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

Percentage Accurate: 41.0% → 90.1%

Time: 9.9s

Alternatives: 8

Speedup: 2.2×

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: 41.0% 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, 0.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* (* im 0.5) (sqrt (/ 1.0 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 = (im * 0.5) * sqrt((1.0 / 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 = (im * 0.5) * Math.sqrt((1.0 / 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 = (im * 0.5) * math.sqrt((1.0 / 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(Float64(im * 0.5) * sqrt(Float64(1.0 / 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 = (im * 0.5) * sqrt((1.0 / 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[(N[(im * 0.5), $MachinePrecision] * N[Sqrt[N[(1.0 / 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 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 9.4%

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

   3. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

      3. lower-*.f64 39.9

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

   5. Simplified 39.9%

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

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 99.6

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

   8. Simplified 99.6%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 51.3%

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

      1. lower-hypot.f64 95.0

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

   4. Applied egg-rr 95.0%

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

4. Final simplification 95.5%

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

Alternative 2: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}\\ \mathbf{if}\;re \leq -5.2 \cdot 10^{+92}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.42 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 - re\right)}\\ \mathbf{elif}\;re \leq 1.22 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, 2 \cdot im\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2}}{\sqrt{re + t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (fma re re (* im im)))))
   (if (<= re -5.2e+92)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re -1.42e-143)
       (* 0.5 (sqrt (* 2.0 (- t_0 re))))
       (if (<= re 1.22e-60)
         (* 0.5 (sqrt (fma re (+ -2.0 (/ re im)) (* 2.0 im))))
         (if (<= re 8e+127)
           (* 0.5 (/ (* im (sqrt 2.0)) (sqrt (+ re t_0))))
           (* (* im 0.5) (sqrt (/ 1.0 re)))))))))

C

double code(double re, double im) {
	double t_0 = sqrt(fma(re, re, (im * im)));
	double tmp;
	if (re <= -5.2e+92) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -1.42e-143) {
		tmp = 0.5 * sqrt((2.0 * (t_0 - re)));
	} else if (re <= 1.22e-60) {
		tmp = 0.5 * sqrt(fma(re, (-2.0 + (re / im)), (2.0 * im)));
	} else if (re <= 8e+127) {
		tmp = 0.5 * ((im * sqrt(2.0)) / sqrt((re + t_0)));
	} else {
		tmp = (im * 0.5) * sqrt((1.0 / re));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = sqrt(fma(re, re, Float64(im * im)))
	tmp = 0.0
	if (re <= -5.2e+92)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -1.42e-143)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 - re))));
	elseif (re <= 1.22e-60)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(-2.0 + Float64(re / im)), Float64(2.0 * im))));
	elseif (re <= 8e+127)
		tmp = Float64(0.5 * Float64(Float64(im * sqrt(2.0)) / sqrt(Float64(re + t_0))));
	else
		tmp = Float64(Float64(im * 0.5) * sqrt(Float64(1.0 / re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -5.2e+92], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.42e-143], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.22e-60], N[(0.5 * N[Sqrt[N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision] + N[(2.0 * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8e+127], N[(0.5 * N[(N[(im * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(re + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -5.1999999999999998e92

   1. Initial program 28.1%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 85.6

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

   5. Simplified 85.6%

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

   if -5.1999999999999998e92 < re < -1.42e-143

   1. Initial program 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.1

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

   4. Applied egg-rr 81.1%

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

   if -1.42e-143 < re < 1.22e-60

   1. Initial program 54.3%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(re, \frac{re}{im} - 2, 2 \cdot im\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, \color{blue}{\frac{re}{im} + \left(\mathsf{neg}\left(2\right)\right)}, 2 \cdot im\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, \frac{re}{im} + \color{blue}{-2}, 2 \cdot im\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, \color{blue}{-2 + \frac{re}{im}}, 2 \cdot im\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, \color{blue}{-2 + \frac{re}{im}}, 2 \cdot im\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, -2 + \color{blue}{\frac{re}{im}}, 2 \cdot im\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, \color{blue}{im \cdot 2}\right)} \]

      9. lower-*.f64 90.8

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

   5. Simplified 90.8%

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

   if 1.22e-60 < re < 7.99999999999999964e127

   1. Initial program 32.6%

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

      1. lower-hypot.f64 61.0

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

   4. Applied egg-rr 61.0%

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

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot 2} \]

      5. flip-- N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right) - re \cdot re}{\mathsf{hypot}\left(re, im\right) + re}} \cdot 2} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right) - re \cdot re\right) \cdot 2}{\mathsf{hypot}\left(re, im\right) + re}}} \]

      7. sqrt-div N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right) - re \cdot re\right) \cdot 2}}{\sqrt{\mathsf{hypot}\left(re, im\right) + re}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right) - re \cdot re\right) \cdot 2}}{\sqrt{\mathsf{hypot}\left(re, im\right) + re}}} \]

   6. Applied egg-rr 29.8%

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

   7. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{im \cdot \sqrt{2}}}{\sqrt{re + \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{im \cdot \sqrt{2}}}{\sqrt{re + \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}} \]

