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

Average Error: 38.1 → 7.7

Time: 11.1s

Precision: binary64

Cost: 13708

\[im > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{if}\;re \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{0.5 \cdot im}{{re}^{0.25}}}{{re}^{0.25}}\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))
   (if (<= re 7.5e-111)
     t_0
     (if (<= re 4.7e-82)
       (/ (/ (* 0.5 im) (pow re 0.25)) (pow re 0.25))
       (if (<= re 8.8e-16) t_0 (* (* 0.5 im) (pow re -0.5)))))))

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 t_0 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	double tmp;
	if (re <= 7.5e-111) {
		tmp = t_0;
	} else if (re <= 4.7e-82) {
		tmp = ((0.5 * im) / pow(re, 0.25)) / pow(re, 0.25);
	} else if (re <= 8.8e-16) {
		tmp = t_0;
	} else {
		tmp = (0.5 * im) * pow(re, -0.5);
	}
	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 t_0 = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	double tmp;
	if (re <= 7.5e-111) {
		tmp = t_0;
	} else if (re <= 4.7e-82) {
		tmp = ((0.5 * im) / Math.pow(re, 0.25)) / Math.pow(re, 0.25);
	} else if (re <= 8.8e-16) {
		tmp = t_0;
	} else {
		tmp = (0.5 * im) * Math.pow(re, -0.5);
	}
	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):
	t_0 = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	tmp = 0
	if re <= 7.5e-111:
		tmp = t_0
	elif re <= 4.7e-82:
		tmp = ((0.5 * im) / math.pow(re, 0.25)) / math.pow(re, 0.25)
	elif re <= 8.8e-16:
		tmp = t_0
	else:
		tmp = (0.5 * im) * math.pow(re, -0.5)
	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)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))))
	tmp = 0.0
	if (re <= 7.5e-111)
		tmp = t_0;
	elseif (re <= 4.7e-82)
		tmp = Float64(Float64(Float64(0.5 * im) / (re ^ 0.25)) / (re ^ 0.25));
	elseif (re <= 8.8e-16)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * im) * (re ^ -0.5));
	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)
	t_0 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	tmp = 0.0;
	if (re <= 7.5e-111)
		tmp = t_0;
	elseif (re <= 4.7e-82)
		tmp = ((0.5 * im) / (re ^ 0.25)) / (re ^ 0.25);
	elseif (re <= 8.8e-16)
		tmp = t_0;
	else
		tmp = (0.5 * im) * (re ^ -0.5);
	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_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 7.5e-111], t$95$0, If[LessEqual[re, 4.7e-82], N[(N[(N[(0.5 * im), $MachinePrecision] / N[Power[re, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[re, 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.8e-16], t$95$0, N[(N[(0.5 * im), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 7.49999999999999965e-111 or 4.7000000000000001e-82 < re < 8.80000000000000001e-16

   1. Initial program 31.1

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

   2. Simplified 3.8

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

      [Start] 31.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 31.1	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 31.1	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 31.1	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 31.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 31.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 3.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   if 7.49999999999999965e-111 < re < 4.7000000000000001e-82

   1. Initial program 39.2

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

   2. Simplified 22.2

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

      [Start] 39.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 39.2	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 39.2	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 39.2	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 39.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 39.2	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 22.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Taylor expanded in im around 0 38.8

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

   4. Applied egg-rr 61.0

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

   5. Simplified 38.5

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

      [Start] 61.0	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} + -1\right) \]
      metadata-eval [<=] 61.0	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} + \color{blue}{\left(-1\right)}\right) \]
      sub-neg [<=] 61.0	\[ 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
      expm1-def [=>] 38.5	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]
      expm1-log1p [=>] 38.5	\[ 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   6. Applied egg-rr 38.6

      \[\leadsto \color{blue}{\frac{\frac{im \cdot 0.5}{{re}^{0.25}}}{{re}^{0.25}}} \]

   if 8.80000000000000001e-16 < re

   1. Initial program 56.8

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

   2. Simplified 38.7

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

      [Start] 56.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 56.8	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 56.8	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 56.8	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 56.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 56.8	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 38.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Taylor expanded in im around 0 15.1

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

   4. Applied egg-rr 45.5

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

   5. Simplified 14.7

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

      [Start] 45.5	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} + -1\right) \]
      metadata-eval [<=] 45.5	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} + \color{blue}{\left(-1\right)}\right) \]
      sub-neg [<=] 45.5	\[ 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
      expm1-def [=>] 15.2	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]
      expm1-log1p [=>] 14.7	\[ 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   6. Applied egg-rr 17.9

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

   7. Applied egg-rr 14.7

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

   8. Simplified 14.7

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

      [Start] 14.7	\[ im \cdot \left({re}^{-0.5} \cdot 0.5\right) \]
      *-commutative [=>] 14.7	\[ im \cdot \color{blue}{\left(0.5 \cdot {re}^{-0.5}\right)} \]
      associate-*l* [<=] 14.7	\[ \color{blue}{\left(im \cdot 0.5\right) \cdot {re}^{-0.5}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{0.5 \cdot im}{{re}^{0.25}}}{{re}^{0.25}}\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.5
Cost	13708

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{if}\;re \leq 7.6 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-84}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]

Alternative 2
Error	16.1
Cost	7576

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -4.5 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.12 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.56 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 7.6 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.26 \cdot 10^{-83}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 1.22 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]

Alternative 3
Error	15.3
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -8.5 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7.6 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{-80}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]

Alternative 4
Error	23.0
Cost	7180

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq 6 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-83}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 5
Error	23.0
Cost	7180

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq 7.6 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{-85}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]

Alternative 6
Error	22.9
Cost	7117

\[\begin{array}{l} \mathbf{if}\;re \leq 7.6 \cdot 10^{-111} \lor \neg \left(re \leq 5.8 \cdot 10^{-86}\right) \land re \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 7
Error	23.0
Cost	7117

\[\begin{array}{l} \mathbf{if}\;re \leq 6 \cdot 10^{-111} \lor \neg \left(re \leq 1.3 \cdot 10^{-85}\right) \land re \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 8
Error	30.9
Cost	6720

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

Reproduce

herbie shell --seed 2023054 
(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)))))