math.sqrt on complex, real part

Percentage Accurate: 41.0% → 84.5%

Time: 10.5s

Alternatives: 9

Speedup: 1.2×

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: 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: 84.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (*
    0.5
    (sqrt (* (/ im -1.0) (/ (* 2.0 im) (- re (sqrt (fma re re (* im 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) * ((2.0 * im) / (re - sqrt(fma(re, re, (im * im)))))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + 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 / -1.0) * Float64(Float64(2.0 * im) / Float64(re - sqrt(fma(re, re, Float64(im * im))))))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return 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 / -1.0), $MachinePrecision] * N[(N[(2.0 * im), $MachinePrecision] / N[(re - N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 12.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. flip-+ N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 12.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 58.1

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

   7. Simplified 58.1%

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

   9. Applied egg-rr 59.9%

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

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

   1. Initial program 43.3%

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

      1. lower-hypot.f64 90.0

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

   4. Applied egg-rr 90.0%

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

4. Final simplification 86.8%

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

Alternative 2: 58.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ t_1 := \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{2 \cdot im}{re - t\_1}}\\ \mathbf{elif}\;t\_0 \leq 10^{+76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{re + im}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))))
        (t_1 (sqrt (fma re re (* im im)))))
   (if (<= t_0 0.0)
     (* 0.5 (sqrt (* (/ im -1.0) (/ (* 2.0 im) (- re t_1)))))
     (if (<= t_0 1e+76)
       (* 0.5 (sqrt (* 2.0 (+ re t_1))))
       (* (* 0.5 (sqrt (+ re im))) (sqrt 2.0))))))

C

double code(double re, double im) {
	double t_0 = sqrt((2.0 * (re + sqrt(((re * re) + (im * im))))));
	double t_1 = sqrt(fma(re, re, (im * im)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * sqrt(((im / -1.0) * ((2.0 * im) / (re - t_1))));
	} else if (t_0 <= 1e+76) {
		tmp = 0.5 * sqrt((2.0 * (re + t_1)));
	} else {
		tmp = (0.5 * sqrt((re + im))) * sqrt(2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im))))))
	t_1 = sqrt(fma(re, re, Float64(im * im)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / -1.0) * Float64(Float64(2.0 * im) / Float64(re - t_1)))));
	elseif (t_0 <= 1e+76)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + t_1))));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(re + im))) * sqrt(2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[Sqrt[N[(N[(im / -1.0), $MachinePrecision] * N[(N[(2.0 * im), $MachinePrecision] / N[(re - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+76], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(re + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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 10.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. flip-+ N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 10.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 51.4

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

   7. Simplified 51.4%

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

   9. Applied egg-rr 56.4%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 1e76

   1. Initial program 92.6%

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

      1. lift-*.f64 N/A

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

      2. lower-fma.f64 92.6

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

   4. Applied egg-rr 92.6%

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

   if 1e76 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 4.6%

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

      1. 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. *-commutative N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. pow1/2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied egg-rr 4.6%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 28.9

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

   7. Simplified 28.9%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{2 \cdot im}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 10^{+76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{re + im}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 48.1% accurate, 0.6× 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 2024218 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

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