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

Average Error: 39.9% → 90.4%

Time: 10.8s

Precision: binary64

Cost: 20356.00

\[im > 0\]

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* 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(((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(((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(((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 (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(((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[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $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 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 9.1

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

   2. Taylor expanded in im around 0 89.9

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

   3. Applied egg-rr 14.7

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

      [Start] 89.9	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      expm1-log1p-u [=>] 89.8	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
      expm1-udef [=>] 14.7	\[ 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right)} \]
      sub-neg [=>] 14.7	\[ 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} + \left(-1\right)\right)} \]
      *-commutative [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{re}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)}\right)} + \left(-1\right)\right) \]
      associate-*r* [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot im\right)}\right)} + \left(-1\right)\right) \]
      associate-*r* [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot im}\right)} + \left(-1\right)\right) \]
      sqrt-unprod [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\sqrt{2 \cdot 0.5}}\right) \cdot im\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\color{blue}{1}}\right) \cdot im\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(\sqrt{\frac{1}{re}} \cdot \color{blue}{1}\right) \cdot im\right)} + \left(-1\right)\right) \]
      *-rgt-identity [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{re}}} \cdot im\right)} + \left(-1\right)\right) \]
      sqrt-div [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \cdot im\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{re}} \cdot im\right)} + \left(-1\right)\right) \]
      associate-*l/ [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot im}{\sqrt{re}}}\right)} + \left(-1\right)\right) \]
      *-un-lft-identity [<=] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{im}}{\sqrt{re}}\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} + \color{blue}{-1}\right) \]

   4. Simplified 90.8

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
      Proof

      [Start] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} + -1\right) \]
      metadata-eval [<=] 14.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} + \color{blue}{\left(-1\right)}\right) \]
      sub-neg [<=] 14.7	\[ 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
      expm1-def [=>] 90.6	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]
      expm1-log1p [=>] 90.8	\[ 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

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

   1. Initial program 45.3

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

   2. Simplified 90.4

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

      [Start] 45.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 45.3	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 45.3	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 45.3	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 45.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 45.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 90.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 90.4

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

Alternatives

Alternative 1
Error	75.9%
Cost	7112.00

\[\begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 2
Error	75.6%
Cost	7048.00

\[\begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3
Error	75.6%
Cost	6984.00

\[\begin{array}{l} \mathbf{if}\;re \leq -36000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4
Error	64.7%
Cost	6852.00

\[\begin{array}{l} \mathbf{if}\;re \leq -4.3 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	51.9%
Cost	6720.00

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

Reproduce

herbie shell --seed 2023121 
(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)))))