math.sqrt on complex, real part

Average Error: 38.8 → 9.5

Time: 9.3s

Precision: binary64

Cost: 27401

Math

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

↓

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-303} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{-re}{im}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ re (sqrt (+ (* re re) (* im im))))))
   (if (or (<= t_0 -1e-303) (not (<= t_0 0.0)))
     (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))
     (* 0.5 (sqrt (/ im (/ (- 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 t_0 = re + sqrt(((re * re) + (im * im)));
	double tmp;
	if ((t_0 <= -1e-303) || !(t_0 <= 0.0)) {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	} else {
		tmp = 0.5 * sqrt((im / (-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 t_0 = re + Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if ((t_0 <= -1e-303) || !(t_0 <= 0.0)) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	} else {
		tmp = 0.5 * Math.sqrt((im / (-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):
	t_0 = re + math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if (t_0 <= -1e-303) or not (t_0 <= 0.0):
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	else:
		tmp = 0.5 * math.sqrt((im / (-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)
	t_0 = Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im))))
	tmp = 0.0
	if ((t_0 <= -1e-303) || !(t_0 <= 0.0))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(im / Float64(Float64(-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)
	t_0 = re + sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if ((t_0 <= -1e-303) || ~((t_0 <= 0.0)))
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	else
		tmp = 0.5 * sqrt((im / (-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_] := Block[{t$95$0 = N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-303], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im / N[((-re) / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	38.8
Target	33.5
Herbie	9.5

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -9.99999999999999931e-304 or 0.0 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 36.1

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

   2. Simplified 7.1

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

      [Start] 36.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 36.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 7.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   if -9.99999999999999931e-304 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 56.5

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

   2. Simplified 55.9

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

      [Start] 56.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 56.5	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 55.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Applied egg-rr 55.9

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

   4. Taylor expanded in re around -inf 29.8

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

   5. Simplified 25.9

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

      [Start] 29.8	\[ 0.5 \cdot {\left({\left(\left(-0.5 \cdot \frac{{im}^{2}}{re}\right) \cdot 2\right)}^{0.25}\right)}^{2} \]
      *-commutative [=>] 29.8	\[ 0.5 \cdot {\left({\left(\color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)} \cdot 2\right)}^{0.25}\right)}^{2} \]
      unpow2 [=>] 29.8	\[ 0.5 \cdot {\left({\left(\left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right) \cdot 2\right)}^{0.25}\right)}^{2} \]
      associate-/l* [=>] 25.9	\[ 0.5 \cdot {\left({\left(\left(\color{blue}{\frac{im}{\frac{re}{im}}} \cdot -0.5\right) \cdot 2\right)}^{0.25}\right)}^{2} \]

   6. Taylor expanded in im around 0 35.2

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

   7. Applied egg-rr 52.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\frac{im \cdot im}{-re}\right)}^{0.25}\right)}^{2}\right)} - 1\right)} \]

   8. Simplified 25.7

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

      [Start] 52.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left({\left(\frac{im \cdot im}{-re}\right)}^{0.25}\right)}^{2}\right)} - 1\right) \]
      expm1-def [=>] 29.9	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\frac{im \cdot im}{-re}\right)}^{0.25}\right)}^{2}\right)\right)} \]
      expm1-log1p [=>] 29.8	\[ 0.5 \cdot \color{blue}{{\left({\left(\frac{im \cdot im}{-re}\right)}^{0.25}\right)}^{2}} \]
      unpow2 [=>] 29.8	\[ 0.5 \cdot \color{blue}{\left({\left(\frac{im \cdot im}{-re}\right)}^{0.25} \cdot {\left(\frac{im \cdot im}{-re}\right)}^{0.25}\right)} \]
      pow-sqr [=>] 29.6	\[ 0.5 \cdot \color{blue}{{\left(\frac{im \cdot im}{-re}\right)}^{\left(2 \cdot 0.25\right)}} \]
      rem-exp-log [<=] 31.6	\[ 0.5 \cdot {\color{blue}{\left(e^{\log \left(\frac{im \cdot im}{-re}\right)}\right)}}^{\left(2 \cdot 0.25\right)} \]
      metadata-eval [=>] 31.6	\[ 0.5 \cdot {\left(e^{\log \left(\frac{im \cdot im}{-re}\right)}\right)}^{\color{blue}{0.5}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -1 \cdot 10^{-303} \lor \neg \left(re + \sqrt{re \cdot re + im \cdot im} \leq 0\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{-re}{im}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.0
Cost	7508

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -8 \cdot 10^{-198}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	26.5
Cost	7508

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	27.2
Cost	7444

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -3.2 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -6.1 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 4
Error	27.2
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -3.3 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{-176}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	37.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 7.4 \cdot 10^{-177}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	48.0
Cost	6720

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

Reproduce

herbie shell --seed 2023018 
(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)))))