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

Average Error: 38.9 → 13.8

Time: 14.9s

Precision: binary64

Cost: 13896

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -7.2 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -5.5e+120)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -7.2e-94)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
     (if (<= re 1e-5)
       (* 0.5 (sqrt (* im 2.0)))
       (* 0.5 (* im (sqrt (/ 1.0 re))))))))

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 tmp;
	if (re <= -5.5e+120) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= -7.2e-94) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 1e-5) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	}
	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) :: tmp
    if (re <= (-5.5d+120)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= (-7.2d-94)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
    else if (re <= 1d-5) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    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 tmp;
	if (re <= -5.5e+120) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= -7.2e-94) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 1e-5) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	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):
	tmp = 0
	if re <= -5.5e+120:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= -7.2e-94:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
	elif re <= 1e-5:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	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)
	tmp = 0.0
	if (re <= -5.5e+120)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= -7.2e-94)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))));
	elseif (re <= 1e-5)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	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)
	tmp = 0.0;
	if (re <= -5.5e+120)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= -7.2e-94)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	elseif (re <= 1e-5)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	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_] := If[LessEqual[re, -5.5e+120], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -7.2e-94], 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, 1e-5], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.50000000000000003e120

   1. Initial program 55.5

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

   2. Taylor expanded in re around -inf 9.8

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

   3. Simplified 9.8

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

      [Start] 9.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)} \]
      rational_best-simplify-2 [=>] 9.8	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

   if -5.50000000000000003e120 < re < -7.2e-94

   1. Initial program 16.5

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

   if -7.2e-94 < re < 1.00000000000000008e-5

   1. Initial program 30.8

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

   2. Taylor expanded in re around 0 13.3

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

   3. Simplified 13.0

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

      [Start] 13.3	\[ 0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right) \]
      exponential-simplify-19 [=>] 13.0	\[ 0.5 \cdot \color{blue}{\sqrt{im \cdot 2}} \]

   if 1.00000000000000008e-5 < re

   1. Initial program 57.3

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

   2. Taylor expanded in im around 0 15.8

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

   3. Simplified 15.5

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

      [Start] 15.8	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational_best-simplify-2 [=>] 15.8	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\left(im \cdot \sqrt{0.5}\right)}\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational_best-simplify-44 [=>] 16.0	\[ 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)} \cdot \sqrt{\frac{1}{re}}\right) \]
      exponential-simplify-19 [=>] 15.5	\[ 0.5 \cdot \left(\left(im \cdot \color{blue}{\sqrt{0.5 \cdot 2}}\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      metadata-eval [=>] 15.5	\[ 0.5 \cdot \left(\left(im \cdot \sqrt{\color{blue}{1}}\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      metadata-eval [=>] 15.5	\[ 0.5 \cdot \left(\left(im \cdot \color{blue}{1}\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational_best-simplify-5 [=>] 15.5	\[ 0.5 \cdot \left(\color{blue}{im} \cdot \sqrt{\frac{1}{re}}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -7.2 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	15.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -0.102:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]

Alternative 2
Error	22.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;re \leq -0.092:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 3
Error	29.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{-215}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 4
Error	60.0
Cost	6592

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

Reproduce

herbie shell --seed 2023092 
(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)))))