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

Average Error: 38.7 → 6.6

Time: 19.1s

Precision: binary64

Cost: 20356

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(\begin{array}{l} \mathbf{if}\;\frac{1}{re} \ne 0:\\ \;\;\;\;\frac{1}{{\left(\frac{1}{re}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}}\\ \end{array} \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(re - \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (*
    0.5
    (*
     (if (!= (/ 1.0 re) 0.0) (/ 1.0 (pow (/ 1.0 re) -0.5)) (sqrt (/ 1.0 re)))
     im))
   (* 0.5 (sqrt (* -2.0 (- re (hypot re im)))))))

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 tmp_1;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		double tmp_2;
		if ((1.0 / re) != 0.0) {
			tmp_2 = 1.0 / pow((1.0 / re), -0.5);
		} else {
			tmp_2 = sqrt((1.0 / re));
		}
		tmp_1 = 0.5 * (tmp_2 * im);
	} else {
		tmp_1 = 0.5 * sqrt((-2.0 * (re - hypot(re, im))));
	}
	return tmp_1;
}

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 tmp_1;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		double tmp_2;
		if ((1.0 / re) != 0.0) {
			tmp_2 = 1.0 / Math.pow((1.0 / re), -0.5);
		} else {
			tmp_2 = Math.sqrt((1.0 / re));
		}
		tmp_1 = 0.5 * (tmp_2 * im);
	} else {
		tmp_1 = 0.5 * Math.sqrt((-2.0 * (re - Math.hypot(re, im))));
	}
	return tmp_1;
}

Python

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

↓

def code(re, im):
	tmp_1 = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp_2 = 0
		if (1.0 / re) != 0.0:
			tmp_2 = 1.0 / math.pow((1.0 / re), -0.5)
		else:
			tmp_2 = math.sqrt((1.0 / re))
		tmp_1 = 0.5 * (tmp_2 * im)
	else:
		tmp_1 = 0.5 * math.sqrt((-2.0 * (re - math.hypot(re, im))))
	return tmp_1

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)
	tmp_1 = 0.0
	if (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp_2 = 0.0
		if (Float64(1.0 / re) != 0.0)
			tmp_2 = Float64(1.0 / (Float64(1.0 / re) ^ -0.5));
		else
			tmp_2 = sqrt(Float64(1.0 / re));
		end
		tmp_1 = Float64(0.5 * Float64(tmp_2 * im));
	else
		tmp_1 = Float64(0.5 * sqrt(Float64(-2.0 * Float64(re - hypot(re, im)))));
	end
	return tmp_1
end

MATLAB

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

↓

function tmp_4 = code(re, im)
	tmp_2 = 0.0;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0)
		tmp_3 = 0.0;
		if ((1.0 / re) ~= 0.0)
			tmp_3 = 1.0 / ((1.0 / re) ^ -0.5);
		else
			tmp_3 = sqrt((1.0 / re));
		end
		tmp_2 = 0.5 * (tmp_3 * im);
	else
		tmp_2 = 0.5 * sqrt((-2.0 * (re - hypot(re, im))));
	end
	tmp_4 = tmp_2;
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_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(If[Unequal[N[(1.0 / re), $MachinePrecision], 0.0], N[(1.0 / N[Power[N[(1.0 / re), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]] * im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(-2.0 * N[(re - N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 58.3

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

   2. Simplified 51.5

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

   3. Taylor expanded in re around inf 37.4

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

   4. Simplified 37.5

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

   5. Taylor expanded in im around 0 6.0

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

   6. Applied egg-rr 6.1

      \[\leadsto 0.5 \cdot \left(\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{1}{re} \ne 0:\\ \;\;\;\;\frac{1}{{\left(\frac{1}{re}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}}\\ } \end{array}} \cdot im\right) \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 35.2

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

   2. Simplified 6.7

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{-2 \cdot \left(re - \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	15.7
Cost	7564

\[\begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(2 \cdot re\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\begin{array}{l} \mathbf{if}\;\frac{1}{re} \ne 0:\\ \;\;\;\;\frac{1}{{\left(\frac{1}{re}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}}\\ \end{array} \cdot im\right)\\ \end{array} \]

Alternative 2
Error	18.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(2 \cdot re\right)}\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{re}}\\ \end{array} \]

Alternative 3
Error	15.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(2 \cdot re\right)}\\ \mathbf{elif}\;re \leq 7.6 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\\ \end{array} \]

Alternative 4
Error	15.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(2 \cdot re\right)}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-109}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{re}}\right) \cdot im\\ \end{array} \]

Alternative 5
Error	23.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot \left(2 \cdot re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6
Error	64.0
Cost	6720

\[0.5 \cdot \sqrt{-2 \cdot im} \]

Alternative 7
Error	59.8
Cost	6720

\[0.5 \cdot \sqrt{-2 \cdot re} \]

Alternative 8
Error	30.4
Cost	6720

\[0.5 \cdot \sqrt{2 \cdot im} \]

Reproduce

herbie shell --seed 2023010 
(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)))))