math.sqrt on complex, real part

Percentage Accurate: 40.6% → 85.1%

Time: 6.5s

Alternatives: 6

Speedup: 2.0×

Specification

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: 40.6% 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: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \frac{-1}{re}}{\frac{1}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (/ (* im (/ -1.0 re)) (/ 1.0 im))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * sqrt(((im * (-1.0 / re)) / (1.0 / im)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * Math.sqrt(((im * (-1.0 / re)) / (1.0 / im)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im * im)))))) <= 0.0:
		tmp = 0.5 * math.sqrt(((im * (-1.0 / re)) / (1.0 / im)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))))) <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * Float64(-1.0 / re)) / Float64(1.0 / im))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0)
		tmp = 0.5 * sqrt(((im * (-1.0 / re)) / (1.0 / im)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(N[(im * N[(-1.0 / re), $MachinePrecision]), $MachinePrecision] / N[(1.0 / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $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 8.9%

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

      1. +-commutative 8.9%

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

      2. hypot-def 8.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in re around -inf 45.1%

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

      1. *-commutative 45.1%

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

      2. unpow2 45.1%

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

      3. associate-/l* 55.0%

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

   6. Simplified 55.0%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 18.6%

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

      3. *-commutative 18.6%

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

      4. associate-*l* 18.6%

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

      5. metadata-eval 18.6%

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

   8. Applied egg-rr 18.6%

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

      1. expm1-def 54.6%

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

      2. expm1-log1p 55.0%

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

      3. *-commutative 55.0%

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

      4. mul-1-neg 55.0%

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

      5. associate-/r/ 55.0%

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

      6. distribute-rgt-neg-in 55.0%

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

   10. Simplified 55.0%

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

       1. distribute-rgt-neg-out 55.0%

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

       2. associate-/r/ 55.0%

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

       3. *-un-lft-identity 55.0%

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

       4. div-inv 55.1%

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

       5. frac-times 45.4%

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

       6. associate-*r/ 55.1%

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

       7. distribute-neg-frac 55.1%

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

   12. Applied egg-rr 55.1%

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

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

   1. Initial program 46.2%

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

      1. +-commutative 46.2%

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

      2. hypot-def 91.3%

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

   3. Simplified 91.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \frac{-1}{re}}{\frac{1}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 48.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{re} \cdot \left(-im\right)}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 80000 \lor \neg \left(re \leq 1.16 \cdot 10^{+50}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.5e+176)
   (* 0.5 (sqrt (* (/ im re) (- im))))
   (if (<= re 1.9e-71)
     (* 0.5 (sqrt (* 2.0 im)))
     (if (or (<= re 80000.0) (not (<= re 1.16e+50)))
       (* 0.5 (* 2.0 (sqrt re)))
       (* 0.5 (sqrt (* 2.0 (+ re im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.5e+176) {
		tmp = 0.5 * sqrt(((im / re) * -im));
	} else if (re <= 1.9e-71) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else if ((re <= 80000.0) || !(re <= 1.16e+50)) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + 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 <= (-8.5d+176)) then
        tmp = 0.5d0 * sqrt(((im / re) * -im))
    else if (re <= 1.9d-71) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else if ((re <= 80000.0d0) .or. (.not. (re <= 1.16d+50))) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.5e+176) {
		tmp = 0.5 * Math.sqrt(((im / re) * -im));
	} else if (re <= 1.9e-71) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else if ((re <= 80000.0) || !(re <= 1.16e+50)) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.5e+176:
		tmp = 0.5 * math.sqrt(((im / re) * -im))
	elif re <= 1.9e-71:
		tmp = 0.5 * math.sqrt((2.0 * im))
	elif (re <= 80000.0) or not (re <= 1.16e+50):
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.5e+176)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / re) * Float64(-im))));
	elseif (re <= 1.9e-71)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	elseif ((re <= 80000.0) || !(re <= 1.16e+50))
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.5e+176)
		tmp = 0.5 * sqrt(((im / re) * -im));
	elseif (re <= 1.9e-71)
		tmp = 0.5 * sqrt((2.0 * im));
	elseif ((re <= 80000.0) || ~((re <= 1.16e+50)))
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.5e+176], N[(0.5 * N[Sqrt[N[(N[(im / re), $MachinePrecision] * (-im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e-71], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 80000.0], N[Not[LessEqual[re, 1.16e+50]], $MachinePrecision]], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.4999999999999995e176

