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

Average Error: 38.6 → 13.7

Time: 9.5s

Precision: binary64

Cost: 13896

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+121}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.15 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 10^{-174}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e+121)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -1.15e-158)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
     (if (<= re 1e-174)
       (* 0.5 (* (sqrt 2.0) (sqrt im)))
       (if (<= re 3e-21)
         (* 0.5 (sqrt (* 2.0 (- im re))))
         (* (sqrt (/ 1.0 re)) (* im 0.5)))))))

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 <= -7.5e+121) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= -1.15e-158) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 1e-174) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(im));
	} else if (re <= 3e-21) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = sqrt((1.0 / re)) * (im * 0.5);
	}
	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 <= (-7.5d+121)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= (-1.15d-158)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
    else if (re <= 1d-174) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(im))
    else if (re <= 3d-21) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = sqrt((1.0d0 / re)) * (im * 0.5d0)
    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 <= -7.5e+121) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= -1.15e-158) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 1e-174) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(im));
	} else if (re <= 3e-21) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = Math.sqrt((1.0 / re)) * (im * 0.5);
	}
	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 <= -7.5e+121:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= -1.15e-158:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
	elif re <= 1e-174:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(im))
	elif re <= 3e-21:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = math.sqrt((1.0 / re)) * (im * 0.5)
	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 <= -7.5e+121)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= -1.15e-158)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))));
	elseif (re <= 1e-174)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(im)));
	elseif (re <= 3e-21)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(sqrt(Float64(1.0 / re)) * Float64(im * 0.5));
	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 <= -7.5e+121)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= -1.15e-158)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	elseif (re <= 1e-174)
		tmp = 0.5 * (sqrt(2.0) * sqrt(im));
	elseif (re <= 3e-21)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = sqrt((1.0 / re)) * (im * 0.5);
	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, -7.5e+121], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.15e-158], 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-174], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e-21], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -7.49999999999999965e121

   1. Initial program 55.4

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

   2. Taylor expanded in re around -inf 9.2

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

   3. Simplified 9.2

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

      [Start] 9.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 9.2	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

   if -7.49999999999999965e121 < re < -1.1499999999999999e-158

   1. Initial program 15.7

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

   if -1.1499999999999999e-158 < re < 1e-174

   1. Initial program 30.4

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

   2. Taylor expanded in re around 0 7.5

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

   if 1e-174 < re < 2.99999999999999991e-21

   1. Initial program 36.0

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

   2. Taylor expanded in re around 0 20.9

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

   if 2.99999999999999991e-21 < re

   1. Initial program 56.2

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

   2. Taylor expanded in re around inf 35.7

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

   3. Taylor expanded in im around 0 16.5

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

   4. Simplified 16.5

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

      [Start] 16.5	\[ 0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 16.5	\[ \color{blue}{\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 16.5	\[ \sqrt{\frac{1}{re}} \cdot \color{blue}{\left(im \cdot 0.5\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+121}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.15 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 10^{-174}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -0.026:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \]

Alternative 2
Error	15.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 3
Error	23.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;re \leq -0.9:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]

Alternative 4
Error	47.0
Cost	6848

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

Reproduce

herbie shell --seed 2023090 
(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)))))