math.sqrt on complex, real part

Average Error: 38.6 → 9.9

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im \cdot \frac{im}{re}\right) \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ re (sqrt (+ (* re re) (* im im)))))
        (t_1 (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))
   (if (<= t_0 -5e-287)
     t_1
     (if (<= t_0 0.0) (* 0.5 (sqrt (* 2.0 (* (* im (/ im re)) -0.5)))) t_1))))

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 = re + sqrt(((re * re) + (im * im)));
	double t_1 = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	double tmp;
	if (t_0 <= -5e-287) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * sqrt((2.0 * ((im * (im / re)) * -0.5)));
	} else {
		tmp = t_1;
	}
	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 = re + Math.sqrt(((re * re) + (im * im)));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	double tmp;
	if (t_0 <= -5e-287) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im * (im / re)) * -0.5)));
	} else {
		tmp = t_1;
	}
	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 = re + math.sqrt(((re * re) + (im * im)))
	t_1 = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	tmp = 0
	if t_0 <= -5e-287:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = 0.5 * math.sqrt((2.0 * ((im * (im / re)) * -0.5)))
	else:
		tmp = t_1
	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(re + sqrt(Float64(Float64(re * re) + Float64(im * im))))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))))
	tmp = 0.0
	if (t_0 <= -5e-287)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im * Float64(im / re)) * -0.5))));
	else
		tmp = t_1;
	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 = re + sqrt(((re * re) + (im * im)));
	t_1 = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	tmp = 0.0;
	if (t_0 <= -5e-287)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = 0.5 * sqrt((2.0 * ((im * (im / re)) * -0.5)));
	else
		tmp = t_1;
	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[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-287], t$95$1, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus re

Bits error versus im

Target

Original	38.6
Target	33.7
Herbie	9.9

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -5.00000000000000025e-287 or 0.0 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 35.9

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

   2. Simplified 7.3

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

   if -5.00000000000000025e-287 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 57.7

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

   2. Simplified 55.6

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

   3. Taylor expanded in re around -inf 32.1

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

   4. Simplified 28.1

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -5 \cdot 10^{-287}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im \cdot \frac{im}{re}\right) \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(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)))))