math.sqrt on complex, real part

Average Error: 38.8 → 10.2

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ t_1 := \sqrt[3]{\sqrt[3]{re}}\\ \mathbf{if}\;t_0 \leq -2.3723730750189226 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left({t_1}^{4} \cdot {t_1}^{2}, \sqrt[3]{\sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)}, \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

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 (+ re (sqrt (+ (* re re) (* im im))))) (t_1 (cbrt (cbrt re))))
   (if (<= t_0 -2.3723730750189226e-300)
     (*
      0.5
      (sqrt
       (*
        2.0
        (fma
         (* (pow t_1 4.0) (pow t_1 2.0))
         (cbrt (* (cbrt re) (* (cbrt re) (cbrt re))))
         (hypot re im)))))
     (if (<= t_0 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 = re + sqrt(((re * re) + (im * im)));
	double t_1 = cbrt(cbrt(re));
	double tmp;
	if (t_0 <= -2.3723730750189226e-300) {
		tmp = 0.5 * sqrt((2.0 * fma((pow(t_1, 4.0) * pow(t_1, 2.0)), cbrt((cbrt(re) * (cbrt(re) * cbrt(re)))), hypot(re, im))));
	} else if (t_0 <= 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 = Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im))))
	t_1 = cbrt(cbrt(re))
	tmp = 0.0
	if (t_0 <= -2.3723730750189226e-300)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * fma(Float64((t_1 ^ 4.0) * (t_1 ^ 2.0)), cbrt(Float64(cbrt(re) * Float64(cbrt(re) * cbrt(re)))), hypot(re, im)))));
	elseif (t_0 <= 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[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[re, 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$0, -2.3723730750189226e-300], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(N[Power[t$95$1, 4.0], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[re, 1/3], $MachinePrecision] * N[(N[Power[re, 1/3], $MachinePrecision] * N[Power[re, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 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]]]]]

Error

Bits error versus re

Bits error versus im

Target

Original	38.8
Target	33.7
Herbie	10.2

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -2.37237307501892263e-300

   1. Initial program 64.0

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

   2. Simplified 27.6

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

   3. Applied add-cube-cbrt_binary64 27.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}} + \mathsf{hypot}\left(re, im\right)\right)} \]

   4. Applied fma-def_binary64 27.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

   5. Applied add-cbrt-cube_binary64 27.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \color{blue}{\sqrt[3]{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}}, \mathsf{hypot}\left(re, im\right)\right)} \]

   6. Applied add-cube-cbrt_binary64 27.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sqrt[3]{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right) \cdot \sqrt[3]{\sqrt[3]{re}}\right)}, \sqrt[3]{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}, \mathsf{hypot}\left(re, im\right)\right)} \]

   7. Applied add-cube-cbrt_binary64 27.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right) \cdot \sqrt[3]{\sqrt[3]{re}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right) \cdot \sqrt[3]{\sqrt[3]{re}}\right), \sqrt[3]{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}, \mathsf{hypot}\left(re, im\right)\right)} \]

   8. Applied swap-sqr_binary64 27.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right)}, \sqrt[3]{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}, \mathsf{hypot}\left(re, im\right)\right)} \]

   9. Simplified 27.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{re}}\right)}^{4}} \cdot \left(\sqrt[3]{\sqrt[3]{re}} \cdot \sqrt[3]{\sqrt[3]{re}}\right), \sqrt[3]{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}, \mathsf{hypot}\left(re, im\right)\right)} \]

   10. Simplified 27.2

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{re}}\right)}^{4} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{re}}\right)}^{2}}, \sqrt[3]{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}, \mathsf{hypot}\left(re, im\right)\right)} \]

   if -2.37237307501892263e-300 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 58.0

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

   2. Simplified 56.8

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

   3. Taylor expanded in re around -inf 31.6

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

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

   1. Initial program 35.1

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

   2. Simplified 6.4

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq -2.3723730750189226 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{re}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{re}}\right)}^{2}, \sqrt[3]{\sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)}, \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(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)))))