math.sqrt on complex, real part

Percentage Accurate: 41.8% → 82.3%

Time: 11.8s

Precision: binary64

Cost: 13444

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+98}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e+98)
   (* 0.5 (sqrt (* im (/ (- im) re))))
   (* 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 (re <= -2.2e+98) {
		tmp = 0.5 * sqrt((im * (-im / re)));
	} 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 (re <= -2.2e+98) {
		tmp = 0.5 * Math.sqrt((im * (-im / re)));
	} 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 re <= -2.2e+98:
		tmp = 0.5 * math.sqrt((im * (-im / re)))
	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 (re <= -2.2e+98)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(-im) / re))));
	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 (re <= -2.2e+98)
		tmp = 0.5 * sqrt((im * (-im / re)));
	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[re, -2.2e+98], N[(0.5 * N[Sqrt[N[(im * N[((-im) / re), $MachinePrecision]), $MachinePrecision]], $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	41.8%
Target	49.1%
Herbie	82.3%

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

Derivation

1. Split input into 2 regimes

2. if re < -2.20000000000000009e98

   1. Initial program 6.8%

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

   2. Taylor expanded in re around -inf 19.9%

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

   3. Simplified 20.6%

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

      [Start] 19.9%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(-1 \cdot re + -0.5 \cdot \frac{{im}^{2}}{re}\right) + re\right)} \]
      +-commutative [=>] 19.9%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + -1 \cdot re\right)} + re\right)} \]
      mul-1-neg [=>] 19.9%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(-0.5 \cdot \frac{{im}^{2}}{re} + \color{blue}{\left(-re\right)}\right) + re\right)} \]
      unsub-neg [=>] 19.9%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} - re\right)} + re\right)} \]
      *-commutative [=>] 19.9%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\frac{{im}^{2}}{re} \cdot -0.5} - re\right) + re\right)} \]
      unpow2 [=>] 19.9%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5 - re\right) + re\right)} \]
      associate-/l* [=>] 20.6%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\frac{im}{\frac{re}{im}}} \cdot -0.5 - re\right) + re\right)} \]

   4. Applied egg-rr 54.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{\mathsf{fma}\left(\frac{im \cdot im}{re}, -0.5, 0\right) \cdot 2}\right)} \]
      Step-by-step derivation

      [Start] 20.6%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) + re\right)} \]
      add-log-exp [=>] 19.6%	\[ 0.5 \cdot \color{blue}{\log \left(e^{\sqrt{2 \cdot \left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) + re\right)}}\right)} \]
      *-un-lft-identity [=>] 19.6%	\[ 0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{2 \cdot \left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) + re\right)}}\right)} \]
      log-prod [=>] 19.6%	\[ 0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{2 \cdot \left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) + re\right)}}\right)\right)} \]
      metadata-eval [=>] 19.6%	\[ 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{2 \cdot \left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) + re\right)}}\right)\right) \]
      add-log-exp [<=] 20.6%	\[ 0.5 \cdot \left(0 + \color{blue}{\sqrt{2 \cdot \left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) + re\right)}}\right) \]
      *-commutative [=>] 20.6%	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) + re\right) \cdot 2}}\right) \]
      associate-+l- [=>] 66.1%	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - \left(re - re\right)\right)} \cdot 2}\right) \]
      fma-neg [=>] 66.1%	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{im}{\frac{re}{im}}, -0.5, -\left(re - re\right)\right)} \cdot 2}\right) \]
      associate-/r/ [=>] 66.2%	\[ 0.5 \cdot \left(0 + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{im}{re} \cdot im}, -0.5, -\left(re - re\right)\right) \cdot 2}\right) \]
      associate-*l/ [=>] 54.9%	\[ 0.5 \cdot \left(0 + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{im \cdot im}{re}}, -0.5, -\left(re - re\right)\right) \cdot 2}\right) \]
      +-inverses [=>] 54.9%	\[ 0.5 \cdot \left(0 + \sqrt{\mathsf{fma}\left(\frac{im \cdot im}{re}, -0.5, -\color{blue}{0}\right) \cdot 2}\right) \]
      metadata-eval [=>] 54.9%	\[ 0.5 \cdot \left(0 + \sqrt{\mathsf{fma}\left(\frac{im \cdot im}{re}, -0.5, \color{blue}{0}\right) \cdot 2}\right) \]

   5. Simplified 66.2%

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

      [Start] 54.9%	\[ 0.5 \cdot \left(0 + \sqrt{\mathsf{fma}\left(\frac{im \cdot im}{re}, -0.5, 0\right) \cdot 2}\right) \]
      +-lft-identity [=>] 54.9%	\[ 0.5 \cdot \color{blue}{\sqrt{\mathsf{fma}\left(\frac{im \cdot im}{re}, -0.5, 0\right) \cdot 2}} \]
      fma-udef [=>] 54.9%	\[ 0.5 \cdot \sqrt{\color{blue}{\left(\frac{im \cdot im}{re} \cdot -0.5 + 0\right)} \cdot 2} \]
      +-rgt-identity [=>] 54.9%	\[ 0.5 \cdot \sqrt{\color{blue}{\left(\frac{im \cdot im}{re} \cdot -0.5\right)} \cdot 2} \]
      *-commutative [=>] 54.9%	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}} \]
      *-commutative [=>] 54.9%	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{im \cdot im}{re}\right)}} \]
      associate-*r* [=>] 54.9%	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot -0.5\right) \cdot \frac{im \cdot im}{re}}} \]
      metadata-eval [=>] 54.9%	\[ 0.5 \cdot \sqrt{\color{blue}{-1} \cdot \frac{im \cdot im}{re}} \]
      neg-mul-1 [<=] 54.9%	\[ 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
      associate-*r/ [<=] 66.2%	\[ 0.5 \cdot \sqrt{-\color{blue}{im \cdot \frac{im}{re}}} \]
      distribute-rgt-neg-in [=>] 66.2%	\[ 0.5 \cdot \sqrt{\color{blue}{im \cdot \left(-\frac{im}{re}\right)}} \]

   if -2.20000000000000009e98 < re

   1. Initial program 49.3%

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

   2. Simplified 88.7%

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

      [Start] 49.3%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 49.3%	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 88.7%	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+98}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.3%
Cost	13444

\[\begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+98}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	58.7%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;im \leq -2.9 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(\frac{re}{\frac{im}{re}} \cdot -0.5 - im\right)\right)}\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-112}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Accuracy	58.5%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(-2\right)}\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Accuracy	58.7%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Accuracy	58.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -7 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(-2\right)}\\ \mathbf{elif}\;im \leq 1.22 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Accuracy	42.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-108}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Accuracy	26.5%
Cost	6720

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

Reproduce

herbie shell --seed 2023277 
(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)))))