   1. Initial program 2.4%

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

      1. +-commutative 2.4%

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

      2. hypot-def 20.6%

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

   3. Simplified 20.6%

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

   4. Taylor expanded in re around -inf 48.1%

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

      1. *-commutative 48.1%

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

      2. unpow2 48.1%

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

      3. associate-/l* 57.3%

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

   6. Simplified 57.3%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 33.1%

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

      3. *-commutative 33.1%

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

      4. associate-*l* 33.1%

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

      5. metadata-eval 33.1%

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

   8. Applied egg-rr 33.1%

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

      1. expm1-def 55.9%

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

      2. expm1-log1p 57.3%

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

      3. *-commutative 57.3%

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

      4. mul-1-neg 57.3%

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

      5. associate-/r/ 57.2%

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

      6. distribute-rgt-neg-in 57.2%

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

   10. Simplified 57.2%

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

   if -8.4999999999999995e176 < re < 1.89999999999999996e-71

   1. Initial program 46.9%

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

      1. +-commutative 46.9%

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

      2. hypot-def 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in re around 0 31.8%

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

   if 1.89999999999999996e-71 < re < 8e4 or 1.16e50 < re

   1. Initial program 41.1%

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

      1. +-commutative 41.1%

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

      2. hypot-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in im around 0 85.2%

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

      1. unpow2 85.2%

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

      2. rem-square-sqrt 86.9%

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

   6. Simplified 86.9%

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

   if 8e4 < re < 1.16e50

   1. Initial program 52.1%

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

      1. +-commutative 52.1%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 26.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{re} \cdot \left(-im\right)}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 80000 \lor \neg \left(re \leq 1.16 \cdot 10^{+50}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3: 48.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 660000 \lor \neg \left(re \leq 2.5 \cdot 10^{+51}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.2e+176)
   (* 0.5 (sqrt (/ (- im) (/ re im))))
   (if (<= re 4.5e-70)
     (* 0.5 (sqrt (* 2.0 im)))
     (if (or (<= re 660000.0) (not (<= re 2.5e+51)))
       (* 0.5 (* 2.0 (sqrt re)))
       (* 0.5 (sqrt (* 2.0 (+ re im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.2e+176) {
		tmp = 0.5 * sqrt((-im / (re / im)));
	} else if (re <= 4.5e-70) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else if ((re <= 660000.0) || !(re <= 2.5e+51)) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + 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 <= (-8.2d+176)) then
        tmp = 0.5d0 * sqrt((-im / (re / im)))
    else if (re <= 4.5d-70) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else if ((re <= 660000.0d0) .or. (.not. (re <= 2.5d+51))) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.2e+176) {
		tmp = 0.5 * Math.sqrt((-im / (re / im)));
	} else if (re <= 4.5e-70) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else if ((re <= 660000.0) || !(re <= 2.5e+51)) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.2e+176:
		tmp = 0.5 * math.sqrt((-im / (re / im)))
	elif re <= 4.5e-70:
		tmp = 0.5 * math.sqrt((2.0 * im))
	elif (re <= 660000.0) or not (re <= 2.5e+51):
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.2e+176)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) / Float64(re / im))));
	elseif (re <= 4.5e-70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	elseif ((re <= 660000.0) || !(re <= 2.5e+51))
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.2e+176)
		tmp = 0.5 * sqrt((-im / (re / im)));
	elseif (re <= 4.5e-70)
		tmp = 0.5 * sqrt((2.0 * im));
	elseif ((re <= 660000.0) || ~((re <= 2.5e+51)))
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.2e+176], N[(0.5 * N[Sqrt[N[((-im) / N[(re / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e-70], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 660000.0], N[Not[LessEqual[re, 2.5e+51]], $MachinePrecision]], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.1999999999999998e176

