math.sqrt on complex, real part

Average Error: 38.7 → 10.1

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\mathsf{hypot}\left(re, im\right)}\\ t_1 := re + \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(t_0, t_0, re\right)}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (hypot re im))) (t_1 (+ re (sqrt (+ (* re re) (* im im))))))
   (if (<= t_1 -5e-305)
     (* 0.5 (sqrt (* 2.0 (fma t_0 t_0 re))))
     (if (<= t_1 0.0)
       (* 0.5 (sqrt (* 2.0 (* -0.5 (/ (pow im 2.0) 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 t_0 = sqrt(hypot(re, im));
	double t_1 = re + sqrt(((re * re) + (im * im)));
	double tmp;
	if (t_1 <= -5e-305) {
		tmp = 0.5 * sqrt((2.0 * fma(t_0, t_0, re)));
	} else if (t_1 <= 0.0) {
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (pow(im, 2.0) / re))));
	} 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 = sqrt(hypot(re, im))
	t_1 = Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im))))
	tmp = 0.0
	if (t_1 <= -5e-305)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * fma(t_0, t_0, re))));
	elseif (t_1 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(-0.5 * Float64((im ^ 2.0) / re)))));
	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[Sqrt[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-305], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 * t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(-0.5 * N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]), $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]]]]]

Target

Original	38.7
Target	33.4
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -4.99999999999999985e-305

   1. Initial program 64.0

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

   2. Simplified 29.1

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

   3. Applied egg-rr 29.5

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

   if -4.99999999999999985e-305 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 57.1

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

   2. Simplified 56.7

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

   3. Taylor expanded in re around -inf 29.2

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

   if 0.0 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 35.0

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

   2. Simplified 6.6

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -5 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sqrt{\mathsf{hypot}\left(re, im\right)}, \sqrt{\mathsf{hypot}\left(re, im\right)}, re\right)}\\ \mathbf{elif}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022207 
(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)))))