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

Average Error: 38.7 → 14.5

Time: 20.3s

Precision: binary64

Cost: 13896

\[im > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{if}\;re \leq -4.5 \cdot 10^{+137}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -0.66:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 35000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (* im (sqrt (/ 1.0 re))))))
   (if (<= re -4.5e+137)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re -0.66)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
       (if (<= re 1.6e-86)
         (* 0.5 (sqrt (* 2.0 (- im re))))
         (if (<= re 1.4e-38)
           t_0
           (if (<= re 35000.0) (* 0.5 (sqrt (* im 2.0))) 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 * (im * sqrt((1.0 / re)));
	double tmp;
	if (re <= -4.5e+137) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -0.66) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 1.6e-86) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 1.4e-38) {
		tmp = t_0;
	} else if (re <= 35000.0) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 * (im * sqrt((1.0d0 / re)))
    if (re <= (-4.5d+137)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= (-0.66d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
    else if (re <= 1.6d-86) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 1.4d-38) then
        tmp = t_0
    else if (re <= 35000.0d0) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 * (im * Math.sqrt((1.0 / re)));
	double tmp;
	if (re <= -4.5e+137) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= -0.66) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 1.6e-86) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 1.4e-38) {
		tmp = t_0;
	} else if (re <= 35000.0) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} 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 * (im * math.sqrt((1.0 / re)))
	tmp = 0
	if re <= -4.5e+137:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= -0.66:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
	elif re <= 1.6e-86:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 1.4e-38:
		tmp = t_0
	elif re <= 35000.0:
		tmp = 0.5 * math.sqrt((im * 2.0))
	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(im * sqrt(Float64(1.0 / re))))
	tmp = 0.0
	if (re <= -4.5e+137)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -0.66)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))));
	elseif (re <= 1.6e-86)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 1.4e-38)
		tmp = t_0;
	elseif (re <= 35000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 * (im * sqrt((1.0 / re)));
	tmp = 0.0;
	if (re <= -4.5e+137)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= -0.66)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	elseif (re <= 1.6e-86)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 1.4e-38)
		tmp = t_0;
	elseif (re <= 35000.0)
		tmp = 0.5 * sqrt((im * 2.0));
	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[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4.5e+137], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -0.66], 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, 1.6e-86], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e-38], t$95$0, If[LessEqual[re, 35000.0], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -4.5000000000000001e137

   1. Initial program 60.1

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

   2. Taylor expanded in re around -inf 17.5

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

   3. Simplified 17.5

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

      [Start] 17.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re} + -2 \cdot re\right)} \]
      rational.json-simplify-2 [=>] 17.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{{im}^{2}}{re} \cdot -0.5} + -2 \cdot re\right)} \]
      rational.json-simplify-2 [=>] 17.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{{im}^{2}}{re} \cdot -0.5 + \color{blue}{re \cdot -2}\right)} \]

   4. Taylor expanded in im around 0 64.0

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

   5. Simplified 9.2

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{re \cdot -4}} \]
      Proof

      [Start] 64.0	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-2}\right) \cdot \sqrt{re}\right) \]
      exponential.json-simplify-20 [=>] 64.0	\[ 0.5 \cdot \left(\color{blue}{\sqrt{-2 \cdot 2}} \cdot \sqrt{re}\right) \]
      exponential.json-simplify-20 [=>] 9.2	\[ 0.5 \cdot \color{blue}{\sqrt{re \cdot \left(-2 \cdot 2\right)}} \]
      metadata-eval [=>] 9.2	\[ 0.5 \cdot \sqrt{re \cdot \color{blue}{-4}} \]

   if -4.5000000000000001e137 < re < -0.660000000000000031

   1. Initial program 15.9

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

   if -0.660000000000000031 < re < 1.60000000000000003e-86

   1. Initial program 26.7

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

   2. Taylor expanded in re around 0 12.5

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

   if 1.60000000000000003e-86 < re < 1.4e-38 or 35000 < re

   1. Initial program 54.4

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

   2. Taylor expanded in im around 0 18.1

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

   3. Simplified 17.8

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

      [Start] 18.1	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational.json-simplify-43 [=>] 18.1	\[ 0.5 \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right)} \cdot \sqrt{\frac{1}{re}}\right) \]
      rational.json-simplify-43 [=>] 18.3	\[ 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)} \cdot \sqrt{\frac{1}{re}}\right) \]
      rational.json-simplify-2 [=>] 18.3	\[ 0.5 \cdot \left(\color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot im\right)} \cdot \sqrt{\frac{1}{re}}\right) \]
      exponential.json-simplify-20 [=>] 17.8	\[ 0.5 \cdot \left(\left(\color{blue}{\sqrt{0.5 \cdot 2}} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      metadata-eval [=>] 17.8	\[ 0.5 \cdot \left(\left(\sqrt{\color{blue}{1}} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      metadata-eval [=>] 17.8	\[ 0.5 \cdot \left(\left(\color{blue}{1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational.json-simplify-6 [=>] 17.8	\[ 0.5 \cdot \left(\color{blue}{im} \cdot \sqrt{\frac{1}{re}}\right) \]

   if 1.4e-38 < re < 35000

   1. Initial program 45.2

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

   2. Taylor expanded in re around 0 27.9

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

   3. Simplified 27.6

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

      [Start] 27.9	\[ 0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right) \]
      exponential.json-simplify-20 [=>] 27.6	\[ 0.5 \cdot \color{blue}{\sqrt{im \cdot 2}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+137}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -0.66:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 35000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	15.4
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{if}\;re \leq -8.2 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.55 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 370000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	23.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 3
Error	30.9
Cost	6720

\[0.5 \cdot \sqrt{im \cdot 2} \]

Reproduce

herbie shell --seed 2023075 
(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)))))