math.sqrt on complex, real part

Percentage Accurate: 41.7% → 84.0%

Time: 8.2s

Precision: binary64

Cost: 46916

Math

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

↓

\[\begin{array}{l} t_0 := {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\\ \mathbf{if}\;re \leq -1.26 \cdot 10^{+128}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-0.125, \frac{im \cdot im}{re} \cdot \frac{t_0}{re}, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (pow (* (pow (* im im) 0.25) (pow (/ -1.0 re) 0.25)) 2.0)))
   (if (<= re -1.26e+128)
     (* 0.5 (fma -0.125 (* (/ (* im im) re) (/ t_0 re)) t_0))
     (sqrt (* 0.5 (+ 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 t_0 = pow((pow((im * im), 0.25) * pow((-1.0 / re), 0.25)), 2.0);
	double tmp;
	if (re <= -1.26e+128) {
		tmp = 0.5 * fma(-0.125, (((im * im) / re) * (t_0 / re)), t_0);
	} else {
		tmp = sqrt((0.5 * (re + 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)
	t_0 = Float64((Float64(im * im) ^ 0.25) * (Float64(-1.0 / re) ^ 0.25)) ^ 2.0
	tmp = 0.0
	if (re <= -1.26e+128)
		tmp = Float64(0.5 * fma(-0.125, Float64(Float64(Float64(im * im) / re) * Float64(t_0 / re)), t_0));
	else
		tmp = sqrt(Float64(0.5 * 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_] := Block[{t$95$0 = N[Power[N[(N[Power[N[(im * im), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(-1.0 / re), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[re, -1.26e+128], N[(0.5 * N[(-0.125 * N[(N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision] * N[(t$95$0 / re), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	41.7%
Target	48.8%
Herbie	84.0%

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

Derivation

1. Split input into 2 regimes

2. if re < -1.26000000000000009e128

   1. Initial program 3.8%

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

   2. Simplified 28.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Step-by-step derivation

      [Start] 3.8%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 3.8%	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 28.5%	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Applied egg-rr 28.6%

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

      [Start] 28.5%	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]
      add-sqr-sqrt [=>] 28.6%	\[ 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \cdot \sqrt{\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}}\right)} \]
      pow2 [=>] 28.6%	\[ 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}}\right)}^{2}} \]
      pow1/2 [=>] 28.6%	\[ 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{0.5}}}\right)}^{2} \]
      sqrt-pow1 [=>] 28.6%	\[ 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]
      *-commutative [=>] 28.6%	\[ 0.5 \cdot {\left({\color{blue}{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]
      metadata-eval [=>] 28.6%	\[ 0.5 \cdot {\left({\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Taylor expanded in re around -inf 76.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.125 \cdot \frac{{\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot {im}^{2}}{{re}^{2}} + {\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2}\right)} \]

   5. Simplified 86.2%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{im \cdot im}{re} \cdot \frac{{\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}}{re}, {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\right)} \]
      Step-by-step derivation

      [Start] 76.3%	\[ 0.5 \cdot \left(-0.125 \cdot \frac{{\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot {im}^{2}}{{re}^{2}} + {\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2}\right) \]
      fma-def [=>] 76.3%	\[ 0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot {im}^{2}}{{re}^{2}}, {\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2}\right)} \]

   if -1.26000000000000009e128 < re

   1. Initial program 50.0%

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

   2. Simplified 90.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Step-by-step derivation

      [Start] 50.0%	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 50.0%	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 90.9%	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25}} \]
      Step-by-step derivation

      [Start] 90.9%	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]
      add-sqr-sqrt [=>] 90.2%	\[ \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}}} \]
      sqrt-unprod [=>] 90.9%	\[ \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}} \]
      *-commutative [=>] 90.9%	\[ \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)} \]
      *-commutative [=>] 90.9%	\[ \sqrt{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right)}} \]
      swap-sqr [=>] 90.9%	\[ \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]
      add-sqr-sqrt [<=] 90.9%	\[ \sqrt{\color{blue}{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]
      *-commutative [=>] 90.9%	\[ \sqrt{\color{blue}{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]
      metadata-eval [=>] 90.9%	\[ \sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Simplified 90.9%

      \[\leadsto \color{blue}{\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5}} \]
      Step-by-step derivation

      [Start] 90.9%	\[ \sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25} \]
      associate-*l* [=>] 90.9%	\[ \sqrt{\color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot \left(2 \cdot 0.25\right)}} \]
      metadata-eval [=>] 90.9%	\[ \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot \color{blue}{0.5}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.26 \cdot 10^{+128}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-0.125, \frac{im \cdot im}{re} \cdot \frac{{\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}}{re}, {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.0%
Cost	46916

\[\begin{array}{l} t_0 := {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\\ \mathbf{if}\;re \leq -1.26 \cdot 10^{+128}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-0.125, \frac{im \cdot im}{re} \cdot \frac{t_0}{re}, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	84.3%
Cost	20100

\[\begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	82.8%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	57.9%
Cost	7512

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -1.6 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.5 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Accuracy	59.8%
Cost	7376

\[\begin{array}{l} \mathbf{if}\;im \leq -1.16 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -8 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6
Accuracy	58.8%
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{if}\;im \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Accuracy	59.3%
Cost	7248

\[\begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -8 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 8
Accuracy	42.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 4.5 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 9
Accuracy	26.2%
Cost	6464

\[\sqrt{re} \]

Reproduce

herbie shell --seed 2023178 
(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)))))