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

Average Accuracy: 40.7% → 90.7%

Time: 13.3s

Precision: binary64

Cost: 20356

\[im > 0\]

Math

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

↓

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

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)
   (/ (* im 0.5) (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 = (im * 0.5) / 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 = (im * 0.5) / 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 = (im * 0.5) / 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(Float64(im * 0.5) / 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 = (im * 0.5) / 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[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $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 8.4%

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

   2. Simplified 18.6%

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

      [Start] 8.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 8.4	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 8.4	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 8.4	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 8.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 8.4	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 18.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Taylor expanded in re around inf 43.9%

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

   4. Simplified 43.9%

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

      [Start] 43.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      associate-*r/ [=>] 43.9	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{0.5 \cdot {im}^{2}}{re}}} \]
      associate-/l* [=>] 43.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{0.5}{\frac{re}{{im}^{2}}}}} \]
      associate-/r/ [=>] 43.9	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{0.5}{re} \cdot {im}^{2}\right)}} \]
      unpow2 [=>] 43.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \color{blue}{\left(im \cdot im\right)}\right)} \]

   5. Applied egg-rr 34.8%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\frac{im}{\frac{re}{im}}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
      Proof

      [Start] 43.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)} \]
      add-cbrt-cube [=>] 34.7	\[ 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)} \cdot \sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)}\right) \cdot \sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)}}} \]
      pow1/3 [=>] 32.9	\[ 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)} \cdot \sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)}\right) \cdot \sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)}\right)}^{0.3333333333333333}} \]
      add-sqr-sqrt [<=] 32.9	\[ 0.5 \cdot {\left(\color{blue}{\left(2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)}\right)}^{0.3333333333333333} \]
      pow1 [=>] 32.9	\[ 0.5 \cdot {\left(\color{blue}{{\left(2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)}\right)}^{0.3333333333333333} \]
      pow1/2 [=>] 32.9	\[ 0.5 \cdot {\left({\left(2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]
      pow-prod-up [=>] 32.9	\[ 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]
      associate-*l/ [=>] 32.9	\[ 0.5 \cdot {\left({\left(2 \cdot \color{blue}{\frac{0.5 \cdot \left(im \cdot im\right)}{re}}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]
      associate-/l* [=>] 32.9	\[ 0.5 \cdot {\left({\left(2 \cdot \color{blue}{\frac{0.5}{\frac{re}{im \cdot im}}}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]
      associate-*r/ [=>] 32.9	\[ 0.5 \cdot {\left({\color{blue}{\left(\frac{2 \cdot 0.5}{\frac{re}{im \cdot im}}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]
      metadata-eval [=>] 32.9	\[ 0.5 \cdot {\left({\left(\frac{\color{blue}{1}}{\frac{re}{im \cdot im}}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]
      associate-/l* [<=] 32.9	\[ 0.5 \cdot {\left({\color{blue}{\left(\frac{1 \cdot \left(im \cdot im\right)}{re}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]
      *-un-lft-identity [<=] 32.9	\[ 0.5 \cdot {\left({\left(\frac{\color{blue}{im \cdot im}}{re}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]
      associate-/l* [=>] 34.8	\[ 0.5 \cdot {\left({\color{blue}{\left(\frac{im}{\frac{re}{im}}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]
      metadata-eval [=>] 34.8	\[ 0.5 \cdot {\left({\left(\frac{im}{\frac{re}{im}}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 13.9%

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

      [Start] 34.8	\[ 0.5 \cdot {\left({\left(\frac{im}{\frac{re}{im}}\right)}^{1.5}\right)}^{0.3333333333333333} \]
      expm1-log1p-u [=>] 34.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {\left({\left(\frac{im}{\frac{re}{im}}\right)}^{1.5}\right)}^{0.3333333333333333}\right)\right)} \]
      expm1-udef [=>] 13.9	\[ \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {\left({\left(\frac{im}{\frac{re}{im}}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} - 1} \]
      *-commutative [=>] 13.9	\[ e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{im}{\frac{re}{im}}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot 0.5}\right)} - 1 \]
      pow-pow [=>] 13.9	\[ e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{im}{\frac{re}{im}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot 0.5\right)} - 1 \]
      metadata-eval [=>] 13.9	\[ e^{\mathsf{log1p}\left({\left(\frac{im}{\frac{re}{im}}\right)}^{\color{blue}{0.5}} \cdot 0.5\right)} - 1 \]
      unpow1/2 [=>] 13.9	\[ e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{im}{\frac{re}{im}}}} \cdot 0.5\right)} - 1 \]
      sqrt-div [=>] 13.9	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{im}}{\sqrt{\frac{re}{im}}}} \cdot 0.5\right)} - 1 \]
      sqrt-div [=>] 13.9	\[ e^{\mathsf{log1p}\left(\frac{\sqrt{im}}{\color{blue}{\frac{\sqrt{re}}{\sqrt{im}}}} \cdot 0.5\right)} - 1 \]
      associate-/l* [<=] 13.9	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{im} \cdot \sqrt{im}}{\sqrt{re}}} \cdot 0.5\right)} - 1 \]
      add-sqr-sqrt [<=] 13.9	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{im}}{\sqrt{re}} \cdot 0.5\right)} - 1 \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
      Proof

      [Start] 13.9	\[ e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}} \cdot 0.5\right)} - 1 \]
      expm1-def [=>] 90.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}} \cdot 0.5\right)\right)} \]
      expm1-log1p [=>] 91.1	\[ \color{blue}{\frac{im}{\sqrt{re}} \cdot 0.5} \]
      associate-*l/ [=>] 91.1	\[ \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]

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

   1. Initial program 46.6%

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

   2. Simplified 90.6%

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

      [Start] 46.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 46.6	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 46.6	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 46.6	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 46.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 46.6	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 90.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

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

Alternatives

Alternative 1
Accuracy	76.1%
Cost	7248

\[\begin{array}{l} \mathbf{if}\;re \leq -3.7 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 126:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 2
Accuracy	76.4%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 3
Accuracy	64.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 980:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 4
Accuracy	64.5%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 1.42 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 5
Accuracy	64.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 0.09:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 6
Accuracy	52.8%
Cost	6720

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

Reproduce

herbie shell --seed 2023133 
(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)))))