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

Percentage Accurate: 41.8% → 90.3%

Time: 11.0s

Precision: binary64

Cost: 26756

\[im > 0\]

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (/ 0.5 (/ (sqrt re) im))
   (sqrt (* 0.5 (- (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((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = 0.5 / (sqrt(re) / im);
	} else {
		tmp = sqrt((0.5 * (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((2.0 * (Math.sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = 0.5 / (Math.sqrt(re) / im);
	} else {
		tmp = Math.sqrt((0.5 * (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((2.0 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = 0.5 / (math.sqrt(re) / im)
	else:
		tmp = math.sqrt((0.5 * (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 (sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))) <= 0.0)
		tmp = Float64(0.5 / Float64(sqrt(re) / im));
	else
		tmp = sqrt(Float64(0.5 * 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((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = 0.5 / (sqrt(re) / im);
	else
		tmp = sqrt((0.5 * (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[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.4%

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

   2. Taylor expanded in im around 0 98.5%

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

   3. Simplified 98.7%

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

      [Start] 98.5	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      associate-*l* [=>] 98.7	\[ 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\sqrt{0.5} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
      *-commutative [=>] 98.7	\[ 0.5 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(im \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
      Step-by-step derivation

      [Start] 98.7	\[ 0.5 \cdot \left(\sqrt{2} \cdot \left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)\right) \]
      add-log-exp [=>] 8.2	\[ 0.5 \cdot \color{blue}{\log \left(e^{\sqrt{2} \cdot \left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)} \]
      *-un-lft-identity [=>] 8.2	\[ 0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{2} \cdot \left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)} \]
      log-prod [=>] 8.2	\[ 0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{2} \cdot \left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right)} \]
      metadata-eval [=>] 8.2	\[ 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{2} \cdot \left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]
      add-log-exp [<=] 98.7	\[ 0.5 \cdot \left(0 + \color{blue}{\sqrt{2} \cdot \left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
      associate-*r* [=>] 98.5	\[ 0.5 \cdot \left(0 + \color{blue}{\left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \]
      sqrt-div [=>] 98.5	\[ 0.5 \cdot \left(0 + \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right) \]
      metadata-eval [=>] 98.5	\[ 0.5 \cdot \left(0 + \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right) \]
      un-div-inv [=>] 98.5	\[ 0.5 \cdot \left(0 + \color{blue}{\frac{\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)}{\sqrt{re}}}\right) \]
      *-commutative [=>] 98.5	\[ 0.5 \cdot \left(0 + \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{0.5} \cdot im\right)}}{\sqrt{re}}\right) \]
      associate-*r* [=>] 97.9	\[ 0.5 \cdot \left(0 + \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot im}}{\sqrt{re}}\right) \]
      pow1/2 [=>] 97.9	\[ 0.5 \cdot \left(0 + \frac{\left(\color{blue}{{2}^{0.5}} \cdot \sqrt{0.5}\right) \cdot im}{\sqrt{re}}\right) \]
      pow1/2 [=>] 97.9	\[ 0.5 \cdot \left(0 + \frac{\left({2}^{0.5} \cdot \color{blue}{{0.5}^{0.5}}\right) \cdot im}{\sqrt{re}}\right) \]
      pow-prod-down [=>] 99.6	\[ 0.5 \cdot \left(0 + \frac{\color{blue}{{\left(2 \cdot 0.5\right)}^{0.5}} \cdot im}{\sqrt{re}}\right) \]
      metadata-eval [=>] 99.6	\[ 0.5 \cdot \left(0 + \frac{{\color{blue}{1}}^{0.5} \cdot im}{\sqrt{re}}\right) \]
      pow-to-exp [=>] 99.6	\[ 0.5 \cdot \left(0 + \frac{\color{blue}{e^{\log 1 \cdot 0.5}} \cdot im}{\sqrt{re}}\right) \]
      metadata-eval [=>] 99.6	\[ 0.5 \cdot \left(0 + \frac{e^{\color{blue}{0} \cdot 0.5} \cdot im}{\sqrt{re}}\right) \]
      metadata-eval [=>] 99.6	\[ 0.5 \cdot \left(0 + \frac{e^{\color{blue}{0}} \cdot im}{\sqrt{re}}\right) \]
      1-exp [<=] 99.6	\[ 0.5 \cdot \left(0 + \frac{\color{blue}{1} \cdot im}{\sqrt{re}}\right) \]
      *-un-lft-identity [<=] 99.6	\[ 0.5 \cdot \left(0 + \frac{\color{blue}{im}}{\sqrt{re}}\right) \]

