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

Average Accuracy: 39.7% → 89.4%

Time: 10.3s

Precision: binary64

Cost: 20356

\[im > 0\]

Math

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

↓

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

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)
   (/ (pow re -0.5) (/ 2.0 im))
   (* 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 = pow(re, -0.5) / (2.0 / im);
	} 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 = Math.pow(re, -0.5) / (2.0 / im);
	} 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 = math.pow(re, -0.5) / (2.0 / im)
	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((re ^ -0.5) / Float64(2.0 / im));
	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 = (re ^ -0.5) / (2.0 / im);
	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[Power[re, -0.5], $MachinePrecision] / N[(2.0 / im), $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 8.1%

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

   2. Taylor expanded in re around inf 41.5%

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

   3. Simplified 50.3%

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

      [Start] 41.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      unpow2 [=>] 41.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]
      associate-/l* [=>] 50.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \color{blue}{\frac{im}{\frac{re}{im}}}\right)} \]

   4. Applied egg-rr 14.7%

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

      [Start] 50.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)} \]
      expm1-log1p-u [=>] 50.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}\right)\right)} \]
      expm1-udef [=>] 14.7	\[ \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}\right)} - 1} \]
      associate-*r* [=>] 14.7	\[ e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{im}{\frac{re}{im}}}}\right)} - 1 \]
      metadata-eval [=>] 14.7	\[ e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{im}{\frac{re}{im}}}\right)} - 1 \]
      *-un-lft-identity [<=] 14.7	\[ e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}}}\right)} - 1 \]
      sqrt-div [=>] 14.7	\[ e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{\frac{\sqrt{im}}{\sqrt{\frac{re}{im}}}}\right)} - 1 \]
      sqrt-div [=>] 14.7	\[ e^{\mathsf{log1p}\left(0.5 \cdot \frac{\sqrt{im}}{\color{blue}{\frac{\sqrt{re}}{\sqrt{im}}}}\right)} - 1 \]
      associate-/l* [<=] 14.7	\[ e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{\frac{\sqrt{im} \cdot \sqrt{im}}{\sqrt{re}}}\right)} - 1 \]
      add-sqr-sqrt [<=] 14.7	\[ e^{\mathsf{log1p}\left(0.5 \cdot \frac{\color{blue}{im}}{\sqrt{re}}\right)} - 1 \]

   5. Simplified 90.1%

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

      [Start] 14.7	\[ e^{\mathsf{log1p}\left(0.5 \cdot \frac{im}{\sqrt{re}}\right)} - 1 \]
      expm1-def [=>] 89.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{im}{\sqrt{re}}\right)\right)} \]
      expm1-log1p [=>] 90.1	\[ \color{blue}{0.5 \cdot \frac{im}{\sqrt{re}}} \]
      associate-*r/ [=>] 90.1	\[ \color{blue}{\frac{0.5 \cdot im}{\sqrt{re}}} \]
      *-commutative [=>] 90.1	\[ \frac{\color{blue}{im \cdot 0.5}}{\sqrt{re}} \]

   6. Applied egg-rr 90.1%

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

      [Start] 90.1	\[ \frac{im \cdot 0.5}{\sqrt{re}} \]
      div-inv [=>] 89.9	\[ \color{blue}{\left(im \cdot 0.5\right) \cdot \frac{1}{\sqrt{re}}} \]
      *-commutative [=>] 89.9	\[ \color{blue}{\frac{1}{\sqrt{re}} \cdot \left(im \cdot 0.5\right)} \]
      pow1/2 [=>] 89.9	\[ \frac{1}{\color{blue}{{re}^{0.5}}} \cdot \left(im \cdot 0.5\right) \]
      pow-flip [=>] 90.1	\[ \color{blue}{{re}^{\left(-0.5\right)}} \cdot \left(im \cdot 0.5\right) \]
      metadata-eval [=>] 90.1	\[ {re}^{\color{blue}{-0.5}} \cdot \left(im \cdot 0.5\right) \]

   7. Taylor expanded in im around 0 90.0%

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

   8. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{{re}^{-0.5}}{\frac{2}{im}}} \]
      Proof

      [Start] 90.0	\[ 0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right) \]
      *-commutative [=>] 90.0	\[ 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
      associate-*r* [=>] 90.0	\[ \color{blue}{\left(0.5 \cdot im\right) \cdot \sqrt{\frac{1}{re}}} \]
      metadata-eval [<=] 90.0	\[ \left(\color{blue}{\frac{1}{2}} \cdot im\right) \cdot \sqrt{\frac{1}{re}} \]
      associate-/r/ [<=] 89.9	\[ \color{blue}{\frac{1}{\frac{2}{im}}} \cdot \sqrt{\frac{1}{re}} \]
      unpow-1 [<=] 89.9	\[ \frac{1}{\frac{2}{im}} \cdot \sqrt{\color{blue}{{re}^{-1}}} \]
      metadata-eval [<=] 89.9	\[ \frac{1}{\frac{2}{im}} \cdot \sqrt{{re}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]
      pow-sqr [<=] 90.0	\[ \frac{1}{\frac{2}{im}} \cdot \sqrt{\color{blue}{{re}^{-0.5} \cdot {re}^{-0.5}}} \]
      rem-sqrt-square [=>] 90.0	\[ \frac{1}{\frac{2}{im}} \cdot \color{blue}{\left|{re}^{-0.5}\right|} \]
      sqr-pow [=>] 89.6	\[ \frac{1}{\frac{2}{im}} \cdot \left|\color{blue}{{re}^{\left(\frac{-0.5}{2}\right)} \cdot {re}^{\left(\frac{-0.5}{2}\right)}}\right| \]
      fabs-sqr [=>] 89.6	\[ \frac{1}{\frac{2}{im}} \cdot \color{blue}{\left({re}^{\left(\frac{-0.5}{2}\right)} \cdot {re}^{\left(\frac{-0.5}{2}\right)}\right)} \]
      sqr-pow [<=] 90.0	\[ \frac{1}{\frac{2}{im}} \cdot \color{blue}{{re}^{-0.5}} \]
      associate-*l/ [=>] 89.9	\[ \color{blue}{\frac{1 \cdot {re}^{-0.5}}{\frac{2}{im}}} \]
      *-lft-identity [=>] 89.9	\[ \frac{\color{blue}{{re}^{-0.5}}}{\frac{2}{im}} \]

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

   1. Initial program 45.1%

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

   2. Simplified 89.3%

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

      [Start] 45.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 45.1	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 45.1	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 45.1	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 45.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 45.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      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 89.4%

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

Alternatives

Alternative 1
Accuracy	76.3%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.04 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 2
Accuracy	75.7%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;re \leq -52000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 3
Accuracy	59.6%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;im \leq 2.95 \cdot 10^{-86}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 4
Accuracy	59.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 1.62 \cdot 10^{-87}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Accuracy	59.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 1.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Accuracy	52.7%
Cost	6720

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

Reproduce

herbie shell --seed 2023135 
(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)))))