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

Average Error: 39.1 → 6.3

Time: 11.9s

Precision: binary64

Cost: 20356

\[im > 0\]

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (/ (pow re -0.5) (/ 2.0 im))
   (* 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(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = pow(re, -0.5) / (2.0 / im);
	} 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(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = Math.pow(re, -0.5) / (2.0 / im);
	} 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(((re * re) + (im * im))) - re) <= 0.0:
		tmp = math.pow(re, -0.5) / (2.0 / im)
	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 (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64((re ^ -0.5) / Float64(2.0 / im));
	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(((re * re) + (im * im))) - re) <= 0.0)
		tmp = (re ^ -0.5) / (2.0 / im);
	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[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(N[Power[re, -0.5], $MachinePrecision] / N[(2.0 / im), $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 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 59.0

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

   2. Taylor expanded in re around inf 37.1

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

   3. Simplified 29.3

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

      [Start] 37.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      unpow2 [=>] 37.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]
      associate-/l* [=>] 29.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \color{blue}{\frac{im}{\frac{re}{im}}}\right)} \]

   4. Taylor expanded in im around 0 5.7

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

   5. Applied egg-rr 55.1

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\right)} - 1} \]

   6. Simplified 5.7

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

      [Start] 55.1	\[ e^{\mathsf{log1p}\left(0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\right)} - 1 \]
      expm1-def [=>] 5.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\right)\right)} \]
      expm1-log1p [=>] 5.7	\[ \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
      *-commutative [=>] 5.7	\[ \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5} \]
      associate-*l* [=>] 5.7	\[ \color{blue}{\sqrt{\frac{1}{re}} \cdot \left(im \cdot 0.5\right)} \]
      unpow-1 [<=] 5.7	\[ \sqrt{\color{blue}{{re}^{-1}}} \cdot \left(im \cdot 0.5\right) \]
      metadata-eval [<=] 5.7	\[ \sqrt{{re}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(im \cdot 0.5\right) \]
      pow-sqr [<=] 5.7	\[ \sqrt{\color{blue}{{re}^{-0.5} \cdot {re}^{-0.5}}} \cdot \left(im \cdot 0.5\right) \]
      rem-sqrt-square [=>] 5.7	\[ \color{blue}{\left|{re}^{-0.5}\right|} \cdot \left(im \cdot 0.5\right) \]
      sqr-pow [=>] 6.0	\[ \left|\color{blue}{{re}^{\left(\frac{-0.5}{2}\right)} \cdot {re}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot \left(im \cdot 0.5\right) \]
      fabs-sqr [=>] 6.0	\[ \color{blue}{\left({re}^{\left(\frac{-0.5}{2}\right)} \cdot {re}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot \left(im \cdot 0.5\right) \]
      sqr-pow [<=] 5.7	\[ \color{blue}{{re}^{-0.5}} \cdot \left(im \cdot 0.5\right) \]

   7. Taylor expanded in im around 0 5.7

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

   8. Simplified 5.8

      \[\leadsto \color{blue}{\frac{{re}^{-0.5}}{\frac{2}{im}}} \]
      Proof

      [Start] 5.7	\[ 0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right) \]
      *-commutative [=>] 5.7	\[ 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
      associate-*r* [=>] 5.7	\[ \color{blue}{\left(0.5 \cdot im\right) \cdot \sqrt{\frac{1}{re}}} \]
      metadata-eval [<=] 5.7	\[ \left(\color{blue}{\frac{1}{2}} \cdot im\right) \cdot \sqrt{\frac{1}{re}} \]
      associate-/r/ [<=] 5.8	\[ \color{blue}{\frac{1}{\frac{2}{im}}} \cdot \sqrt{\frac{1}{re}} \]
      unpow-1 [<=] 5.8	\[ \frac{1}{\frac{2}{im}} \cdot \sqrt{\color{blue}{{re}^{-1}}} \]
      metadata-eval [<=] 5.8	\[ \frac{1}{\frac{2}{im}} \cdot \sqrt{{re}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]
      pow-sqr [<=] 5.8	\[ \frac{1}{\frac{2}{im}} \cdot \sqrt{\color{blue}{{re}^{-0.5} \cdot {re}^{-0.5}}} \]
      rem-sqrt-square [=>] 5.8	\[ \frac{1}{\frac{2}{im}} \cdot \color{blue}{\left|{re}^{-0.5}\right|} \]
      sqr-pow [=>] 6.0	\[ \frac{1}{\frac{2}{im}} \cdot \left|\color{blue}{{re}^{\left(\frac{-0.5}{2}\right)} \cdot {re}^{\left(\frac{-0.5}{2}\right)}}\right| \]
      fabs-sqr [=>] 6.0	\[ \frac{1}{\frac{2}{im}} \cdot \color{blue}{\left({re}^{\left(\frac{-0.5}{2}\right)} \cdot {re}^{\left(\frac{-0.5}{2}\right)}\right)} \]
      sqr-pow [<=] 5.8	\[ \frac{1}{\frac{2}{im}} \cdot \color{blue}{{re}^{-0.5}} \]
      associate-*l/ [=>] 5.8	\[ \color{blue}{\frac{1 \cdot {re}^{-0.5}}{\frac{2}{im}}} \]
      *-lft-identity [=>] 5.8	\[ \frac{\color{blue}{{re}^{-0.5}}}{\frac{2}{im}} \]

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

   1. Initial program 35.5

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

   2. Simplified 6.4

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

      [Start] 35.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 35.5	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 35.5	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 35.5	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 35.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 35.5	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 6.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 6.3

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

Alternatives

Alternative 1
Error	15.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{re}^{-0.5}}{\frac{2}{im}}\\ \end{array} \]

Alternative 2
Error	15.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{re}^{-0.5}}{\frac{2}{im}}\\ \end{array} \]

Alternative 3
Error	23.4
Cost	6916

\[\begin{array}{l} \mathbf{if}\;re \leq 3.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{re}^{-0.5}}{\frac{2}{im}}\\ \end{array} \]

Alternative 4
Error	23.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5
Error	31.1
Cost	6720

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

Reproduce

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