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

Percentage Accurate: 41.4% → 90.1%

Time: 6.5s

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: 41.4% 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: 90.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (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((2.0 * (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((2.0 * (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((2.0 * (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 (sqrt(Float64(2.0 * 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((2.0 * (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[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $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 (sqrt.f64 (*.f64 2 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 5.4%

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

   2. Taylor expanded in re around inf 54.8%

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

      1. unpow2 54.8%

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

      2. associate-/l* 58.0%

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

   4. Simplified 58.0%

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

   5. Taylor expanded in im around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. unpow1/2 99.6%

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

      3. unpow-1 99.6%

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

      4. exp-to-pow 93.5%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative 93.5%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-1 93.5%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod 93.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out 93.6%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in 93.6%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval 93.6%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow 99.6%

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

   7. Simplified 99.6%

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

      1. add-sqr-sqrt 99.2%

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

      2. associate-*l* 99.0%

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

      3. add-sqr-sqrt 98.8%

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

      4. sqrt-unprod 99.0%

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

      5. pow-prod-up 98.9%

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

      6. metadata-eval 98.9%

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

      7. inv-pow 98.9%

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

      8. sqrt-prod 90.1%

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

      9. div-inv 90.3%

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

      10. sqrt-div 99.1%

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

      11. associate-*r/ 99.3%

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

      12. add-sqr-sqrt 99.6%

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

   9. Applied egg-rr 99.6%

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

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

   1. Initial program 48.1%

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

      1. hypot-def 90.7%

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

   3. Simplified 90.7%

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

4. Final simplification 91.8%

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

Alternative 2: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6700:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-29} \lor \neg \left(re \leq 3.15 \cdot 10^{+68}\right) \land re \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6700.0)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 4.6e-29) (and (not (<= re 3.15e+68)) (<= re 3.2e+113)))
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6700.0) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 4.6e-29) || (!(re <= 3.15e+68) && (re <= 3.2e+113))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6700.0d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 4.6d-29) .or. (.not. (re <= 3.15d+68)) .and. (re <= 3.2d+113)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6700.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 4.6e-29) || (!(re <= 3.15e+68) && (re <= 3.2e+113))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6700.0:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 4.6e-29) or (not (re <= 3.15e+68) and (re <= 3.2e+113)):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6700.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 4.6e-29) || (!(re <= 3.15e+68) && (re <= 3.2e+113)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6700.0)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 4.6e-29) || (~((re <= 3.15e+68)) && (re <= 3.2e+113)))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6700.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 4.6e-29], And[N[Not[LessEqual[re, 3.15e+68]], $MachinePrecision], LessEqual[re, 3.2e+113]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6700

   1. Initial program 38.3%

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

   2. Taylor expanded in re around -inf 77.5%

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

      1. *-commutative 77.5%

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

   4. Simplified 77.5%

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

   if -6700 < re < 4.59999999999999982e-29 or 3.15000000000000013e68 < re < 3.1999999999999998e113

   1. Initial program 57.8%

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

   2. Taylor expanded in re around 0 83.5%

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

      1. expm1-log1p-u 79.7%

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

      2. expm1-udef 57.5%

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

      3. sqrt-unprod 57.5%

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

   4. Applied egg-rr 57.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot im}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 79.9%

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

      2. expm1-log1p 84.0%

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

      3. *-commutative 84.0%

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

   6. Simplified 84.0%

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

   if 4.59999999999999982e-29 < re < 3.15000000000000013e68 or 3.1999999999999998e113 < re

   1. Initial program 14.7%

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

   2. Taylor expanded in re around inf 50.3%

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

      1. unpow2 50.3%

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

      2. associate-/l* 55.7%

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

   4. Simplified 55.7%

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

   5. Taylor expanded in im around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. unpow1/2 76.5%

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

      3. unpow-1 76.5%

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

      4. exp-to-pow 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-1 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval 71.7%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow 76.5%

