math.sqrt on complex, real part

Average Error: 39.2 → 9.5

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ t_2 := {\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot im\right)}^{0.25}\\ \mathbf{if}\;t_0 \leq -3.3919108234128158 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot {\left(\mathsf{fma}\left(\left(\frac{im}{re} \cdot \frac{im}{re}\right) \cdot t_2, -0.0625, t_2\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))))))
        (t_2 (* (pow (/ -1.0 re) 0.25) (pow (* im im) 0.25))))
   (if (<= t_0 -3.3919108234128158e-307)
     t_1
     (if (<= t_0 0.0)
       (* 0.5 (pow (fma (* (* (/ im re) (/ im re)) t_2) -0.0625 t_2) 2.0))
       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 t_2 = pow((-1.0 / re), 0.25) * pow((im * im), 0.25);
	double tmp;
	if (t_0 <= -3.3919108234128158e-307) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * pow(fma((((im / re) * (im / re)) * t_2), -0.0625, t_2), 2.0);
	} 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)))))
	t_2 = Float64((Float64(-1.0 / re) ^ 0.25) * (Float64(im * im) ^ 0.25))
	tmp = 0.0
	if (t_0 <= -3.3919108234128158e-307)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 * (fma(Float64(Float64(Float64(im / re) * Float64(im / re)) * t_2), -0.0625, t_2) ^ 2.0));
	else
		tmp = t_1;
	end
	return 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]}, Block[{t$95$2 = N[(N[Power[N[(-1.0 / re), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(im * im), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -3.3919108234128158e-307], t$95$1, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[Power[N[(N[(N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * -0.0625 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Error

Bits error versus re

Bits error versus im

Target

Original	39.2
Target	33.7
Herbie	9.5

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -3.3919108234128158e-307 or 0.0 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 36.5

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

   2. Simplified 7.1

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

   if -3.3919108234128158e-307 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 57.8

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

   2. Simplified 57.6

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

   3. Applied egg-rr 57.6

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in re around -inf 28.2

      \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)} - 0.0625 \cdot \frac{e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)} \cdot {im}^{2}}{{re}^{2}}\right)}}^{2} \]

   5. Simplified 25.7

      \[\leadsto 0.5 \cdot {\color{blue}{\left(\mathsf{fma}\left(\left(\frac{im}{re} \cdot \frac{im}{re}\right) \cdot \left({\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot im\right)}^{0.25}\right), -0.0625, {\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot im\right)}^{0.25}\right)\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -3.3919108234128158 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot {\left(\mathsf{fma}\left(\left(\frac{im}{re} \cdot \frac{im}{re}\right) \cdot \left({\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot im\right)}^{0.25}\right), -0.0625, {\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot im\right)}^{0.25}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))))