math.sqrt on complex, real part

Average Error: 38.9 → 10.3

Time: 11.9s

Precision: binary64

Cost: 26884

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (/ (* im 0.5) (sqrt (- 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 (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = (im * 0.5) / sqrt(-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 (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = (im * 0.5) / Math.sqrt(-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 math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im * im)))))) <= 0.0:
		tmp = (im * 0.5) / math.sqrt(-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 (sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))))) <= 0.0)
		tmp = Float64(Float64(im * 0.5) / sqrt(Float64(-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 (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0)
		tmp = (im * 0.5) / sqrt(-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[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[(im * 0.5), $MachinePrecision] / N[Sqrt[(-re)], $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	38.9
Target	33.8
Herbie	10.3

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.1

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

   2. Simplified 57.1

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

      [Start] 57.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 57.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 57.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 29.1

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

   4. Simplified 29.1

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

      [Start] 29.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 29.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 29.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 29.3

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

   6. Applied egg-rr 29.4

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

   7. Simplified 29.4

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

      [Start] 29.4	\[ \frac{0.5 \cdot im}{\sqrt{-re}} \]
      *-commutative [<=] 29.4	\[ \frac{\color{blue}{im \cdot 0.5}}{\sqrt{-re}} \]

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

   1. Initial program 36.4

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

   2. Simplified 7.7

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

      [Start] 36.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 36.4	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 7.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 10.3

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

Alternatives

Alternative 1
Error	27.5
Cost	7772

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := \frac{im \cdot 0.5}{\sqrt{-re}}\\ t_2 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -1.02 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4.4 \cdot 10^{-306}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	27.3
Cost	7772

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ t_2 := \frac{im \cdot 0.5}{\sqrt{-re}}\\ \mathbf{if}\;im \leq -1.2 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.25 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.26 \cdot 10^{-306}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	27.3
Cost	7772

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ t_2 := \sqrt{-re}\\ t_3 := \frac{im \cdot 0.5}{t_2}\\ \mathbf{if}\;im \leq -3.9 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -4.8 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3.1 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{im}{t_2}\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{-166}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+35}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	27.6
Cost	7708

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := \frac{im \cdot 0.5}{\sqrt{-re}}\\ t_2 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.4 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.05 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -9.4 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1 \cdot 10^{-306}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{-78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 5
Error	26.6
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -9.2 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.05 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6
Error	37.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 3 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 7
Error	62.9
Cost	6720

\[-0.5 \cdot \frac{im}{\sqrt{re}} \]

Alternative 8
Error	48.0
Cost	6720

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

Reproduce

herbie shell --seed 2023046 
(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)))))