math.sqrt on complex, real part

Average Error: 38.9 → 11.0

Time: 9.3s

Precision: binary64

Cost: 13444

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+155}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{re} \cdot \left(-im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e+155)
   (* 0.5 (sqrt (* (/ im re) (- im))))
   (* 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 <= -3.6e+155) {
		tmp = 0.5 * sqrt(((im / re) * -im));
	} 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 <= -3.6e+155) {
		tmp = 0.5 * Math.sqrt(((im / re) * -im));
	} 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 <= -3.6e+155:
		tmp = 0.5 * math.sqrt(((im / re) * -im))
	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 <= -3.6e+155)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / re) * Float64(-im))));
	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 <= -3.6e+155)
		tmp = 0.5 * sqrt(((im / re) * -im));
	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, -3.6e+155], N[(0.5 * N[Sqrt[N[(N[(im / re), $MachinePrecision] * (-im)), $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	38.9
Target	34.0
Herbie	11.0

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

Derivation

1. Split input into 2 regimes

2. if re < -3.60000000000000007e155

   1. Initial program 64.0

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

   2. Simplified 42.3

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

      [Start] 64.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 64.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 42.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 39.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.125 \cdot \frac{{im}^{4}}{{re}^{3}} + -0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]

   4. Simplified 39.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{im}{\frac{re}{im}}, -0.5, {im}^{4} \cdot \frac{0.125}{{re}^{3}}\right)}} \]
      Proof

      [Start] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.125 \cdot \frac{{im}^{4}}{{re}^{3}} + -0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      +-commutative [=>] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}} \]
      *-commutative [=>] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{{im}^{2}}{re} \cdot -0.5} + 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)} \]
      fma-def [=>] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{{im}^{2}}{re}, -0.5, 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}} \]
      unpow2 [=>] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{\color{blue}{im \cdot im}}{re}, -0.5, 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)} \]
      associate-/l* [=>] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\frac{im}{\frac{re}{im}}}, -0.5, 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)} \]
      associate-*r/ [=>] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{im}{\frac{re}{im}}, -0.5, \color{blue}{\frac{0.125 \cdot {im}^{4}}{{re}^{3}}}\right)} \]
      *-commutative [=>] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{im}{\frac{re}{im}}, -0.5, \frac{\color{blue}{{im}^{4} \cdot 0.125}}{{re}^{3}}\right)} \]
      associate-*r/ [<=] 39.8	\[ 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{im}{\frac{re}{im}}, -0.5, \color{blue}{{im}^{4} \cdot \frac{0.125}{{re}^{3}}}\right)} \]

   5. Taylor expanded in im around 0 32.1

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

   6. Simplified 23.2

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

      [Start] 32.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      unpow2 [=>] 32.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]
      associate-/l* [=>] 23.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \color{blue}{\frac{im}{\frac{re}{im}}}\right)} \]
      associate-/r/ [=>] 23.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \color{blue}{\left(\frac{im}{re} \cdot im\right)}\right)} \]

   7. Taylor expanded in im around 0 32.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]

   8. Simplified 23.2

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

      [Start] 32.1	\[ 0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re}} \]
      mul-1-neg [=>] 32.1	\[ 0.5 \cdot \sqrt{\color{blue}{-\frac{{im}^{2}}{re}}} \]
      unpow2 [=>] 32.1	\[ 0.5 \cdot \sqrt{-\frac{\color{blue}{im \cdot im}}{re}} \]
      associate-*l/ [<=] 23.2	\[ 0.5 \cdot \sqrt{-\color{blue}{\frac{im}{re} \cdot im}} \]
      distribute-rgt-neg-in [=>] 23.2	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{im}{re} \cdot \left(-im\right)}} \]

   if -3.60000000000000007e155 < re

   1. Initial program 35.4

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

   2. Simplified 9.4

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

      [Start] 35.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 35.4	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 9.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+155}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{re} \cdot \left(-im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.4
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.05 \cdot 10^{-137}:\\ \;\;\;\;\frac{im}{\sqrt{-re}} \cdot -0.5\\ \mathbf{elif}\;im \leq -6.6 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{-283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-235}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	25.8
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.8 \cdot 10^{-177}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	26.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -3.1 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 7 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	25.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{-176}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	26.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	36.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Error	46.9
Cost	6720

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

Reproduce

herbie shell --seed 2023060 
(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)))))