math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.7 → 7.9

Time: 9.6s

Precision: binary64

Cost: 13708

\[im > 0\]

Math

\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{if}\;re \leq 1.1991846674608896 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 5.721386932251573 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 28024986079.47817:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))
   (if (<= re 1.1991846674608896e-72)
     t_0
     (if (<= re 5.721386932251573e-48)
       (* 0.5 (* im (sqrt (/ 1.0 re))))
       (if (<= re 28024986079.47817) t_0 (* 0.5 (/ im (sqrt re))))))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

↓

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	double tmp;
	if (re <= 1.1991846674608896e-72) {
		tmp = t_0;
	} else if (re <= 5.721386932251573e-48) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else if (re <= 28024986079.47817) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

↓

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	double tmp;
	if (re <= 1.1991846674608896e-72) {
		tmp = t_0;
	} else if (re <= 5.721386932251573e-48) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else if (re <= 28024986079.47817) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

↓

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	tmp = 0
	if re <= 1.1991846674608896e-72:
		tmp = t_0
	elif re <= 5.721386932251573e-48:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	elif re <= 28024986079.47817:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

↓

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))))
	tmp = 0.0
	if (re <= 1.1991846674608896e-72)
		tmp = t_0;
	elseif (re <= 5.721386932251573e-48)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	elseif (re <= 28024986079.47817)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

↓

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	tmp = 0.0;
	if (re <= 1.1991846674608896e-72)
		tmp = t_0;
	elseif (re <= 5.721386932251573e-48)
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	elseif (re <= 28024986079.47817)
		tmp = t_0;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.1991846674608896e-72], t$95$0, If[LessEqual[re, 5.721386932251573e-48], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 28024986079.47817], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.19918466746088959e-72 or 5.72138693225157319e-48 < re < 28024986079.4781685

   1. Initial program 32.3

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Simplified 5.3

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   if 1.19918466746088959e-72 < re < 5.72138693225157319e-48

   1. Initial program 40.0

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Simplified 26.4

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   3. Taylor expanded in re around inf 47.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]

   4. Applied egg-rr 34.0

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]

   if 28024986079.4781685 < re

   1. Initial program 57.7

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Simplified 39.7

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   3. Taylor expanded in re around inf 34.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]

   4. Applied egg-rr 13.5

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.1991846674608896 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 5.721386932251573 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 28024986079.47817:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.6
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -5.072551663792424 \cdot 10^{+76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.1991846674608896 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 5.721386932251573 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 28024986079.47817:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 2
Error	21.2
Cost	7244

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq 1.1991846674608896 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 5.721386932251573 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 28024986079.47817:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3
Error	23.3
Cost	7116

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{if}\;re \leq 1.1991846674608896 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 5.721386932251573 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 28024986079.47817:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	23.3
Cost	7116

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq 1.1991846674608896 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 5.721386932251573 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 28024986079.47817:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5
Error	46.3
Cost	6720

\[0.5 \cdot \frac{im}{\sqrt{re}} \]

Reproduce

herbie shell --seed 2022284 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))