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

Average Error: 38.4 → 7.8

Time: 10.6s

Precision: binary64

Cost: 13444

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq 1.2219160605447044 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re 1.2219160605447044e-39)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* (pow re -0.5) (* 0.5 im))))

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 <= 1.2219160605447044e-39) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = pow(re, -0.5) * (0.5 * im);
	}
	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 <= 1.2219160605447044e-39) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = Math.pow(re, -0.5) * (0.5 * im);
	}
	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 <= 1.2219160605447044e-39:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = math.pow(re, -0.5) * (0.5 * im)
	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 <= 1.2219160605447044e-39)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64((re ^ -0.5) * Float64(0.5 * im));
	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 <= 1.2219160605447044e-39)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = (re ^ -0.5) * (0.5 * im);
	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, 1.2219160605447044e-39], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.2219160605447044e-39

   1. Initial program 32.0

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

   2. Simplified 3.5

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      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)))): 130 points increase in error, 1 points decrease in error

   if 1.2219160605447044e-39 < re

   1. Initial program 55.1

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

   2. Taylor expanded in im around 0 19.2

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

   3. Simplified 19.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
      Proof
      (*.f64 im (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 1 re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 im (*.f64 (sqrt.f64 2) (sqrt.f64 1/2))) (sqrt.f64 (/.f64 1 re)))): 23 points increase in error, 12 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) im)) (sqrt.f64 (/.f64 1 re))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) im))) (sqrt.f64 (/.f64 1 re))): 0 points increase in error, 51 points decrease in error

   4. Applied egg-rr 18.8

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

   5. Applied egg-rr 18.8

      \[\leadsto \color{blue}{{re}^{-0.5} \cdot \left(0.5 \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.2219160605447044 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	15.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -1.5200070050150921 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.2219160605447044 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 2
Error	16.0
Cost	7048

\[\begin{array}{l} \mathbf{if}\;re \leq -1.5200070050150921 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.2219160605447044 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 3
Error	23.4
Cost	6916

\[\begin{array}{l} \mathbf{if}\;re \leq 1.2219160605447044 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 4
Error	23.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 1.2219160605447044 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 5
Error	48.2
Cost	6720

\[im \cdot \sqrt{\frac{0.25}{re}} \]

Alternative 6
Error	48.2
Cost	6720

\[\frac{0.5 \cdot im}{\sqrt{re}} \]

Reproduce

herbie shell --seed 2022311 
(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)))))