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

Percentage Accurate: 41.5% → 90.3%

Time: 6.6s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

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

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

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

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

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

Initial Program: 41.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

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

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

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

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

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

Alternative 1: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (/ im (sqrt re)))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0)
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 5.9%

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

   2. Taylor expanded in re around inf 38.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 38.8%

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

      2. associate-/l* 50.4%

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

   4. Simplified 50.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 50.4%

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

      2. metadata-eval 50.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{im}{\frac{re}{im}}} \]

      3. *-un-lft-identity 50.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}}} \]

      4. associate-/l* 38.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]

      5. div-inv 38.8%

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

      6. sqrt-prod 41.1%

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

      7. sqrt-prod 89.7%

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

      8. add-sqr-sqrt 89.9%

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

      9. *-un-lft-identity 89.9%

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

      10. sqrt-prod 89.9%

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

      11. metadata-eval 89.9%

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

      12. sqrt-unprod 88.7%

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

      13. associate-*r* 89.2%

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

      14. expm1-log1p-u 88.9%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right)} \]

      15. expm1-udef 14.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} - 1\right)} \]

   6. Applied egg-rr 14.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 89.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 90.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   8. Simplified 90.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 47.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 47.3%

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

      2. sqr-neg 47.3%

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

      3. hypot-def 91.3%

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

   3. Simplified 91.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]

Alternative 2: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 10^{-144}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-112} \lor \neg \left(re \leq 6.9 \cdot 10^{+106}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.1e-17)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1e-144)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 1.2e-112) (not (<= re 6.9e+106)))
       (* 0.5 (/ im (sqrt re)))
       (* 0.5 (* (sqrt im) (sqrt 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.1e-17) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1e-144) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 1.2e-112) || !(re <= 6.9e+106)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * (sqrt(im) * sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5.1d-17)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1d-144) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 1.2d-112) .or. (.not. (re <= 6.9d+106))) then
        tmp = 0.5d0 * (im / sqrt(re))
    else
        tmp = 0.5d0 * (sqrt(im) * sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.1e-17) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1e-144) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 1.2e-112) || !(re <= 6.9e+106)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * (Math.sqrt(im) * Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.1e-17:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1e-144:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 1.2e-112) or not (re <= 6.9e+106):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * (math.sqrt(im) * math.sqrt(2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.1e-17)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1e-144)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 1.2e-112) || !(re <= 6.9e+106))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * Float64(sqrt(im) * sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.1e-17)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1e-144)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 1.2e-112) || ~((re <= 6.9e+106)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * (sqrt(im) * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.1e-17], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-144], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.2e-112], N[Not[LessEqual[re, 6.9e+106]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[im], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.1000000000000003e-17

   1. Initial program 51.4%

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

   2. Taylor expanded in re around -inf 79.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative 79.1%

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

   4. Simplified 79.1%

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

   if -5.1000000000000003e-17 < re < 9.9999999999999995e-145

   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 0 80.5%

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

   if 9.9999999999999995e-145 < re < 1.2e-112 or 6.8999999999999998e106 < re

   1. Initial program 8.6%

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

   2. Taylor expanded in re around inf 50.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 50.6%

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

      2. associate-/l* 54.7%

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

   4. Simplified 54.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 54.7%

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

      2. metadata-eval 54.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{im}{\frac{re}{im}}} \]

      3. *-un-lft-identity 54.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}}} \]

      4. associate-/l* 50.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]

      5. div-inv 50.5%

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

      6. sqrt-prod 63.4%

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

      7. sqrt-prod 84.6%

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

      8. add-sqr-sqrt 84.7%

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

      9. *-un-lft-identity 84.7%

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

      10. sqrt-prod 84.7%

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

      11. metadata-eval 84.7%

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

      12. sqrt-unprod 83.9%

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

      13. associate-*r* 84.2%

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

      14. expm1-log1p-u 83.6%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right)} \]

      15. expm1-udef 35.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} - 1\right)} \]

   6. Applied egg-rr 35.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 84.0%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 84.8%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   8. Simplified 84.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   if 1.2e-112 < re < 6.8999999999999998e106