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

   7. Simplified 76.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6700:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{-29} \lor \neg \left(re \leq 3.15 \cdot 10^{+68}\right) \land re \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+65} \lor \neg \left(re \leq 3.4 \cdot 10^{+113}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e+70)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 8.8e-30)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 8e+65) (not (<= re 3.4e+113)))
       (* 0.5 (* im (pow re -0.5)))
       (* 0.5 (sqrt (* 2.0 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+70) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 8.8e-30) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 8e+65) || !(re <= 3.4e+113)) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d+70)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 8.8d-30) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 8d+65) .or. (.not. (re <= 3.4d+113))) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+70) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 8.8e-30) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 8e+65) || !(re <= 3.4e+113)) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e+70:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 8.8e-30:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 8e+65) or not (re <= 3.4e+113):
		tmp = 0.5 * (im * math.pow(re, -0.5))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e+70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 8.8e-30)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 8e+65) || !(re <= 3.4e+113))
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e+70)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 8.8e-30)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 8e+65) || ~((re <= 3.4e+113)))
		tmp = 0.5 * (im * (re ^ -0.5));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e+70], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.8e-30], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 8e+65], N[Not[LessEqual[re, 3.4e+113]], $MachinePrecision]], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.1000000000000003e70

   1. Initial program 25.1%

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

   2. Taylor expanded in re around -inf 84.8%

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

      1. *-commutative 84.8%

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

   4. Simplified 84.8%

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

   if -3.1000000000000003e70 < re < 8.79999999999999933e-30

   1. Initial program 63.6%

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

   2. Taylor expanded in re around 0 82.8%

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

   if 8.79999999999999933e-30 < re < 7.9999999999999999e65 or 3.40000000000000019e113 < re

   1. Initial program 14.7%

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

   2. Taylor expanded in re around inf 50.3%

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

      1. unpow2 50.3%

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

      2. associate-/l* 55.7%

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

   4. Simplified 55.7%

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

   5. Taylor expanded in im around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. unpow1/2 76.5%

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

      3. unpow-1 76.5%

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

      4. exp-to-pow 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-1 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval 71.7%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow 76.5%

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

   7. Simplified 76.5%

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

   if 7.9999999999999999e65 < re < 3.40000000000000019e113

   1. Initial program 18.1%

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

   2. Taylor expanded in re around 0 73.9%

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

      1. expm1-log1p-u 68.8%

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

      2. expm1-udef 69.3%

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

      3. sqrt-unprod 69.3%

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

   4. Applied egg-rr 69.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot im}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 68.8%

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

      2. expm1-log1p 74.4%

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

      3. *-commutative 74.4%

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

   6. Simplified 74.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+65} \lor \neg \left(re \leq 3.4 \cdot 10^{+113}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 4: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-31} \lor \neg \left(re \leq 3.15 \cdot 10^{+68}\right) \land re \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 3.7e-31) (and (not (<= re 3.15e+68)) (<= re 3.2e+113)))
   (* 0.5 (sqrt (* 2.0 im)))
   (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 3.7e-31) || (!(re <= 3.15e+68) && (re <= 3.2e+113))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 3.7d-31) .or. (.not. (re <= 3.15d+68)) .and. (re <= 3.2d+113)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 3.7e-31) || (!(re <= 3.15e+68) && (re <= 3.2e+113))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 3.7e-31) or (not (re <= 3.15e+68) and (re <= 3.2e+113)):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 3.7e-31) || (!(re <= 3.15e+68) && (re <= 3.2e+113)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 3.7e-31) || (~((re <= 3.15e+68)) && (re <= 3.2e+113)))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 3.7e-31], And[N[Not[LessEqual[re, 3.15e+68]], $MachinePrecision], LessEqual[re, 3.2e+113]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.6999999999999998e-31 or 3.15000000000000013e68 < re < 3.1999999999999998e113

   1. Initial program 51.9%

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

   2. Taylor expanded in re around 0 66.2%

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

      1. expm1-log1p-u 63.1%

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

      2. expm1-udef 47.4%

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

      3. sqrt-unprod 47.4%

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

   4. Applied egg-rr 47.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot im}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 63.3%

