math.sqrt on complex, real part

Average Error: 39.1 → 11.3

Time: 10.3s

Precision: binary64

Cost: 45956

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -520000000000:\\ \;\;\;\;0.5 \cdot {\left({\left(\sqrt{2}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(0.5 \cdot {im}^{2}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -520000000000.0)
   (*
    0.5
    (pow
     (*
      (pow (sqrt 2.0) 0.3333333333333333)
      (exp
       (*
        0.16666666666666666
        (+ (log (/ -1.0 re)) (log (* 0.5 (pow im 2.0)))))))
     3.0))
   (* 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 <= -520000000000.0) {
		tmp = 0.5 * pow((pow(sqrt(2.0), 0.3333333333333333) * exp((0.16666666666666666 * (log((-1.0 / re)) + log((0.5 * pow(im, 2.0))))))), 3.0);
	} 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 <= -520000000000.0) {
		tmp = 0.5 * Math.pow((Math.pow(Math.sqrt(2.0), 0.3333333333333333) * Math.exp((0.16666666666666666 * (Math.log((-1.0 / re)) + Math.log((0.5 * Math.pow(im, 2.0))))))), 3.0);
	} 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 <= -520000000000.0:
		tmp = 0.5 * math.pow((math.pow(math.sqrt(2.0), 0.3333333333333333) * math.exp((0.16666666666666666 * (math.log((-1.0 / re)) + math.log((0.5 * math.pow(im, 2.0))))))), 3.0)
	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 <= -520000000000.0)
		tmp = Float64(0.5 * (Float64((sqrt(2.0) ^ 0.3333333333333333) * exp(Float64(0.16666666666666666 * Float64(log(Float64(-1.0 / re)) + log(Float64(0.5 * (im ^ 2.0))))))) ^ 3.0));
	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 <= -520000000000.0)
		tmp = 0.5 * (((sqrt(2.0) ^ 0.3333333333333333) * exp((0.16666666666666666 * (log((-1.0 / re)) + log((0.5 * (im ^ 2.0))))))) ^ 3.0);
	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, -520000000000.0], N[(0.5 * N[Power[N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] + N[Log[N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $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	39.1
Target	34.2
Herbie	11.3

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

Derivation

1. Split input into 2 regimes

2. if re < -5.2e11

   1. Initial program 57.3

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

   2. Simplified 38.2

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

      [Start] 57.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 57.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 38.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Applied egg-rr 38.5

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}}\right)}^{3}} \]

   4. Taylor expanded in re around -inf 30.0

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(\sqrt{2} \cdot 1\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(0.5 \cdot {im}^{2}\right)\right)}\right)}}^{3} \]

   if -5.2e11 < re

   1. Initial program 33.3

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

   2. Simplified 5.3

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

      [Start] 33.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 33.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 5.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -520000000000:\\ \;\;\;\;0.5 \cdot {\left({\left(\sqrt{2}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(0.5 \cdot {im}^{2}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	26884

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

Alternative 2
Error	27.0
Cost	7837

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := \sqrt{-re}\\ \mathbf{if}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -1.48 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot \frac{im}{t_1}\\ \mathbf{elif}\;im \leq 2.85 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{-157} \lor \neg \left(im \leq 1.85 \cdot 10^{-84}\right) \land im \leq 4.3 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{t_1}{im}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	26.9
Cost	7837

\[\begin{array}{l} t_0 := \sqrt{-re}\\ \mathbf{if}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{-271}:\\ \;\;\;\;0.5 \cdot {\left(im \cdot \frac{im}{re} + 2 \cdot \left(re \cdot 2\right)\right)}^{0.5}\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot \frac{im}{t_0}\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{-227}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{-156} \lor \neg \left(im \leq 5.5 \cdot 10^{-88}\right) \land im \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{t_0}{im}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	27.6
Cost	7773

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-157} \lor \neg \left(im \leq 6 \cdot 10^{-80}\right) \land im \leq 2.25 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	28.3
Cost	7708

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ t_2 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3.2 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{-271}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6
Error	26.9
Cost	7509

\[\begin{array}{l} \mathbf{if}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -3.5 \cdot 10^{-270}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-157} \lor \neg \left(im \leq 5.4 \cdot 10^{-83}\right) \land im \leq 1.7 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 7
Error	27.6
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 8
Error	36.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 3.6 \cdot 10^{-130}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 9
Error	47.3
Cost	6720

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

Reproduce

herbie shell --seed 2023073 
(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)))))