   1. Initial program 32.3%

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

   2. Taylor expanded in re around 0 67.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 10^{-144}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-112} \lor \neg \left(re \leq 6.9 \cdot 10^{+106}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im} \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 3: 74.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 10^{-144}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-112} \lor \neg \left(re \leq 6.9 \cdot 10^{+106}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im - re\right) + 0.5 \cdot \frac{re}{\frac{im}{re}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e-16)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1e-144)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 1.35e-112) (not (<= re 6.9e+106)))
       (* 0.5 (/ im (sqrt re)))
       (* 0.5 (sqrt (* 2.0 (+ (- im re) (* 0.5 (/ re (/ im re)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e-16) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1e-144) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 1.35e-112) || !(re <= 6.9e+106)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * ((im - re) + (0.5 * (re / (im / re))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.6d-16)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1d-144) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 1.35d-112) .or. (.not. (re <= 6.9d+106))) then
        tmp = 0.5d0 * (im / sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * ((im - re) + (0.5d0 * (re / (im / re))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.6e-16) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1e-144) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 1.35e-112) || !(re <= 6.9e+106)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * ((im - re) + (0.5 * (re / (im / re))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e-16:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1e-144:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 1.35e-112) or not (re <= 6.9e+106):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * ((im - re) + (0.5 * (re / (im / re))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e-16)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1e-144)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 1.35e-112) || !(re <= 6.9e+106))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im - re) + Float64(0.5 * Float64(re / Float64(im / re)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.6e-16)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1e-144)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 1.35e-112) || ~((re <= 6.9e+106)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * ((im - re) + (0.5 * (re / (im / re))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e-16], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-144], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.35e-112], N[Not[LessEqual[re, 6.9e+106]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im - re), $MachinePrecision] + N[(0.5 * N[(re / N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.59999999999999983e-16

   1. Initial program 51.4%

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

   2. Taylor expanded in re around -inf 79.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative 79.1%

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

   4. Simplified 79.1%

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

   if -3.59999999999999983e-16 < re < 9.9999999999999995e-145

   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 0 80.5%

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

   if 9.9999999999999995e-145 < re < 1.35e-112 or 6.8999999999999998e106 < re

   1. Initial program 8.6%

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

   2. Taylor expanded in re around inf 50.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 50.6%

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

      2. associate-/l* 54.7%

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

   4. Simplified 54.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 54.7%

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

      2. metadata-eval 54.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{im}{\frac{re}{im}}} \]

      3. *-un-lft-identity 54.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}}} \]

      4. associate-/l* 50.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]

      5. div-inv 50.5%

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

      6. sqrt-prod 63.4%

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

      7. sqrt-prod 84.6%

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

      8. add-sqr-sqrt 84.7%

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

      9. *-un-lft-identity 84.7%

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

      10. sqrt-prod 84.7%

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

      11. metadata-eval 84.7%

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

      12. sqrt-unprod 83.9%

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

      13. associate-*r* 84.2%

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

      14. expm1-log1p-u 83.6%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right)} \]

      15. expm1-udef 35.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} - 1\right)} \]

   6. Applied egg-rr 35.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 84.0%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 84.8%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   8. Simplified 84.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   if 1.35e-112 < re < 6.8999999999999998e106

   1. Initial program 32.3%

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

   2. Taylor expanded in re around 0 66.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im + \left(-1 \cdot re + 0.5 \cdot \frac{{re}^{2}}{im}\right)\right)}} \]
   3. Step-by-step derivation

      1. neg-mul-1 66.7%

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

      2. associate-+r+ 66.7%

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

      3. sub-neg 66.7%

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

      4. +-commutative 66.7%

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

      5. unpow2 66.7%

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

      6. associate-/l* 66.7%

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

   4. Simplified 66.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{re}{\frac{im}{re}} + \left(im - re\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 10^{-144}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-112} \lor \neg \left(re \leq 6.9 \cdot 10^{+106}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im - re\right) + 0.5 \cdot \frac{re}{\frac{im}{re}}\right)}\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-145} \lor \neg \left(re \leq 2.4 \cdot 10^{-112}\right) \land re \leq 6.9 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8e-17)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 6.5e-145) (and (not (<= re 2.4e-112)) (<= re 6.9e+106)))
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8e-17) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 6.5e-145) || (!(re <= 2.4e-112) && (re <= 6.9e+106))) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-8d-17)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 6.5d-145) .or. (.not. (re <= 2.4d-112)) .and. (re <= 6.9d+106)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8e-17) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 6.5e-145) || (!(re <= 2.4e-112) && (re <= 6.9e+106))) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8e-17:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 6.5e-145) or (not (re <= 2.4e-112) and (re <= 6.9e+106)):
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8e-17)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 6.5e-145) || (!(re <= 2.4e-112) && (re <= 6.9e+106)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8e-17)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 6.5e-145) || (~((re <= 2.4e-112)) && (re <= 6.9e+106)))
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8e-17], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 6.5e-145], And[N[Not[LessEqual[re, 2.4e-112]], $MachinePrecision], LessEqual[re, 6.9e+106]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.00000000000000057e-17