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

      2. expm1-log1p 66.6%

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

      3. *-commutative 66.6%

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

   6. Simplified 66.6%

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

   if 3.6999999999999998e-31 < re < 3.15000000000000013e68 or 3.1999999999999998e113 < re

   1. Initial program 14.7%

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

   2. Taylor expanded in re around inf 50.3%

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

      1. unpow2 50.3%

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

      2. associate-/l* 55.7%

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

   4. Simplified 55.7%

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

   5. Taylor expanded in im around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. unpow1/2 76.5%

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

      3. unpow-1 76.5%

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

      4. exp-to-pow 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-1 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval 71.7%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow 76.5%

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

   7. Simplified 76.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-31} \lor \neg \left(re \leq 3.15 \cdot 10^{+68}\right) \land re \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.2 \cdot 10^{-30} \lor \neg \left(re \leq 5.5 \cdot 10^{+65}\right) \land re \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 7.2e-30) (and (not (<= re 5.5e+65)) (<= re 3.2e+113)))
   (* 0.5 (sqrt (* 2.0 im)))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 7.2e-30) || (!(re <= 5.5e+65) && (re <= 3.2e+113))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= 7.2d-30) .or. (.not. (re <= 5.5d+65)) .and. (re <= 3.2d+113)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= 7.2e-30) || (!(re <= 5.5e+65) && (re <= 3.2e+113))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 7.2e-30) or (not (re <= 5.5e+65) and (re <= 3.2e+113)):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 7.2e-30) || (!(re <= 5.5e+65) && (re <= 3.2e+113)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= 7.2e-30) || (~((re <= 5.5e+65)) && (re <= 3.2e+113)))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 7.2e-30], And[N[Not[LessEqual[re, 5.5e+65]], $MachinePrecision], LessEqual[re, 3.2e+113]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.2000000000000006e-30 or 5.4999999999999996e65 < re < 3.1999999999999998e113

   1. Initial program 51.9%

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

   2. Taylor expanded in re around 0 66.2%

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

      1. expm1-log1p-u 63.1%

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

      2. expm1-udef 47.4%

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

      3. sqrt-unprod 47.4%

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

   4. Applied egg-rr 47.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot im}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 63.3%

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

      2. expm1-log1p 66.6%

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

      3. *-commutative 66.6%

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

   6. Simplified 66.6%

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

   if 7.2000000000000006e-30 < re < 5.4999999999999996e65 or 3.1999999999999998e113 < re

   1. Initial program 14.7%

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

   2. Taylor expanded in re around inf 50.3%

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

      1. unpow2 50.3%

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

      2. associate-/l* 55.7%

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

   4. Simplified 55.7%

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

   5. Taylor expanded in im around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. unpow1/2 76.5%

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

      3. unpow-1 76.5%

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

      4. exp-to-pow 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-1 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in 71.7%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval 71.7%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow 76.5%

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

   7. Simplified 76.5%

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

      1. add-sqr-sqrt 76.2%

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

      2. associate-*l* 76.1%

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

      3. add-sqr-sqrt 76.0%

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

      4. sqrt-unprod 76.1%

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

      5. pow-prod-up 76.1%

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

      6. metadata-eval 76.1%

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

      7. inv-pow 76.1%

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

      8. sqrt-prod 66.4%

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

      9. div-inv 66.5%

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

      10. sqrt-div 76.1%

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

      11. associate-*r/ 76.2%

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

      12. add-sqr-sqrt 76.4%

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

   9. Applied egg-rr 76.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.2 \cdot 10^{-30} \lor \neg \left(re \leq 5.5 \cdot 10^{+65}\right) \land re \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 6: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * im))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.9%

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

2. Taylor expanded in re around 0 57.0%

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

   1. expm1-log1p-u 54.3%

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

   2. expm1-udef 44.3%

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

   3. sqrt-unprod 44.3%

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

4. Applied egg-rr 44.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot im}\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-def 54.4%

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

   2. expm1-log1p 57.3%

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

   3. *-commutative 57.3%

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

6. Simplified 57.3%

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

7. Final simplification 57.3%

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

Reproduce

herbie shell --seed 2023215 
(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)))))