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

Percentage Accurate: 41.2% → 89.9%

Time: 8.5s

Alternatives: 5

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.2% 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: 89.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in re around inf 84.3%

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

      1. associate-*l* 84.4%

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

      2. *-commutative 84.4%

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

   5. Simplified 84.4%

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

      1. sqrt-unprod 85.6%

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

      2. metadata-eval 85.6%

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

      3. metadata-eval 85.6%

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

      4. associate-*r* 85.6%

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

      5. metadata-eval 85.6%

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

      6. metadata-eval 85.6%

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

      7. sqrt-unprod 84.3%

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

      8. associate-*l* 84.1%

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

      9. sqrt-div 84.3%

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

      10. metadata-eval 84.3%

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

      11. un-div-inv 84.3%

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

      12. sqrt-unprod 85.5%

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

      13. metadata-eval 85.5%

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

      14. metadata-eval 85.5%

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

      15. *-rgt-identity 85.5%

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

   7. Applied egg-rr 85.5%

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

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

   1. Initial program 44.2%

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

      1. add-sqr-sqrt 43.9%

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

      2. sqrt-unprod 44.2%

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

      3. *-commutative 44.2%

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

      4. *-commutative 44.2%

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

      5. swap-sqr 44.2%

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

      6. add-sqr-sqrt 44.2%

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

      7. *-commutative 44.2%

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

      8. hypot-define 90.5%

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

      9. metadata-eval 90.5%

         \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. associate-*l* 90.5%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 89.9%

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

Alternative 2: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.18 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 46000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.18e-32)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 46000.0)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.18e-32) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 46000.0) {
		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 <= (-1.18d-32)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 46000.0d0) 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 <= -1.18e-32) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 46000.0) {
		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 <= -1.18e-32:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 46000.0:
		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 <= -1.18e-32)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 46000.0)
		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 <= -1.18e-32)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 46000.0)
		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, -1.18e-32], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 46000.0], 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 < -1.17999999999999997e-32

   1. Initial program 42.8%

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

   3. Taylor expanded in re around -inf 85.4%

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

      1. *-commutative 85.4%

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

   5. Simplified 85.4%

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

   if -1.17999999999999997e-32 < re < 46000

   1. Initial program 54.5%

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

   3. Taylor expanded in re around 0 80.7%

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

   if 46000 < re

   1. Initial program 6.5%

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

   3. Taylor expanded in re around inf 71.0%

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

      1. associate-*l* 71.0%

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

      2. *-commutative 71.0%

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

   5. Simplified 71.0%

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

      1. expm1-log1p-u 70.5%

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

      2. expm1-undefine 29.7%

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

      3. sqrt-unprod 29.7%

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

      4. metadata-eval 29.7%

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

      5. metadata-eval 29.7%

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

      6. *-un-lft-identity 29.7%

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

      7. sqrt-div 29.7%

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

      8. metadata-eval 29.7%

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

      9. un-div-inv 29.7%

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

   7. Applied egg-rr 29.7%

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

      1. log1p-undefine 29.7%

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

      2. rem-exp-log 30.3%

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

      3. +-commutative 30.3%

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

      4. associate--l+ 71.8%

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

      5. metadata-eval 71.8%

         \[\leadsto 0.5 \cdot \frac{im}{\sqrt{re}} + \color{blue}{0} \]

      6. +-rgt-identity 71.8%

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

      7. *-commutative 71.8%

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

      8. associate-*l/ 71.9%

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

      9. associate-/l* 72.0%

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

   9. Simplified 72.0%

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

Alternative 3: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 30000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.3e-32)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 30000.0) (* 0.5 (sqrt (* im 2.0))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.3e-32) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 30000.0) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= (-1.3d-32)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 30000.0d0) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= -1.3e-32) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 30000.0) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.3e-32:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 30000.0:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.3e-32)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 30000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 <= -1.3e-32)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 30000.0)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.3e-32], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 30000.0], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2999999999999999e-32