      2. lower-sqrt.f64 70.5

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

   9. Simplified 70.5%

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

   if 7.99999999999999964e127 < re

   1. Initial program 6.4%

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

   3. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

      3. lower-*.f64 54.7

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

   5. Simplified 54.7%

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

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 92.9

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

   8. Simplified 92.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot im\right) \cdot \sqrt{\frac{1}{re}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+92}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.42 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1.22 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, 2 \cdot im\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2}}{\sqrt{re + \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+92}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.42 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, 2 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.2e+92)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re -1.42e-143)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 9e+109)
       (* 0.5 (sqrt (fma re (+ -2.0 (/ re im)) (* 2.0 im))))
       (* (* im 0.5) (sqrt (/ 1.0 re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.2e+92) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -1.42e-143) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 9e+109) {
		tmp = 0.5 * sqrt(fma(re, (-2.0 + (re / im)), (2.0 * im)));
	} else {
		tmp = (im * 0.5) * sqrt((1.0 / re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.2e+92)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -1.42e-143)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 9e+109)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(-2.0 + Float64(re / im)), Float64(2.0 * im))));
	else
		tmp = Float64(Float64(im * 0.5) * sqrt(Float64(1.0 / re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.2e+92], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.42e-143], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+109], N[(0.5 * N[Sqrt[N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision] + N[(2.0 * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.1999999999999998e92

   1. Initial program 28.1%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 85.6

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

   5. Simplified 85.6%

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

   if -5.1999999999999998e92 < re < -1.42e-143

   1. Initial program 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.1

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

   4. Applied egg-rr 81.1%

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

   if -1.42e-143 < re < 8.9999999999999992e109

   1. Initial program 47.2%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(re, \frac{re}{im} - 2, 2 \cdot im\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, \color{blue}{\frac{re}{im} + \left(\mathsf{neg}\left(2\right)\right)}, 2 \cdot im\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, \frac{re}{im} + \color{blue}{-2}, 2 \cdot im\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, \color{blue}{-2 + \frac{re}{im}}, 2 \cdot im\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, \color{blue}{-2 + \frac{re}{im}}, 2 \cdot im\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, -2 + \color{blue}{\frac{re}{im}}, 2 \cdot im\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, \color{blue}{im \cdot 2}\right)} \]

      9. lower-*.f64 80.9

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

   5. Simplified 80.9%

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

   if 8.9999999999999992e109 < re

   1. Initial program 9.7%

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

   3. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

      3. lower-*.f64 49.9

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

   5. Simplified 49.9%

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

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 90.5

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

   8. Simplified 90.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+92}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.42 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, 2 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.85 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.85e+93)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9e+109)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* im 0.5) (sqrt (/ 1.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.85e+93) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9e+109) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) * sqrt((1.0 / 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 <= (-1.85d+93)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 9d+109) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im * 0.5d0) * sqrt((1.0d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.85e+93) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 9e+109) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) * Math.sqrt((1.0 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.85e+93:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 9e+109:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) * math.sqrt((1.0 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.85e+93)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9e+109)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im * 0.5) * sqrt(Float64(1.0 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.85e+93)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 9e+109)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) * sqrt((1.0 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.85e+93], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+109], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.84999999999999994e93

   1. Initial program 28.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 87.4

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

   5. Simplified 87.4%

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

   if -1.84999999999999994e93 < re < 8.9999999999999992e109

   1. Initial program 57.3%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 74.5

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

   5. Simplified 74.5%

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

   if 8.9999999999999992e109 < re

   1. Initial program 9.7%

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

   3. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

      3. lower-*.f64 49.9

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

   5. Simplified 49.9%

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

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 90.5

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

   8. Simplified 90.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.85 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.85 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.85e+93)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9e+109)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.85e+93) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9e+109) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / 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 <= (-1.85d+93)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 9d+109) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.85e+93) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 9e+109) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.85e+93:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 9e+109:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.85e+93)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9e+109)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.85e+93)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 9e+109)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.85e+93], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+109], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.84999999999999994e93

   1. Initial program 28.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 87.4

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

   5. Simplified 87.4%

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

   if -1.84999999999999994e93 < re < 8.9999999999999992e109

   1. Initial program 57.3%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 74.5

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

   5. Simplified 74.5%

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

   if 8.9999999999999992e109 < re

   1. Initial program 9.7%

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

   3. Taylor expanded in re around inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

      3. lower-*.f64 49.9

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

   5. Simplified 49.9%

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

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 90.5

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

   8. Simplified 90.5%

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. un-div-inv N/A

          \[\leadsto im \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{re}}} \]

      11. lower-/.f64 N/A

          \[\leadsto im \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{re}}} \]

      12. lower-sqrt.f64 90.3

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

   10. Applied egg-rr 90.3%

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

Alternative 6: 63.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.85 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.85e+93)
   (* 0.5 (sqrt (* re -4.0)))
   (* 0.5 (sqrt (* 2.0 (- im re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.85e+93) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-1.85d+93)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.85e+93) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.85e+93:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.85e+93)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.85e+93)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.85e+93], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.84999999999999994e93

   1. Initial program 28.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 87.4

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

   5. Simplified 87.4%

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

   if -1.84999999999999994e93 < re

   1. Initial program 50.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 65.8

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

   5. Simplified 65.8%

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

Alternative 7: 64.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+36) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+36) {
		tmp = 0.5 * sqrt((re * -4.0));
	} 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 <= (-1.8d+36)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    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 <= -1.8e+36) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+36:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+36)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	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 <= -1.8e+36)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+36], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.7999999999999999e36

   1. Initial program 37.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   if -1.7999999999999999e36 < re

   1. Initial program 49.8%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{im \cdot 2}} \]

      2. lower-*.f64 65.8

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

   5. Simplified 65.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.8% accurate, 2.2× 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 47.1%

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

3. Taylor expanded in re around 0

   \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   2. lower-*.f64 57.0

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

5. Simplified 57.0%

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

6. Final simplification 57.0%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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)))))