   1. Initial program 51.4%

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

   2. Taylor expanded in re around -inf 79.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative 79.1%

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

   4. Simplified 79.1%

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

   if -8.00000000000000057e-17 < re < 6.5000000000000002e-145 or 2.4000000000000001e-112 < re < 6.8999999999999998e106

   1. Initial program 48.5%

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

   2. Taylor expanded in re around 0 75.8%

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

   if 6.5000000000000002e-145 < re < 2.4000000000000001e-112 or 6.8999999999999998e106 < re

   1. Initial program 8.6%

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

   2. Taylor expanded in re around inf 50.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 50.6%

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

      2. associate-/l* 54.7%

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

   4. Simplified 54.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 54.7%

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

      2. metadata-eval 54.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{im}{\frac{re}{im}}} \]

      3. *-un-lft-identity 54.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}}} \]

      4. associate-/l* 50.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]

      5. div-inv 50.5%

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

      6. sqrt-prod 63.4%

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

      7. sqrt-prod 84.6%

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

      8. add-sqr-sqrt 84.7%

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

      9. *-un-lft-identity 84.7%

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

      10. sqrt-prod 84.7%

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

      11. metadata-eval 84.7%

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

      12. sqrt-unprod 83.9%

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

      13. associate-*r* 84.2%

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

      14. expm1-log1p-u 83.6%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right)} \]

      15. expm1-udef 35.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} - 1\right)} \]

   6. Applied egg-rr 35.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 84.0%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 84.8%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   8. Simplified 84.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-145} \lor \neg \left(re \leq 2.4 \cdot 10^{-112}\right) \land re \leq 6.9 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5d-310)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e-310)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 55.9%

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

   2. Taylor expanded in re around -inf 55.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative 55.8%

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

   4. Simplified 55.8%

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

   if -4.999999999999985e-310 < re

   1. Initial program 26.1%

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

   2. Taylor expanded in re around inf 29.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 29.9%

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

      2. associate-/l* 35.2%

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

   4. Simplified 35.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 35.2%

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

      2. metadata-eval 35.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{im}{\frac{re}{im}}} \]

      3. *-un-lft-identity 35.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}}} \]

      4. associate-/l* 29.9%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]

      5. div-inv 29.8%

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

      6. sqrt-prod 35.3%

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

      7. sqrt-prod 53.6%

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

      8. add-sqr-sqrt 53.7%

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

      9. *-un-lft-identity 53.7%

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

      10. sqrt-prod 53.7%

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

      11. metadata-eval 53.7%

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

      12. sqrt-unprod 53.2%

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

      13. associate-*r* 53.4%

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

      14. expm1-log1p-u 53.2%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right)} \]

      15. expm1-udef 18.1%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} - 1\right)} \]

   6. Applied egg-rr 18.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 53.5%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 53.8%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   8. Simplified 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 6: 26.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{im}{\sqrt{re}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (/ im (sqrt re))))

C

double code(double re, double im) {
	return 0.5 * (im / sqrt(re));
}

Fortran

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

Java

public static double code(double re, double im) {
	return 0.5 * (im / Math.sqrt(re));
}

Python

def code(re, im):
	return 0.5 * (im / math.sqrt(re))

Julia

function code(re, im)
	return Float64(0.5 * Float64(im / sqrt(re)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im / sqrt(re));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

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

2. Taylor expanded in re around inf 15.4%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
3. Step-by-step derivation

   1. unpow2 15.4%

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

   2. associate-/l* 18.0%

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

4. Simplified 18.0%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 18.0%

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

   2. metadata-eval 18.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{1} \cdot \frac{im}{\frac{re}{im}}} \]

   3. *-un-lft-identity 18.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}}} \]

   4. associate-/l* 15.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]

   5. div-inv 15.4%

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

   6. sqrt-prod 17.8%

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

   7. sqrt-prod 27.0%

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

   8. add-sqr-sqrt 27.1%

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

   9. *-un-lft-identity 27.1%

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

   10. sqrt-prod 27.1%

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

   11. metadata-eval 27.1%

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

   12. sqrt-unprod 26.8%

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

   13. associate-*r* 26.9%

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

   14. expm1-log1p-u 26.8%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right)} \]

   15. expm1-udef 9.1%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} - 1\right)} \]

6. Applied egg-rr 9.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 27.0%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

   2. expm1-log1p 27.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

8. Simplified 27.1%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

9. Final simplification 27.1%

   \[\leadsto 0.5 \cdot \frac{im}{\sqrt{re}} \]

Reproduce

herbie shell --seed 2023293 
(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)))))