math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.6 → 6.7

Time: 13.3s

Precision: binary64

Cost: 26884

\[im > 0\]

Math

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

↓

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

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 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (* im (pow re -0.5)))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

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 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	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 * (Math.sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	}
	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 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	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(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))) <= 0.0)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	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 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = 0.5 * (im * (re ^ -0.5));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	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[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

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 57.7

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

   2. Simplified 57.7

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (-.f64 (hypot.f64 re im) re)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (-.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) re)))): 131 points increase in error, 0 points decrease in error

   3. Taylor expanded in re around inf 29.9

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

   4. Simplified 29.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)}} \]
      Proof
      (*.f64 (/.f64 1/2 re) (*.f64 im im)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 1/2 re) (Rewrite<= unpow2_binary64 (pow.f64 im 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 1/2 (/.f64 re (pow.f64 im 2)))): 22 points increase in error, 14 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1/2 (pow.f64 im 2)) re)): 13 points increase in error, 20 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (pow.f64 im 2) re))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 26.6

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

   6. Taylor expanded in im around 0 0.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

   7. Simplified 0.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]
      Proof
      (*.f64 im (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 1 re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 im (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 1 re))))): 40 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 im (*.f64 (sqrt.f64 2) (sqrt.f64 1/2))) (sqrt.f64 (/.f64 1 re)))): 23 points increase in error, 11 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) im)) (sqrt.f64 (/.f64 1 re))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) im))) (sqrt.f64 (/.f64 1 re))): 0 points increase in error, 40 points decrease in error

   8. Applied egg-rr 0.3

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

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

   1. Initial program 36.2

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

   2. Simplified 7.5

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (-.f64 (hypot.f64 re im) re)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (-.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) re)))): 131 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 6.7

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

Alternatives

Alternative 1
Error	16.4
Cost	13912

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -2.1329180973142227 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.442788729239936 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -810983252512360700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -6.5963888333294415 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.2054652127399668 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.041582101861977 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \mathbf{elif}\;re \leq 2.2592245848169113 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 6.821039101000745 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \frac{1}{\sqrt{re}}\right)\\ \mathbf{elif}\;re \leq 1.0527031489691478 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 2
Error	16.5
Cost	13776

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{if}\;re \leq -2.1329180973142227 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.442788729239936 \cdot 10^{+135}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -810983252512360700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.2592245848169113 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im - re} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 6.821039101000745 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \frac{1}{\sqrt{re}}\right)\\ \mathbf{elif}\;re \leq 1.0527031489691478 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3
Error	16.3
Cost	7576

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ t_2 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -2.1329180973142227 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.442788729239936 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;re \leq -810983252512360700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.2592245848169113 \cdot 10^{-58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;re \leq 6.821039101000745 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 1.0527031489691478 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.3
Cost	7576

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -2.1329180973142227 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.442788729239936 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -810983252512360700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.2592245848169113 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 6.821039101000745 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \frac{1}{\sqrt{re}}\right)\\ \mathbf{elif}\;re \leq 1.0527031489691478 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 5
Error	16.3
Cost	7312

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -1.828707653476246 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.442788729239936 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1056878323960.99:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.0527031489691478 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 6
Error	23.9
Cost	7116

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -1.828707653476246 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.442788729239936 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1056878323960.99:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	46.8
Cost	6720

\[0.5 \cdot \sqrt{re \cdot -4} \]

Reproduce

herbie shell --seed 2022308 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))