math.sqrt on complex, real part

Average Error: 39.2 → 10.8

Time: 4.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := {\left({\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot \left(0.5 \cdot im\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{if}\;re \leq -3.0096314164618805 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(-0.125, \frac{t_0}{re \cdot re} \cdot \left(im \cdot im\right), t_0\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (pow (* (pow (/ -1.0 re) 0.25) (pow (* im (* 0.5 im)) 0.25)) 2.0)))
   (if (<= re -3.0096314164618805e+112)
     (* 0.5 (* (fma -0.125 (* (/ t_0 (* re re)) (* im im)) t_0) (sqrt 2.0)))
     (* 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 t_0 = pow((pow((-1.0 / re), 0.25) * pow((im * (0.5 * im)), 0.25)), 2.0);
	double tmp;
	if (re <= -3.0096314164618805e+112) {
		tmp = 0.5 * (fma(-0.125, ((t_0 / (re * re)) * (im * im)), t_0) * sqrt(2.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + 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)
	t_0 = Float64((Float64(-1.0 / re) ^ 0.25) * (Float64(im * Float64(0.5 * im)) ^ 0.25)) ^ 2.0
	tmp = 0.0
	if (re <= -3.0096314164618805e+112)
		tmp = Float64(0.5 * Float64(fma(-0.125, Float64(Float64(t_0 / Float64(re * re)) * Float64(im * im)), t_0) * sqrt(2.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	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[Power[N[(N[Power[N[(-1.0 / re), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[re, -3.0096314164618805e+112], N[(0.5 * N[(N[(-0.125 * N[(N[(t$95$0 / N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	39.2
Target	34.0
Herbie	10.8

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

Derivation

1. Split input into 2 regimes

2. if re < -3.00963141646188048e112

   1. Initial program 61.9

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

   2. Simplified 41.2

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

   3. Applied egg-rr 41.2

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

   4. Applied egg-rr 41.3

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

   5. Taylor expanded in re around -inf 27.3

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.125 \cdot \frac{{\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(0.5 \cdot {im}^{2}\right)\right)}\right)}^{2} \cdot {im}^{2}}{{re}^{2}} + {\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(0.5 \cdot {im}^{2}\right)\right)}\right)}^{2}\right)} \cdot \sqrt{2}\right) \]

   6. Simplified 23.7

      \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left({\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot \left(im \cdot 0.5\right)\right)}^{0.25}\right)}^{2}}{re \cdot re} \cdot \left(im \cdot im\right), {\left({\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot \left(im \cdot 0.5\right)\right)}^{0.25}\right)}^{2}\right)} \cdot \sqrt{2}\right) \]

   if -3.00963141646188048e112 < re

   1. Initial program 34.8

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

   2. Simplified 8.3

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

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.0096314164618805 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(-0.125, \frac{{\left({\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot \left(0.5 \cdot im\right)\right)}^{0.25}\right)}^{2}}{re \cdot re} \cdot \left(im \cdot im\right), {\left({\left(\frac{-1}{re}\right)}^{0.25} \cdot {\left(im \cdot \left(0.5 \cdot im\right)\right)}^{0.25}\right)}^{2}\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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)))))