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

Average Error: 38.4 → 14.6

Time: 11.5s

Precision: binary64

Cost: 13896

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -6.5 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -2.05 \cdot 10^{-157}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -6.5e+123)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -2.05e-157)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
     (if (<= re 4e-108)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (* 0.5 (* (sqrt (/ 1.0 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 tmp;
	if (re <= -6.5e+123) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= -2.05e-157) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 4e-108) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (sqrt((1.0 / re)) * im);
	}
	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 <= (-6.5d+123)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= (-2.05d-157)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
    else if (re <= 4d-108) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (sqrt((1.0d0 / re)) * im)
    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 <= -6.5e+123) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= -2.05e-157) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 4e-108) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (Math.sqrt((1.0 / re)) * im);
	}
	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 <= -6.5e+123:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= -2.05e-157:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
	elif re <= 4e-108:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (math.sqrt((1.0 / 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)
	tmp = 0.0
	if (re <= -6.5e+123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= -2.05e-157)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))));
	elseif (re <= 4e-108)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / re)) * im));
	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 <= -6.5e+123)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= -2.05e-157)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	elseif (re <= 4e-108)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (sqrt((1.0 / re)) * im);
	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, -6.5e+123], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -2.05e-157], 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, 4e-108], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -6.5000000000000001e123

   1. Initial program 55.7

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

   2. Taylor expanded in re around -inf 9.1

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

   3. Simplified 9.1

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

      [Start] 9.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)} \]
      rational.json-simplify-2 [=>] 9.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

   if -6.5000000000000001e123 < re < -2.0500000000000001e-157

   1. Initial program 16.4

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

   if -2.0500000000000001e-157 < re < 4.00000000000000016e-108

   1. Initial program 29.6

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

   2. Taylor expanded in re around 0 8.8

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

   if 4.00000000000000016e-108 < re

   1. Initial program 53.3

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

   2. Taylor expanded in re around inf 39.1

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

   3. Taylor expanded in im around 0 20.9

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.5 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -2.05 \cdot 10^{-157}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -1.58 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \]

Alternative 2
Error	15.6
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -3.15 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot im\right)\\ \end{array} \]

Alternative 3
Error	22.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]

Alternative 4
Error	47.4
Cost	6848

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

Reproduce

herbie shell --seed 2023073 
(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)))))