   1. Initial program 2.4%

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

      1. +-commutative 2.4%

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

      2. hypot-def 20.6%

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

   3. Simplified 20.6%

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

   4. Taylor expanded in re around -inf 48.1%

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

      1. *-commutative 48.1%

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

      2. unpow2 48.1%

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

      3. associate-/l* 57.3%

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

   6. Simplified 57.3%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 33.1%

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

      3. *-commutative 33.1%

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

      4. associate-*l* 33.1%

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

      5. metadata-eval 33.1%

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

   8. Applied egg-rr 33.1%

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

      1. expm1-def 55.9%

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

      2. expm1-log1p 57.3%

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

      3. *-commutative 57.3%

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

      4. mul-1-neg 57.3%

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

      5. associate-/r/ 57.2%

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

      6. distribute-rgt-neg-in 57.2%

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

   10. Simplified 57.2%

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

       1. distribute-rgt-neg-out 57.2%

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

       2. associate-/r/ 57.3%

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

       3. neg-mul-1 57.3%

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

       4. *-commutative 57.3%

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

       5. associate-*l/ 57.3%

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

   12. Applied egg-rr 57.3%

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

   if -8.1999999999999998e176 < re < 4.50000000000000022e-70

   1. Initial program 46.9%

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

      1. +-commutative 46.9%

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

      2. hypot-def 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in re around 0 31.8%

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

   if 4.50000000000000022e-70 < re < 6.6e5 or 2.5e51 < re

   1. Initial program 41.1%

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

      1. +-commutative 41.1%

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

      2. hypot-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in im around 0 85.2%

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

      1. unpow2 85.2%

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

      2. rem-square-sqrt 86.9%

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

   6. Simplified 86.9%

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

   if 6.6e5 < re < 2.5e51

   1. Initial program 52.1%

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

      1. +-commutative 52.1%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 26.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 660000 \lor \neg \left(re \leq 2.5 \cdot 10^{+51}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4: 42.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 13000 \lor \neg \left(re \leq 1.02 \cdot 10^{+50}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e-70)
   (* 0.5 (sqrt (* 2.0 im)))
   (if (or (<= re 13000.0) (not (<= re 1.02e+50)))
     (* 0.5 (* 2.0 (sqrt re)))
     (* 0.5 (sqrt (* 2.0 (+ re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-70) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else if ((re <= 13000.0) || !(re <= 1.02e+50)) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + 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 <= 6.2d-70) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else if ((re <= 13000.0d0) .or. (.not. (re <= 1.02d+50))) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-70) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else if ((re <= 13000.0) || !(re <= 1.02e+50)) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e-70:
		tmp = 0.5 * math.sqrt((2.0 * im))
	elif (re <= 13000.0) or not (re <= 1.02e+50):
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e-70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	elseif ((re <= 13000.0) || !(re <= 1.02e+50))
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e-70)
		tmp = 0.5 * sqrt((2.0 * im));
	elseif ((re <= 13000.0) || ~((re <= 1.02e+50)))
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e-70], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 13000.0], N[Not[LessEqual[re, 1.02e+50]], $MachinePrecision]], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 6.2e-70

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. hypot-def 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in re around 0 28.9%

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

   if 6.2e-70 < re < 13000 or 1.01999999999999991e50 < re

   1. Initial program 41.1%

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

      1. +-commutative 41.1%

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

      2. hypot-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in im around 0 85.2%

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

      1. unpow2 85.2%

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

      2. rem-square-sqrt 86.9%

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

   6. Simplified 86.9%

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

   if 13000 < re < 1.01999999999999991e50

   1. Initial program 52.1%

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

      1. +-commutative 52.1%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 26.1%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 13000 \lor \neg \left(re \leq 1.02 \cdot 10^{+50}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5: 43.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e-70) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-70) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * 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 <= 6.2d-70) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-70) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e-70:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e-70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e-70)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e-70], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.2e-70

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. hypot-def 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in re around 0 28.9%

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

   if 6.2e-70 < re

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. hypot-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in im around 0 78.4%

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

      1. unpow2 78.4%

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

      2. rem-square-sqrt 79.9%

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

   6. Simplified 79.9%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6: 25.6% 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 42.0%

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

   1. +-commutative 42.0%

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

   2. hypot-def 82.0%

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

3. Simplified 82.0%

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

4. Taylor expanded in re around 0 23.0%

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

5. Final simplification 23.0%

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

Developer target: 48.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023229 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))