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

Percentage Accurate: 41.3% → 90.6%

Time: 14.1s

Precision: binary64

Cost: 20356

\[im > 0\]

Math

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

↓

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

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 (/ im (sqrt re)))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) 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 tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - 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 tmp;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - 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):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - 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)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - 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)
	tmp = 0.0;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0)
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - 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_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $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 7.0%

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

   2. Taylor expanded in im around 0 95.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

   3. Simplified 95.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
      Step-by-step derivation

      [Start] 95.8%	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      associate-*l* [=>] 95.7%	\[ 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\sqrt{0.5} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
      *-commutative [=>] 95.7%	\[ 0.5 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(im \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

   4. Applied egg-rr 44.8%

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

      [Start] 95.7%	\[ 0.5 \cdot \left(\sqrt{2} \cdot \left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)\right) \]
      *-commutative [=>] 95.7%	\[ 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(im \cdot \sqrt{0.5}\right)\right)}\right) \]
      associate-*r* [=>] 95.6%	\[ 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right) \cdot \left(im \cdot \sqrt{0.5}\right)\right)} \]
      *-commutative [<=] 95.6%	\[ 0.5 \cdot \left(\color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \]
      add-sqr-sqrt [=>] 95.6%	\[ 0.5 \cdot \color{blue}{\left(\sqrt{\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(im \cdot \sqrt{0.5}\right)} \cdot \sqrt{\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(im \cdot \sqrt{0.5}\right)}\right)} \]
      sqrt-unprod [=>] 59.8%	\[ 0.5 \cdot \color{blue}{\sqrt{\left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(im \cdot \sqrt{0.5}\right)\right)}} \]
      *-commutative [=>] 59.8%	\[ 0.5 \cdot \sqrt{\color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(im \cdot \sqrt{0.5}\right)\right)} \]
      *-commutative [=>] 59.8%	\[ 0.5 \cdot \sqrt{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right)}} \]
      swap-sqr [=>] 44.2%	\[ 0.5 \cdot \sqrt{\color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right)}} \]
      *-commutative [=>] 44.2%	\[ 0.5 \cdot \sqrt{\left(\color{blue}{\left(\sqrt{0.5} \cdot im\right)} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right)} \]
      *-commutative [=>] 44.2%	\[ 0.5 \cdot \sqrt{\left(\left(\sqrt{0.5} \cdot im\right) \cdot \color{blue}{\left(\sqrt{0.5} \cdot im\right)}\right) \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right)} \]
      swap-sqr [=>] 44.2%	\[ 0.5 \cdot \sqrt{\color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{0.5}\right) \cdot \left(im \cdot im\right)\right)} \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right)} \]
      add-sqr-sqrt [<=] 44.4%	\[ 0.5 \cdot \sqrt{\left(\color{blue}{0.5} \cdot \left(im \cdot im\right)\right) \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right)} \]

   5. Simplified 60.4%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{im \cdot \frac{im}{re}}} \]
      Step-by-step derivation

      [Start] 44.8%	\[ 0.5 \cdot \sqrt{\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \frac{2}{re}} \]
      associate-*r/ [=>] 44.7%	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot 2}{re}}} \]
      *-commutative [=>] 44.7%	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{\left(\left(im \cdot im\right) \cdot 0.5\right)} \cdot 2}{re}} \]
      associate-*l* [=>] 44.8%	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{\left(im \cdot im\right) \cdot \left(0.5 \cdot 2\right)}}{re}} \]
      metadata-eval [=>] 44.8%	\[ 0.5 \cdot \sqrt{\frac{\left(im \cdot im\right) \cdot \color{blue}{1}}{re}} \]
      *-rgt-identity [=>] 44.8%	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]
      associate-*r/ [<=] 60.4%	\[ 0.5 \cdot \sqrt{\color{blue}{im \cdot \frac{im}{re}}} \]

   6. Applied egg-rr 12.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
      Step-by-step derivation

      [Start] 60.4%	\[ 0.5 \cdot \sqrt{im \cdot \frac{im}{re}} \]
      associate-*r/ [=>] 44.8%	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]
      sqrt-div [=>] 50.9%	\[ 0.5 \cdot \color{blue}{\frac{\sqrt{im \cdot im}}{\sqrt{re}}} \]
      sqrt-unprod [<=] 96.3%	\[ 0.5 \cdot \frac{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}{\sqrt{re}} \]
      add-sqr-sqrt [<=] 96.6%	\[ 0.5 \cdot \frac{\color{blue}{im}}{\sqrt{re}} \]
      expm1-log1p-u [=>] 96.4%	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]
      expm1-udef [=>] 12.9%	\[ 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]

   7. Simplified 96.6%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
      Step-by-step derivation

      [Start] 12.9%	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right) \]
      expm1-def [=>] 96.4%	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]
      expm1-log1p [=>] 96.6%	\[ 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

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

   1. Initial program 46.2%

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

   2. Simplified 90.4%

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

      [Start] 46.2%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      hypot-def [=>] 90.4%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

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

Alternatives

Alternative 1
Accuracy	90.6%
Cost	20356

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

Alternative 2
Accuracy	75.4%
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -2.35 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.8 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3
Accuracy	75.1%
Cost	7312

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -2.35 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.8 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 4
Accuracy	63.6%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 5
Accuracy	63.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 6
Accuracy	51.4%
Cost	6720

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

Reproduce

herbie shell --seed 2023258 
(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)))))