math.sqrt on complex, real part

Percentage Accurate: 41.4% → 84.2%

Time: 9.8s

Precision: binary64

Cost: 33732

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

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 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (- (* (/ (pow im 4.0) (pow re 3.0)) 0.25) (/ im (/ re im)))))
   (sqrt (* 0.5 (+ re (hypot re im))))))

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 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * sqrt((((pow(im, 4.0) / pow(re, 3.0)) * 0.25) - (im / (re / im))));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	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 * (re + Math.sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * Math.sqrt((((Math.pow(im, 4.0) / Math.pow(re, 3.0)) * 0.25) - (im / (re / im))));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	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 * (re + math.sqrt(((re * re) + (im * im)))))) <= 0.0:
		tmp = 0.5 * math.sqrt((((math.pow(im, 4.0) / math.pow(re, 3.0)) * 0.25) - (im / (re / im))))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	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(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))))) <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64((im ^ 4.0) / (re ^ 3.0)) * 0.25) - Float64(im / Float64(re / im)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	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 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0)
		tmp = 0.5 * sqrt(((((im ^ 4.0) / (re ^ 3.0)) * 0.25) - (im / (re / im))));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	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[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(N[(N[(N[Power[im, 4.0], $MachinePrecision] / N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] - N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	41.4%
Target	49.2%
Herbie	84.2%

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

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 19.3%

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

   2. Simplified 19.3%

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

      [Start] 19.3%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 19.3%	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 19.3%	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 49.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]

   4. Simplified 52.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}} \]
      Step-by-step derivation

      [Start] 49.2%	\[ 0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}} \]
      +-commutative [=>] 49.2%	\[ 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]
      mul-1-neg [=>] 49.2%	\[ 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]
      unsub-neg [=>] 49.2%	\[ 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]
      *-commutative [=>] 49.2%	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]
      unpow2 [=>] 49.2%	\[ 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{\color{blue}{im \cdot im}}{re}} \]
      associate-/l* [=>] 52.4%	\[ 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \color{blue}{\frac{im}{\frac{re}{im}}}} \]

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

   1. Initial program 45.0%

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

   2. Simplified 90.8%

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

      [Start] 45.0%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 45.0%	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 90.8%	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25}} \]
      Step-by-step derivation

      [Start] 90.8%	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]
      add-sqr-sqrt [=>] 90.0%	\[ \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}}} \]
      sqrt-unprod [=>] 90.8%	\[ \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}} \]
      *-commutative [=>] 90.8%	\[ \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)} \]
      *-commutative [=>] 90.8%	\[ \sqrt{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right)}} \]
      swap-sqr [=>] 90.8%	\[ \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]
      add-sqr-sqrt [<=] 90.8%	\[ \sqrt{\color{blue}{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]
      *-commutative [=>] 90.8%	\[ \sqrt{\color{blue}{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]
      metadata-eval [=>] 90.8%	\[ \sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Simplified 90.8%

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

      [Start] 90.8%	\[ \sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25} \]
      associate-*l* [=>] 90.8%	\[ \sqrt{\color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot \left(2 \cdot 0.25\right)}} \]
      metadata-eval [=>] 90.8%	\[ \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot \color{blue}{0.5}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.2%
Cost	33732

\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	84.5%
Cost	26756

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

Alternative 3
Accuracy	59.7%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{im \cdot \left(-0.5\right)}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Accuracy	60.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.65 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Accuracy	59.1%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;im \leq -7 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{im \cdot \left(-0.5\right)}\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 6
Accuracy	42.6%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;im \leq 8.6 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 7
Accuracy	26.6%
Cost	6464

\[\sqrt{re} \]

Reproduce

herbie shell --seed 2023272 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))