[N[Sqrt[2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[0.5, 1/3], $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 30.3

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

   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

   4. Applied egg-rr 0.3

      \[\leadsto -\color{blue}{\frac{x \cdot \sqrt[3]{\sqrt{2}}}{\sqrt[3]{0.5}}} \]

   if -4.999999999999985e-310 < x

   1. Initial program 30.1

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

   2. Taylor expanded in x around 0 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	6788

\[\begin{array}{l} t_0 := \sqrt{2} \cdot x\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.4
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\frac{x}{\sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \]

Alternative 3
Error	31.1
Cost	6592

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

Alternative 4
Error	61.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))