math.sqrt on complex, real part

Average Error: 60.66% → 15.23%

Time: 8.7s

Precision: binary64

Cost: 26884

Math

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

↓

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

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)
   (sqrt (* (* im (/ im re)) -0.25))
   (* 0.5 (sqrt (* 2.0 (+ 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 = sqrt(((im * (im / re)) * -0.25));
	} else {
		tmp = 0.5 * sqrt((2.0 * (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.sqrt(((im * (im / re)) * -0.25));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (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.sqrt(((im * (im / re)) * -0.25))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (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 = sqrt(Float64(Float64(im * Float64(im / re)) * -0.25));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 = sqrt(((im * (im / re)) * -0.25));
	else
		tmp = 0.5 * sqrt((2.0 * (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[Sqrt[N[(N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	60.66%
Target	53.06%
Herbie	15.23%

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

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 89.13

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

   2. Simplified 89.13

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

      [Start] 89.13	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 89.13	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 89.13	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 47.87

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

   4. Simplified 42.27

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

      [Start] 47.87	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 47.87	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 47.87	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]
      associate-/l* [=>] 42.27	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{im}{\frac{re}{im}}} \cdot -0.5\right)} \]

   5. Applied egg-rr 71.17

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

   6. Applied egg-rr 71.17

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

   7. Simplified 42.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{\frac{im}{re}}{\frac{1}{im}}} \cdot -0.5\right)} \]
      Proof

      [Start] 71.17	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\frac{im}{re}}{{im}^{-0.5} \cdot {im}^{-0.5}} \cdot -0.5\right)} \]
      pow-sqr [=>] 42.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\frac{im}{re}}{\color{blue}{{im}^{\left(2 \cdot -0.5\right)}}} \cdot -0.5\right)} \]
      metadata-eval [=>] 42.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\frac{im}{re}}{{im}^{\color{blue}{-1}}} \cdot -0.5\right)} \]
      unpow-1 [=>] 42.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\frac{im}{re}}{\color{blue}{\frac{1}{im}}} \cdot -0.5\right)} \]

   8. Applied egg-rr 42.32

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

   9. Simplified 42.32

      \[\leadsto \color{blue}{\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -0.25}} \]
      Proof

      [Start] 42.32	\[ \sqrt{0.25 \cdot \left(\frac{im}{re} \cdot \left(-im\right)\right)} \]
      *-commutative [=>] 42.32	\[ \sqrt{\color{blue}{\left(\frac{im}{re} \cdot \left(-im\right)\right) \cdot 0.25}} \]
      distribute-rgt-neg-out [=>] 42.32	\[ \sqrt{\color{blue}{\left(-\frac{im}{re} \cdot im\right)} \cdot 0.25} \]
      *-commutative [<=] 42.32	\[ \sqrt{\left(-\color{blue}{im \cdot \frac{im}{re}}\right) \cdot 0.25} \]
      neg-mul-1 [=>] 42.32	\[ \sqrt{\color{blue}{\left(-1 \cdot \left(im \cdot \frac{im}{re}\right)\right)} \cdot 0.25} \]
      *-commutative [=>] 42.32	\[ \sqrt{\color{blue}{\left(\left(im \cdot \frac{im}{re}\right) \cdot -1\right)} \cdot 0.25} \]
      associate-*l* [=>] 42.32	\[ \sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right) \cdot \left(-1 \cdot 0.25\right)}} \]
      metadata-eval [=>] 42.32	\[ \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-0.25}} \]

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

   1. Initial program 56.83

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

   2. Simplified 11.58

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

      [Start] 56.83	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 56.83	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 11.58	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 15.23

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

Alternatives

Alternative 1
Error	40.07%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{-112}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	39.65%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -2.5 \cdot 10^{-112}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	40.62%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{-112}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 4
Error	56.99%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5
Error	73.63%
Cost	6720

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

Reproduce

herbie shell --seed 2023115 
(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)))))