math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.6 → 15.6

Time: 35.6s

Precision: binary64

Cost: 13896

\[im > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\\ \mathbf{if}\;re \leq -9.5 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -3.05 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (/ 1.0 re)) im))))
   (if (<= re -9.5e+108)
     (* 0.5 (sqrt (* 2.0 (* re -2.0))))
     (if (<= re -3.05e-117)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
       (if (<= re 2.9e-146)
         (* 0.5 (sqrt (* 2.0 (- im re))))
         (if (<= re 5.5e-46)
           t_0
           (if (<= re 2.15e+66) (* 0.5 (sqrt (* 2.0 im))) 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((1.0 / re)) * im);
	double tmp;
	if (re <= -9.5e+108) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= -3.05e-117) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 2.9e-146) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 5.5e-46) {
		tmp = t_0;
	} else if (re <= 2.15e+66) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (sqrt((1.0d0 / re)) * im)
    if (re <= (-9.5d+108)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= (-3.05d-117)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
    else if (re <= 2.9d-146) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 5.5d-46) then
        tmp = t_0
    else if (re <= 2.15d+66) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = t_0
    end if
    code = tmp
end function

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((1.0 / re)) * im);
	double tmp;
	if (re <= -9.5e+108) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= -3.05e-117) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 2.9e-146) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 5.5e-46) {
		tmp = t_0;
	} else if (re <= 2.15e+66) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

↓

def code(re, im):
	t_0 = 0.5 * (math.sqrt((1.0 / re)) * im)
	tmp = 0
	if re <= -9.5e+108:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= -3.05e-117:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
	elif re <= 2.9e-146:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 5.5e-46:
		tmp = t_0
	elif re <= 2.15e+66:
		tmp = 0.5 * math.sqrt((2.0 * im))
	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 * Float64(sqrt(Float64(1.0 / re)) * im))
	tmp = 0.0
	if (re <= -9.5e+108)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= -3.05e-117)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))));
	elseif (re <= 2.9e-146)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 5.5e-46)
		tmp = t_0;
	elseif (re <= 2.15e+66)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

↓

function tmp_2 = code(re, im)
	t_0 = 0.5 * (sqrt((1.0 / re)) * im);
	tmp = 0.0;
	if (re <= -9.5e+108)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= -3.05e-117)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	elseif (re <= 2.9e-146)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 5.5e-46)
		tmp = t_0;
	elseif (re <= 2.15e+66)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = t_0;
	end
	tmp_2 = 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[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -9.5e+108], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -3.05e-117], 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], If[LessEqual[re, 2.9e-146], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.5e-46], t$95$0, If[LessEqual[re, 2.15e+66], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -9.50000000000000097e108

   1. Initial program 52.8

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

   2. Taylor expanded in re around -inf 9.5

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

   3. Simplified 9.5

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

      [Start] 9.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)} \]
      rational_best-simplify-1 [<=] 9.5	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

   if -9.50000000000000097e108 < re < -3.05000000000000001e-117

   1. Initial program 16.4

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

   if -3.05000000000000001e-117 < re < 2.90000000000000011e-146

   1. Initial program 29.5

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

   2. Taylor expanded in re around 0 8.7

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

   if 2.90000000000000011e-146 < re < 5.49999999999999983e-46 or 2.15000000000000013e66 < re

   1. Initial program 52.2

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

   2. Taylor expanded in re around inf 38.9

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

   3. Taylor expanded in im around 0 20.7

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

   if 5.49999999999999983e-46 < re < 2.15000000000000013e66

   1. Initial program 46.2

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

   2. Taylor expanded in re around 0 32.3

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

   3. Taylor expanded in re around 0 30.9

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

   4. Simplified 30.6

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]
      Proof

      [Start] 30.9	\[ 0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right) \]
      exponential-simplify-21 [=>] 30.6	\[ 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.5 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -3.05 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-146}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.8
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ t_2 := 0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\\ \mathbf{if}\;re \leq -4.8 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.6 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 4.3 \cdot 10^{-45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	23.4
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -1.75 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -6.7 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -3.05 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 3.7 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \end{array} \]

Alternative 3
Error	23.2
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -4 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.1 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -4.1 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.7 \cdot 10^{+137}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \end{array} \]

Alternative 4
Error	30.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 3.55 \cdot 10^{+137}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \end{array} \]

Alternative 5
Error	60.2
Cost	6592

\[0.5 \cdot \sqrt{0} \]

Reproduce

herbie shell --seed 2023099 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))