   5. Simplified 99.6%

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

      [Start] 99.6	\[ 0.5 \cdot \left(0 + \frac{im}{\sqrt{re}}\right) \]
      +-lft-identity [=>] 99.6	\[ 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   6. Applied egg-rr 99.7%

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

      [Start] 99.6	\[ 0.5 \cdot \frac{im}{\sqrt{re}} \]
      clear-num [=>] 99.7	\[ 0.5 \cdot \color{blue}{\frac{1}{\frac{\sqrt{re}}{im}}} \]
      un-div-inv [=>] 99.7	\[ \color{blue}{\frac{0.5}{\frac{\sqrt{re}}{im}}} \]

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

   1. Initial program 42.8%

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

   2. Simplified 86.5%

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

      [Start] 42.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      hypot-def [=>] 86.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{0 + \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5} \]
      Step-by-step derivation

      [Start] 86.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \]
      add-log-exp [=>] 10.0	\[ \color{blue}{\log \left(e^{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)} \]
      *-un-lft-identity [=>] 10.0	\[ \log \color{blue}{\left(1 \cdot e^{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)} \]
      log-prod [=>] 10.0	\[ \color{blue}{\log 1 + \log \left(e^{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)} \]
      metadata-eval [=>] 10.0	\[ \color{blue}{0} + \log \left(e^{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right) \]
      add-log-exp [<=] 86.5	\[ 0 + \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      *-commutative [=>] 86.5	\[ 0 + \color{blue}{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot 0.5} \]
      *-commutative [=>] 86.5	\[ 0 + \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}} \cdot 0.5 \]

   4. Simplified 86.5%

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

      [Start] 86.5	\[ 0 + \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5 \]
      +-lft-identity [=>] 86.5	\[ \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5} \]
      rem-cube-cbrt [<=] 84.8	\[ \color{blue}{{\left(\sqrt[3]{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5}\right)}^{3}} \]
      sqr-pow [=>] 84.7	\[ \color{blue}{{\left(\sqrt[3]{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5}\right)}^{\left(\frac{3}{2}\right)}} \]
      fabs-sqr [<=] 84.7	\[ \color{blue}{\left|{\left(\sqrt[3]{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5}\right)}^{\left(\frac{3}{2}\right)}\right|} \]
      sqr-pow [<=] 84.8	\[ \left|\color{blue}{{\left(\sqrt[3]{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5}\right)}^{3}}\right| \]
      rem-cube-cbrt [=>] 86.5	\[ \left|\color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5}\right| \]
      rem-sqrt-square [<=] 86.5	\[ \color{blue}{\sqrt{\left(\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\right) \cdot \left(\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\right)}} \]
      swap-sqr [=>] 86.5	\[ \sqrt{\color{blue}{\left(\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]
      rem-square-sqrt [=>] 86.5	\[ \sqrt{\color{blue}{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]
      metadata-eval [=>] 86.5	\[ \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]
      associate-*l* [=>] 86.5	\[ \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]
      metadata-eval [=>] 86.5	\[ \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]
      *-commutative [=>] 86.5	\[ \sqrt{\color{blue}{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

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

Alternatives

Alternative 1
Accuracy	75.4%
Cost	7249

\[\begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{+15} \lor \neg \left(re \leq 8 \cdot 10^{+88}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 2
Accuracy	75.3%
Cost	7248

\[\begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+74}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 3
Accuracy	63.8%
Cost	7116

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{if}\;re \leq -7.6 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 4
Accuracy	55.1%
Cost	6720

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

Alternative 5
Accuracy	52.8%
Cost	6592

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

Reproduce

herbie shell --seed 2023161 
(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)))))