math.sqrt on complex, real part

Average Error: 38.5 → 9.6

Time: 8.8s

Precision: binary64

Cost: 27401

Math

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

↓

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-302} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -0.25}\\ \end{array} \]

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 -5e-302) (not (<= t_0 0.0)))
     (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))
     (sqrt (* (* im (/ im re)) -0.25)))))

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 <= -5e-302) || !(t_0 <= 0.0)) {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	} else {
		tmp = sqrt(((im * (im / re)) * -0.25));
	}
	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 <= -5e-302) || !(t_0 <= 0.0)) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	} else {
		tmp = Math.sqrt(((im * (im / re)) * -0.25));
	}
	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 <= -5e-302) or not (t_0 <= 0.0):
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	else:
		tmp = math.sqrt(((im * (im / re)) * -0.25))
	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 <= -5e-302) || !(t_0 <= 0.0))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	else
		tmp = sqrt(Float64(Float64(im * Float64(im / re)) * -0.25));
	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 <= -5e-302) || ~((t_0 <= 0.0)))
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	else
		tmp = sqrt(((im * (im / re)) * -0.25));
	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, -5e-302], 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[Sqrt[N[(N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]

Target

Original	38.5
Target	33.5
Herbie	9.6

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -5.00000000000000033e-302 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 7.3

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      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 [=>] 7.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   if -5.00000000000000033e-302 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 56.7

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

   2. Simplified 55.8

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

      [Start] 56.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 56.7	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 55.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 30.1

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

   4. Simplified 30.1

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

      [Start] 30.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 30.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 30.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 30.1

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

   6. Simplified 26.8

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

      [Start] 30.1	\[ \sqrt{\frac{im \cdot \left(-im\right)}{re} \cdot 0.25} \]
      distribute-rgt-neg-out [=>] 30.1	\[ \sqrt{\frac{\color{blue}{-im \cdot im}}{re} \cdot 0.25} \]
      distribute-frac-neg [=>] 30.1	\[ \sqrt{\color{blue}{\left(-\frac{im \cdot im}{re}\right)} \cdot 0.25} \]
      associate-*r/ [<=] 26.8	\[ \sqrt{\left(-\color{blue}{im \cdot \frac{im}{re}}\right) \cdot 0.25} \]
      mul-1-neg [<=] 26.8	\[ \sqrt{\color{blue}{\left(-1 \cdot \left(im \cdot \frac{im}{re}\right)\right)} \cdot 0.25} \]
      *-commutative [=>] 26.8	\[ \sqrt{\color{blue}{\left(\left(im \cdot \frac{im}{re}\right) \cdot -1\right)} \cdot 0.25} \]
      associate-*l* [=>] 26.8	\[ \sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right) \cdot \left(-1 \cdot 0.25\right)}} \]
      metadata-eval [=>] 26.8	\[ \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-0.25}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -5 \cdot 10^{-302} \lor \neg \left(re + \sqrt{re \cdot re + im \cdot im} \leq 0\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -0.25}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.0
Cost	13512

\[\begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -4.25 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{im \cdot -0.25} \cdot \sqrt{\frac{im}{re}}\\ \mathbf{elif}\;im \leq -1.25 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{-226} \lor \neg \left(im \leq 5.6 \cdot 10^{-182}\right) \land im \leq 1.25 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	27.0
Cost	7641

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4.4 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.45 \cdot 10^{-195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{-229} \lor \neg \left(im \leq 5.7 \cdot 10^{-182}\right) \land im \leq 1.2 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	26.6
Cost	7641

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1 \cdot 10^{-83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -3.35 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.25 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{-226} \lor \neg \left(im \leq 1.12 \cdot 10^{-180}\right) \land im \leq 3.1 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	27.2
Cost	7513

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -4.8 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.9 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.25 \cdot 10^{-194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{-227} \lor \neg \left(im \leq 5.8 \cdot 10^{-182}\right) \land im \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	36.6
Cost	7117

\[\begin{array}{l} \mathbf{if}\;re \leq 6.4 \cdot 10^{-65} \lor \neg \left(re \leq 5200000000000\right) \land re \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6
Error	47.2
Cost	6720

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

Reproduce

herbie shell --seed 2023045 
(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)))))