math.sqrt on complex, real part

Average Error: 38.4 → 11.5

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -7.1 \cdot 10^{+207}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\frac{re}{im}}{-0.5}}}\\ \mathbf{elif}\;re \leq -4 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -7.1e+207)
   (* 0.5 (sqrt (* 2.0 (/ im (/ (/ re im) -0.5)))))
   (if (<= re -4e+31)
     (* 0.5 (* im (sqrt (/ -1.0 re))))
     (* 0.5 (sqrt (* 2.0 (+ re (hypot re 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 <= -7.1e+207) {
		tmp = 0.5 * sqrt((2.0 * (im / ((re / im) / -0.5))));
	} else if (re <= -4e+31) {
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, 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 <= -7.1e+207) {
		tmp = 0.5 * Math.sqrt((2.0 * (im / ((re / im) / -0.5))));
	} else if (re <= -4e+31) {
		tmp = 0.5 * (im * Math.sqrt((-1.0 / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, 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 <= -7.1e+207:
		tmp = 0.5 * math.sqrt((2.0 * (im / ((re / im) / -0.5))))
	elif re <= -4e+31:
		tmp = 0.5 * (im * math.sqrt((-1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, 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 <= -7.1e+207)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im / Float64(Float64(re / im) / -0.5)))));
	elseif (re <= -4e+31)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(-1.0 / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, 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 <= -7.1e+207)
		tmp = 0.5 * sqrt((2.0 * (im / ((re / im) / -0.5))));
	elseif (re <= -4e+31)
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, 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, -7.1e+207], N[(0.5 * N[Sqrt[N[(2.0 * N[(im / N[(N[(re / im), $MachinePrecision] / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4e+31], N[(0.5 * N[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus re

Bits error versus im

Target

Original	38.4
Target	33.3
Herbie	11.5

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

Derivation

1. Split input into 3 regimes

2. if re < -7.10000000000000042e207

   1. Initial program 64.0

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

   2. Simplified 42.4

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

   3. Taylor expanded in re around -inf 32.5

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

   4. Simplified 20.4

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

   5. Taylor expanded in im around 0 32.5

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

   6. Simplified 20.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{im}{\frac{\frac{re}{im}}{-0.5}}}} \]

   if -7.10000000000000042e207 < re < -3.9999999999999999e31

   1. Initial program 55.9

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

   2. Simplified 36.2

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

   3. Taylor expanded in re around -inf 35.6

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

   4. Simplified 34.6

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

   5. Taylor expanded in im around 0 36.2

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

   if -3.9999999999999999e31 < re

   1. Initial program 32.6

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

   2. Simplified 6.0

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.1 \cdot 10^{+207}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\frac{re}{im}}{-0.5}}}\\ \mathbf{elif}\;re \leq -4 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))