math.sqrt on complex, real part

Average Accuracy: 39.0% → 84.7%

Time: 8.4s

Precision: binary64

Cost: 26948

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;{\left(\frac{-0.25}{re}\right)}^{0.5} \cdot {\left(\frac{-1}{im}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

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)
   (* (pow (/ -0.25 re) 0.5) (pow (/ -1.0 im) -1.0))
   (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 = pow((-0.25 / re), 0.5) * pow((-1.0 / im), -1.0);
	} 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 = Math.pow((-0.25 / re), 0.5) * Math.pow((-1.0 / im), -1.0);
	} 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 = math.pow((-0.25 / re), 0.5) * math.pow((-1.0 / im), -1.0)
	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((Float64(-0.25 / re) ^ 0.5) * (Float64(-1.0 / im) ^ -1.0));
	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 = ((-0.25 / re) ^ 0.5) * ((-1.0 / im) ^ -1.0);
	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[(N[Power[N[(-0.25 / re), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(-1.0 / im), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	39.0%
Target	46.9%
Herbie	84.7%

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

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 13.6%

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

   2. Simplified 13.6%

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

      [Start] 13.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 13.6	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 13.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 51.2%

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

   4. Simplified 51.2%

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

      [Start] 51.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 51.2	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 51.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\frac{im}{re} \cdot im\right) \cdot -1\right) \cdot 0.25\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 51.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \]
      add-sqr-sqrt [=>] 51.0	\[ \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}} \]
      pow1/2 [=>] 51.0	\[ \color{blue}{{\left(0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)}^{0.5}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}} \]
      pow1/2 [=>] 51.0	\[ {\left(0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)}^{0.5} \cdot \color{blue}{{\left(0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)}^{0.5}} \]
      pow-prod-down [=>] 51.2	\[ \color{blue}{{\left(\left(0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)\right)}^{0.5}} \]

   6. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{im}{re} \cdot \left(im \cdot -0.25\right)}} \]
      Step-by-step derivation

      [Start] 56.9	\[ {\left(\left(\left(\frac{im}{re} \cdot im\right) \cdot -1\right) \cdot 0.25\right)}^{0.5} \]
      unpow1/2 [=>] 56.9	\[ \color{blue}{\sqrt{\left(\left(\frac{im}{re} \cdot im\right) \cdot -1\right) \cdot 0.25}} \]
      associate-*l* [=>] 56.9	\[ \sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right) \cdot \left(-1 \cdot 0.25\right)}} \]
      metadata-eval [=>] 56.9	\[ \sqrt{\left(\frac{im}{re} \cdot im\right) \cdot \color{blue}{-0.25}} \]
      metadata-eval [<=] 56.9	\[ \sqrt{\left(\frac{im}{re} \cdot im\right) \cdot \color{blue}{\frac{-0.5}{2}}} \]
      associate-*l* [=>] 56.9	\[ \sqrt{\color{blue}{\frac{im}{re} \cdot \left(im \cdot \frac{-0.5}{2}\right)}} \]
      metadata-eval [=>] 56.9	\[ \sqrt{\frac{im}{re} \cdot \left(im \cdot \color{blue}{-0.25}\right)} \]

   7. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{{\left(e^{0.5}\right)}^{\log \left(im \cdot \left(-0.25 \cdot \frac{im}{re}\right)\right)}} \]
      Step-by-step derivation

      [Start] 56.9	\[ \sqrt{\frac{im}{re} \cdot \left(im \cdot -0.25\right)} \]
      add-exp-log [=>] 54.0	\[ \color{blue}{e^{\log \left(\sqrt{\frac{im}{re} \cdot \left(im \cdot -0.25\right)}\right)}} \]
      pow1/2 [=>] 54.0	\[ e^{\log \color{blue}{\left({\left(\frac{im}{re} \cdot \left(im \cdot -0.25\right)\right)}^{0.5}\right)}} \]
      log-pow [=>] 54.0	\[ e^{\color{blue}{0.5 \cdot \log \left(\frac{im}{re} \cdot \left(im \cdot -0.25\right)\right)}} \]
      exp-prod [=>] 53.7	\[ \color{blue}{{\left(e^{0.5}\right)}^{\log \left(\frac{im}{re} \cdot \left(im \cdot -0.25\right)\right)}} \]
      *-commutative [=>] 53.7	\[ {\left(e^{0.5}\right)}^{\log \color{blue}{\left(\left(im \cdot -0.25\right) \cdot \frac{im}{re}\right)}} \]
      associate-*l* [=>] 53.7	\[ {\left(e^{0.5}\right)}^{\log \color{blue}{\left(im \cdot \left(-0.25 \cdot \frac{im}{re}\right)\right)}} \]

