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

Average Error: 38.5 → 7.4

Time: 7.7s

Precision: binary64

Cost: 13444

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq 2400000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re 2400000000.0)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (/ (* 0.5 im) (sqrt 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 (re <= 2400000000.0) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / sqrt(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 (re <= 2400000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(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 re <= 2400000000.0:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(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 (re <= 2400000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(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 (re <= 2400000000.0)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = (0.5 * im) / sqrt(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[re, 2400000000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.4e9

   1. Initial program 32.4

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

   2. Simplified 5.2

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

      [Start] 32.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 32.4	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 32.4	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 32.4	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 32.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 32.4	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 5.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   if 2.4e9 < re

   1. Initial program 57.6

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

   2. Simplified 39.3

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

      [Start] 57.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 57.6	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 57.6	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 57.6	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 57.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 57.6	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 39.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Taylor expanded in re around inf 34.5

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

   4. Simplified 29.5

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

      [Start] 34.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      unpow2 [=>] 34.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]
      associate-/l* [=>] 29.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \color{blue}{\frac{im}{\frac{re}{im}}}\right)} \]

   5. Applied egg-rr 44.3

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

   6. Simplified 14.2

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

      [Start] 44.3	\[ e^{\mathsf{log1p}\left(0.5 \cdot \frac{im}{\sqrt{re}}\right)} - 1 \]
      expm1-def [=>] 14.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{im}{\sqrt{re}}\right)\right)} \]
      expm1-log1p [=>] 14.2	\[ \color{blue}{0.5 \cdot \frac{im}{\sqrt{re}}} \]
      associate-*r/ [=>] 14.2	\[ \color{blue}{\frac{0.5 \cdot im}{\sqrt{re}}} \]
      *-commutative [=>] 14.2	\[ \frac{\color{blue}{im \cdot 0.5}}{\sqrt{re}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2400000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.7
Cost	7249

\[\begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-87}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-62} \lor \neg \left(re \leq 1100000000\right):\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 2
Error	14.9
Cost	6984

\[\begin{array}{l} \mathbf{if}\;re \leq -28000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 12500000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternative 3
Error	22.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 4000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 4
Error	22.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 100000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternative 5
Error	30.6
Cost	6720

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

Reproduce

herbie shell --seed 2023002 
(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)))))