math.sqrt on complex, real part

Average Error: 38.3 → 12.2

Time: 3.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;re \leq -1.1586911918034763 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(\left(im \cdot \sqrt[3]{im}\right) \cdot \frac{{\left(\sqrt[3]{im}\right)}^{2}}{re}\right)\right)}\\ \mathbf{elif}\;re \leq -2.719946185401384 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.0966303163833297 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))
   (if (<= re -1.1586911918034763e+193)
     (*
      0.5
      (sqrt (* 2.0 (* -0.5 (* (* im (cbrt im)) (/ (pow (cbrt im) 2.0) re))))))
     (if (<= re -2.719946185401384e+101)
       t_0
       (if (<= re -1.0966303163833297e+42)
         (* 0.5 (* im (sqrt (/ -1.0 re))))
         t_0)))))

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 = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	double tmp;
	if (re <= -1.1586911918034763e+193) {
		tmp = 0.5 * sqrt((2.0 * (-0.5 * ((im * cbrt(im)) * (pow(cbrt(im), 2.0) / re)))));
	} else if (re <= -2.719946185401384e+101) {
		tmp = t_0;
	} else if (re <= -1.0966303163833297e+42) {
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	} else {
		tmp = t_0;
	}
	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 = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	double tmp;
	if (re <= -1.1586911918034763e+193) {
		tmp = 0.5 * Math.sqrt((2.0 * (-0.5 * ((im * Math.cbrt(im)) * (Math.pow(Math.cbrt(im), 2.0) / re)))));
	} else if (re <= -2.719946185401384e+101) {
		tmp = t_0;
	} else if (re <= -1.0966303163833297e+42) {
		tmp = 0.5 * (im * Math.sqrt((-1.0 / re)));
	} else {
		tmp = t_0;
	}
	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(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))))
	tmp = 0.0
	if (re <= -1.1586911918034763e+193)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(-0.5 * Float64(Float64(im * cbrt(im)) * Float64((cbrt(im) ^ 2.0) / re))))));
	elseif (re <= -2.719946185401384e+101)
		tmp = t_0;
	elseif (re <= -1.0966303163833297e+42)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(-1.0 / re))));
	else
		tmp = t_0;
	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[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.1586911918034763e+193], N[(0.5 * N[Sqrt[N[(2.0 * N[(-0.5 * N[(N[(im * N[Power[im, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[im, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -2.719946185401384e+101], t$95$0, If[LessEqual[re, -1.0966303163833297e+42], N[(0.5 * N[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Error

Bits error versus re

Bits error versus im

Target

Original	38.3
Target	33.1
Herbie	12.2

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

Derivation

1. Split input into 3 regimes

2. if re < -1.1586911918034763e193

   1. Initial program 64.0

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

   2. Simplified 43.1

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

   3. Taylor expanded in re around -inf 30.3

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

   4. Applied *-un-lft-identity_binary64 30.3

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

   5. Applied add-cube-cbrt_binary64 30.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right) \cdot \sqrt[3]{im}\right)}}^{2}}{1 \cdot re}\right)} \]

   6. Applied unpow-prod-down_binary64 30.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{\color{blue}{{\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right)}^{2} \cdot {\left(\sqrt[3]{im}\right)}^{2}}}{1 \cdot re}\right)} \]

   7. Applied times-frac_binary64 24.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt[3]{im}\right)}^{2}}{re}\right)}\right)} \]

   8. Simplified 23.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(\color{blue}{\left(im \cdot \sqrt[3]{im}\right)} \cdot \frac{{\left(\sqrt[3]{im}\right)}^{2}}{re}\right)\right)} \]

   if -1.1586911918034763e193 < re < -2.71994618540138402e101 or -1.0966303163833297e42 < re

   1. Initial program 35.0

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

   2. Simplified 9.3

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

   if -2.71994618540138402e101 < re < -1.0966303163833297e42

   1. Initial program 48.3

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

   2. Simplified 34.5

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

   3. Taylor expanded in re around -inf 38.2

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

   4. Taylor expanded in im around 0 41.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{-1}{re}} \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1586911918034763 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(\left(im \cdot \sqrt[3]{im}\right) \cdot \frac{{\left(\sqrt[3]{im}\right)}^{2}}{re}\right)\right)}\\ \mathbf{elif}\;re \leq -2.719946185401384 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;re \leq -1.0966303163833297 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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)))))