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

Average Error: 38.5 → 6.1

Time: 6.7s

Precision: binary64

Cost: 26884

\[im > 0\]

Math

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

↓

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

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 (sqrt re)))
   (* 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 / sqrt(re));
	} 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.sqrt(re));
	} 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.sqrt(re))
	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 / sqrt(re)));
	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 / sqrt(re));
	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[Sqrt[re], $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 57.8

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

   2. Simplified 57.8

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      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)))): 132 points increase in error, 0 points decrease in error

   3. Taylor expanded in im around 0 0.8

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

   4. Simplified 0.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 im (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 1 re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 im (sqrt.f64 1/2)) (sqrt.f64 (/.f64 1 re))))): 18 points increase in error, 22 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 1/2) im)) (sqrt.f64 (/.f64 1 re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) im)) (sqrt.f64 (/.f64 1 re)))): 30 points increase in error, 23 points decrease in error

   5. Applied egg-rr 0.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + {\left(2 \cdot \frac{0.5}{re}\right)}^{0.5} \cdot im\right)} \]

   6. Simplified 0.3

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
      Proof
      (/.f64 im (sqrt.f64 re)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 im)) (sqrt.f64 re)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1 im) (Rewrite<= unpow1/2_binary64 (pow.f64 re 1/2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1 im) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 re) 1/2)))): 61 points increase in error, 65 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (exp.f64 (*.f64 (log.f64 re) 1/2))) im)): 20 points increase in error, 16 points decrease in error
      (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (*.f64 (log.f64 re) 1/2)))) im): 17 points increase in error, 17 points decrease in error
      (*.f64 (exp.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (log.f64 re)) 1/2))) im): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> exp-prod_binary64 (pow.f64 (exp.f64 (neg.f64 (log.f64 re))) 1/2)) im): 2 points increase in error, 8 points decrease in error
      (*.f64 (pow.f64 (Rewrite=> exp-neg_binary64 (/.f64 1 (exp.f64 (log.f64 re)))) 1/2) im): 10 points increase in error, 4 points decrease in error
      (*.f64 (pow.f64 (/.f64 1 (Rewrite=> rem-exp-log_binary64 re)) 1/2) im): 64 points increase in error, 62 points decrease in error
      (*.f64 (pow.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1/2)) re) 1/2) im): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1/2 re))) 1/2) im): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 (pow.f64 (*.f64 2 (/.f64 1/2 re)) 1/2) im))): 0 points increase in error, 0 points decrease in error

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

   1. Initial program 35.8

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

   2. Simplified 6.9

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      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)))): 132 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 6.1

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

Alternatives

Alternative 1
Error	18.4
Cost	13512

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -8.2 \cdot 10^{+174}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im - re} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+144}:\\ \;\;\;\;\frac{0.5}{\sqrt{re} \cdot \frac{1}{im}}\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	18.1
Cost	7644

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ t_2 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -7.4 \cdot 10^{+174}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+144}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+184}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	18.1
Cost	7644

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ t_2 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -7.4 \cdot 10^{+174}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+144}:\\ \;\;\;\;\frac{0.5}{\sqrt{re} \cdot \frac{1}{im}}\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+184}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.9
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{if}\;re \leq -4.9 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	26.2
Cost	7116

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{if}\;re \leq 4.7 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	30.8
Cost	6720

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

Reproduce

herbie shell --seed 2022325 
(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)))))