   8. Taylor expanded in im around -inf 53.7%

      \[\leadsto \color{blue}{e^{0.5 \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right) + \log \left(\frac{-0.25}{re}\right)\right)}} \]

   9. Simplified 63.8%

      \[\leadsto \color{blue}{{\left(\frac{-0.25}{re}\right)}^{0.5} \cdot {\left(\frac{-1}{im}\right)}^{-1}} \]
      Step-by-step derivation

      [Start] 53.7	\[ e^{0.5 \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right) + \log \left(\frac{-0.25}{re}\right)\right)} \]
      +-commutative [=>] 53.7	\[ e^{0.5 \cdot \color{blue}{\left(\log \left(\frac{-0.25}{re}\right) + -2 \cdot \log \left(\frac{-1}{im}\right)\right)}} \]
      distribute-rgt-in [=>] 53.7	\[ e^{\color{blue}{\log \left(\frac{-0.25}{re}\right) \cdot 0.5 + \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right) \cdot 0.5}} \]
      exp-sum [=>] 53.5	\[ \color{blue}{e^{\log \left(\frac{-0.25}{re}\right) \cdot 0.5} \cdot e^{\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right) \cdot 0.5}} \]
      exp-to-pow [=>] 53.6	\[ \color{blue}{{\left(\frac{-0.25}{re}\right)}^{0.5}} \cdot e^{\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right) \cdot 0.5} \]
      *-commutative [=>] 53.6	\[ {\left(\frac{-0.25}{re}\right)}^{0.5} \cdot e^{\color{blue}{\left(\log \left(\frac{-1}{im}\right) \cdot -2\right)} \cdot 0.5} \]
      associate-*l* [=>] 53.6	\[ {\left(\frac{-0.25}{re}\right)}^{0.5} \cdot e^{\color{blue}{\log \left(\frac{-1}{im}\right) \cdot \left(-2 \cdot 0.5\right)}} \]
      metadata-eval [=>] 53.6	\[ {\left(\frac{-0.25}{re}\right)}^{0.5} \cdot e^{\log \left(\frac{-1}{im}\right) \cdot \color{blue}{-1}} \]
      exp-to-pow [=>] 63.8	\[ {\left(\frac{-0.25}{re}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{-1}{im}\right)}^{-1}} \]

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

   1. Initial program 44.9%

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

   2. Simplified 86.9%

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

      [Start] 44.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 44.9	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 86.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Applied egg-rr 86.9%

      \[\leadsto \color{blue}{{\left(\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 86.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]
      add-sqr-sqrt [=>] 86.3	\[ \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}}} \]
      pow1/2 [=>] 86.3	\[ \color{blue}{{\left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}^{0.5}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      pow1/2 [=>] 86.3	\[ {\left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}^{0.5} \cdot \color{blue}{{\left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}^{0.5}} \]
      pow-prod-down [=>] 86.9	\[ \color{blue}{{\left(\left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)\right)}^{0.5}} \]
      *-commutative [=>] 86.9	\[ {\left(\color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)\right)}^{0.5} \]
      *-commutative [=>] 86.9	\[ {\left(\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right)}\right)}^{0.5} \]
      swap-sqr [=>] 86.9	\[ {\color{blue}{\left(\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \left(0.5 \cdot 0.5\right)\right)}}^{0.5} \]
      add-sqr-sqrt [<=] 86.9	\[ {\left(\color{blue}{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)} \cdot \left(0.5 \cdot 0.5\right)\right)}^{0.5} \]
      *-commutative [=>] 86.9	\[ {\left(\color{blue}{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)\right)}^{0.5} \]
      metadata-eval [=>] 86.9	\[ {\left(\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot \color{blue}{0.25}\right)}^{0.5} \]

   4. Simplified 86.9%

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

      [Start] 86.9	\[ {\left(\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25\right)}^{0.5} \]
      unpow1/2 [=>] 86.9	\[ \color{blue}{\sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25}} \]
      associate-*l* [=>] 86.9	\[ \sqrt{\color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot \left(2 \cdot 0.25\right)}} \]
      metadata-eval [=>] 86.9	\[ \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot \color{blue}{0.5}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;{\left(\frac{-0.25}{re}\right)}^{0.5} \cdot {\left(\frac{-1}{im}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.4%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;re \leq -2.6 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\frac{im}{re} \cdot \left(im \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	58.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{im \cdot \left(-0.5\right)}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Accuracy	57.9%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;im \leq -6.2 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{im \cdot \left(-0.5\right)}\\ \mathbf{elif}\;im \leq 6 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 4
Accuracy	42.8%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 5
Accuracy	26.7%
Cost	6464

\[\sqrt{re} \]

Reproduce

herbie shell --seed 2023157 
(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)))))