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

Average Accuracy: 38.9% → 89.6%

Time: 10.0s

Precision: binary64

Cost: 20356

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]

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 (sqrt (/ 0.25 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 * sqrt((0.25 / 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 * Math.sqrt((0.25 / 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 * math.sqrt((0.25 / 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(im * sqrt(Float64(0.25 / 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 * sqrt((0.25 / 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[(im * N[Sqrt[N[(0.25 / 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 8.3%

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

   2. Simplified 18.6%

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

      [Start] 8.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 8.3	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 8.3	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 8.3	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 8.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 8.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      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 43.0%

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

   4. Simplified 43.0%

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

      [Start] 43.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      unpow2 [=>] 43.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   5. Applied egg-rr 53.5%

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

      [Start] 43.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{im \cdot im}{re}\right)} \]
      associate-*r* [=>] 43.1	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{im \cdot im}{re}}} \]
      metadata-eval [=>] 43.1	\[ 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{im \cdot im}{re}} \]
      *-un-lft-identity [<=] 43.1	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]
      pow1/2 [=>] 43.1	\[ 0.5 \cdot \color{blue}{{\left(\frac{im \cdot im}{re}\right)}^{0.5}} \]
      associate-/l* [=>] 53.5	\[ 0.5 \cdot {\color{blue}{\left(\frac{im}{\frac{re}{im}}\right)}}^{0.5} \]
      div-inv [=>] 53.5	\[ 0.5 \cdot {\color{blue}{\left(im \cdot \frac{1}{\frac{re}{im}}\right)}}^{0.5} \]
      associate-/l* [<=] 53.5	\[ 0.5 \cdot {\left(im \cdot \color{blue}{\frac{1 \cdot im}{re}}\right)}^{0.5} \]
      *-un-lft-identity [<=] 53.5	\[ 0.5 \cdot {\left(im \cdot \frac{\color{blue}{im}}{re}\right)}^{0.5} \]

   6. Simplified 53.5%

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

      [Start] 53.5	\[ 0.5 \cdot {\left(im \cdot \frac{im}{re}\right)}^{0.5} \]
      unpow1/2 [=>] 53.5	\[ 0.5 \cdot \color{blue}{\sqrt{im \cdot \frac{im}{re}}} \]
      associate-*r/ [=>] 43.1	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]
      associate-/l* [=>] 53.5	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}}} \]

   7. Applied egg-rr 14.2%

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

      [Start] 53.5	\[ 0.5 \cdot \sqrt{\frac{im}{\frac{re}{im}}} \]
      expm1-log1p-u [=>] 53.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \sqrt{\frac{im}{\frac{re}{im}}}\right)\right)} \]
      expm1-udef [=>] 14.2	\[ \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{\frac{im}{\frac{re}{im}}}\right)} - 1} \]
      *-commutative [=>] 14.2	\[ e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{im}{\frac{re}{im}}} \cdot 0.5}\right)} - 1 \]
      sqrt-div [=>] 14.2	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{im}}{\sqrt{\frac{re}{im}}}} \cdot 0.5\right)} - 1 \]
      sqrt-div [=>] 14.2	\[ e^{\mathsf{log1p}\left(\frac{\sqrt{im}}{\color{blue}{\frac{\sqrt{re}}{\sqrt{im}}}} \cdot 0.5\right)} - 1 \]
      associate-/l* [<=] 14.2	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{im} \cdot \sqrt{im}}{\sqrt{re}}} \cdot 0.5\right)} - 1 \]
      add-sqr-sqrt [<=] 14.2	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{im}}{\sqrt{re}} \cdot 0.5\right)} - 1 \]

   8. Simplified 90.8%

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

      [Start] 14.2	\[ e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}} \cdot 0.5\right)} - 1 \]
      expm1-def [=>] 90.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}} \cdot 0.5\right)\right)} \]
      expm1-log1p [=>] 91.0	\[ \color{blue}{\frac{im}{\sqrt{re}} \cdot 0.5} \]
      associate-*l/ [=>] 91.0	\[ \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
      *-commutative [=>] 91.0	\[ \frac{\color{blue}{0.5 \cdot im}}{\sqrt{re}} \]
      associate-*l/ [<=] 90.8	\[ \color{blue}{\frac{0.5}{\sqrt{re}} \cdot im} \]
      *-commutative [=>] 90.8	\[ \color{blue}{im \cdot \frac{0.5}{\sqrt{re}}} \]

   9. Applied egg-rr 90.9%

      \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]
      Proof

      [Start] 90.8	\[ im \cdot \frac{0.5}{\sqrt{re}} \]
      add-sqr-sqrt [=>] 90.4	\[ im \cdot \color{blue}{\left(\sqrt{\frac{0.5}{\sqrt{re}}} \cdot \sqrt{\frac{0.5}{\sqrt{re}}}\right)} \]
      sqrt-unprod [=>] 90.8	\[ im \cdot \color{blue}{\sqrt{\frac{0.5}{\sqrt{re}} \cdot \frac{0.5}{\sqrt{re}}}} \]
      frac-times [=>] 90.8	\[ im \cdot \sqrt{\color{blue}{\frac{0.5 \cdot 0.5}{\sqrt{re} \cdot \sqrt{re}}}} \]
      metadata-eval [=>] 90.8	\[ im \cdot \sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \]
      add-sqr-sqrt [<=] 90.9	\[ im \cdot \sqrt{\frac{0.25}{\color{blue}{re}}} \]

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

   1. Initial program 44.2%

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

   2. Simplified 89.3%

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

      [Start] 44.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 44.2	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 44.2	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 44.2	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 44.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 44.2	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 89.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	75.9%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 2
Accuracy	75.4%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 3
Accuracy	63.9%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 4
Accuracy	63.9%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 1.3 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 5
Accuracy	51.5%
Cost	6720

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

Reproduce

herbie shell --seed 2023142 
(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)))))