math.sqrt on complex, real part

Average Error: 38.4 → 10.0

Time: 8.0s

Precision: binary64

Cost: 26756

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt[3]{\sqrt{-re}} \cdot \sqrt[3]{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im im)))) 0.0)
   (/ (* im 0.5) (* (cbrt (sqrt (- re))) (cbrt 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 + sqrt(((re * re) + (im * im)))) <= 0.0) {
		tmp = (im * 0.5) / (cbrt(sqrt(-re)) * cbrt(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 + Math.sqrt(((re * re) + (im * im)))) <= 0.0) {
		tmp = (im * 0.5) / (Math.cbrt(Math.sqrt(-re)) * Math.cbrt(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 (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))) <= 0.0)
		tmp = Float64(Float64(im * 0.5) / Float64(cbrt(sqrt(Float64(-re))) * cbrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return 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[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * 0.5), $MachinePrecision] / N[(N[Power[N[Sqrt[(-re)], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[re, 1/3], $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	38.4
Target	33.7
Herbie	10.0

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.7

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

   2. Simplified 52.1

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

      [Start] 58.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 58.7	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 52.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 37.2

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

   4. Simplified 30.5

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

      [Start] 37.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 37.2	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 37.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]
      associate-/l* [=>] 30.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{im}{\frac{re}{im}}} \cdot -0.5\right)} \]
      associate-/r/ [=>] 30.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot -0.5\right)} \]

   5. Applied egg-rr 43.5

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\left(\frac{im}{re} \cdot im\right) \cdot -0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   6. Simplified 42.5

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(im \cdot \frac{-im}{re}\right)}^{1.5}}} \]
      Proof

      [Start] 43.5	\[ 0.5 \cdot {\left({\left(2 \cdot \left(\left(\frac{im}{re} \cdot im\right) \cdot -0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333} \]
      unpow1/3 [=>] 42.5	\[ 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\left(\frac{im}{re} \cdot im\right) \cdot -0.5\right)\right)}^{1.5}}} \]
      *-commutative [=>] 42.5	\[ 0.5 \cdot \sqrt[3]{{\left(2 \cdot \color{blue}{\left(-0.5 \cdot \left(\frac{im}{re} \cdot im\right)\right)}\right)}^{1.5}} \]
      associate-*r* [=>] 42.5	\[ 0.5 \cdot \sqrt[3]{{\color{blue}{\left(\left(2 \cdot -0.5\right) \cdot \left(\frac{im}{re} \cdot im\right)\right)}}^{1.5}} \]
      metadata-eval [=>] 42.5	\[ 0.5 \cdot \sqrt[3]{{\left(\color{blue}{-1} \cdot \left(\frac{im}{re} \cdot im\right)\right)}^{1.5}} \]
      mul-1-neg [=>] 42.5	\[ 0.5 \cdot \sqrt[3]{{\color{blue}{\left(-\frac{im}{re} \cdot im\right)}}^{1.5}} \]
      distribute-lft-neg-in [=>] 42.5	\[ 0.5 \cdot \sqrt[3]{{\color{blue}{\left(\left(-\frac{im}{re}\right) \cdot im\right)}}^{1.5}} \]
      distribute-frac-neg [<=] 42.5	\[ 0.5 \cdot \sqrt[3]{{\left(\color{blue}{\frac{-im}{re}} \cdot im\right)}^{1.5}} \]
      *-commutative [=>] 42.5	\[ 0.5 \cdot \sqrt[3]{{\color{blue}{\left(im \cdot \frac{-im}{re}\right)}}^{1.5}} \]

   7. Applied egg-rr 33.3

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

   8. Applied egg-rr 31.7

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot im}{\sqrt[3]{re}}}{\sqrt[3]{\sqrt{-re}}}} \]

   9. Simplified 31.7

      \[\leadsto \color{blue}{\frac{0.5 \cdot im}{\sqrt[3]{\sqrt{-re}} \cdot \sqrt[3]{re}}} \]
      Proof

      [Start] 31.7	\[ \frac{\frac{0.5 \cdot im}{\sqrt[3]{re}}}{\sqrt[3]{\sqrt{-re}}} \]
      associate-/l/ [=>] 31.7	\[ \color{blue}{\frac{0.5 \cdot im}{\sqrt[3]{\sqrt{-re}} \cdot \sqrt[3]{re}}} \]

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

   1. Initial program 35.0

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

   2. Simplified 6.3

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

      [Start] 35.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 35.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 6.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt[3]{\sqrt{-re}} \cdot \sqrt[3]{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	26756

\[\begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;\frac{im}{\sqrt[3]{\sqrt{-re}}} \cdot \frac{0.5}{\sqrt[3]{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2
Error	9.8
Cost	20356

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

Alternative 3
Error	25.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -3.25 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	25.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 8 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	29.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 7.4 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	45.0
Cost	6720

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

Alternative 7
Error	47.5
Cost	6592

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

Reproduce

herbie shell --seed 2023187 
(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)))))