   1. Initial program 42.8%

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

   3. Taylor expanded in re around -inf 85.4%

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

      1. *-commutative 85.4%

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

   5. Simplified 85.4%

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

   if -1.2999999999999999e-32 < re < 3e4

   1. Initial program 54.5%

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

   3. Taylor expanded in re around 0 79.2%

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

      1. expm1-log1p-u 75.7%

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

      2. expm1-undefine 55.1%

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

      3. sqrt-unprod 55.1%

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

   5. Applied egg-rr 55.1%

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

      1. log1p-undefine 55.1%

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

      2. rem-exp-log 59.0%

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

      3. +-commutative 59.0%

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

      4. associate--l+ 79.8%

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

      5. metadata-eval 79.8%

         \[\leadsto 0.5 \cdot \left(\sqrt{im \cdot 2} + \color{blue}{0}\right) \]

      6. +-rgt-identity 79.8%

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

   7. Simplified 79.8%

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

   if 3e4 < re

   1. Initial program 6.5%

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

   3. Taylor expanded in re around inf 71.0%

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

      1. associate-*l* 71.0%

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

      2. *-commutative 71.0%

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

   5. Simplified 71.0%

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

      1. expm1-log1p-u 70.5%

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

      2. expm1-undefine 29.7%

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

      3. sqrt-unprod 29.7%

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

      4. metadata-eval 29.7%

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

      5. metadata-eval 29.7%

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

      6. *-un-lft-identity 29.7%

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

      7. sqrt-div 29.7%

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

      8. metadata-eval 29.7%

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

      9. un-div-inv 29.7%

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

   7. Applied egg-rr 29.7%

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

      1. log1p-undefine 29.7%

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

      2. rem-exp-log 30.3%

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

      3. +-commutative 30.3%

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

      4. associate--l+ 71.8%

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

      5. metadata-eval 71.8%

         \[\leadsto 0.5 \cdot \frac{im}{\sqrt{re}} + \color{blue}{0} \]

      6. +-rgt-identity 71.8%

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

      7. *-commutative 71.8%

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

      8. associate-*l/ 71.9%

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

      9. associate-/l* 72.0%

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

   9. Simplified 72.0%

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

Alternative 4: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 235000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 235000000.0) (* 0.5 (sqrt (* im 2.0))) (* im (/ 0.5 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 235000000.0) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= 235000000.0d0) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= 235000000.0) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 235000000.0:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 235000000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 <= 235000000.0)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 235000000.0], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.35e8

   1. Initial program 50.4%

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

   3. Taylor expanded in re around 0 58.4%

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

      1. expm1-log1p-u 55.7%

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

      2. expm1-undefine 41.9%

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

      3. sqrt-unprod 41.9%

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

   5. Applied egg-rr 41.9%

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

      1. log1p-undefine 41.9%

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

      2. rem-exp-log 44.8%

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

      3. +-commutative 44.8%

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

      4. associate--l+ 58.8%

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

      5. metadata-eval 58.8%

         \[\leadsto 0.5 \cdot \left(\sqrt{im \cdot 2} + \color{blue}{0}\right) \]

      6. +-rgt-identity 58.8%

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

   7. Simplified 58.8%

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

   if 2.35e8 < re

   1. Initial program 6.5%

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

   3. Taylor expanded in re around inf 71.0%

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

      1. associate-*l* 71.0%

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

      2. *-commutative 71.0%

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

   5. Simplified 71.0%

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

      1. expm1-log1p-u 70.5%

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

      2. expm1-undefine 29.7%

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

      3. sqrt-unprod 29.7%

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

      4. metadata-eval 29.7%

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

      5. metadata-eval 29.7%

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

      6. *-un-lft-identity 29.7%

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

      7. sqrt-div 29.7%

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

      8. metadata-eval 29.7%

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

      9. un-div-inv 29.7%

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

   7. Applied egg-rr 29.7%

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

      1. log1p-undefine 29.7%

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

      2. rem-exp-log 30.3%

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

      3. +-commutative 30.3%

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

      4. associate--l+ 71.8%

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

      5. metadata-eval 71.8%

         \[\leadsto 0.5 \cdot \frac{im}{\sqrt{re}} + \color{blue}{0} \]

      6. +-rgt-identity 71.8%

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

      7. *-commutative 71.8%

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

      8. associate-*l/ 71.9%

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

      9. associate-/l* 72.0%

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

   9. Simplified 72.0%

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

Alternative 5: 52.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in re around 0 51.9%

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

   1. expm1-log1p-u 49.2%

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

   2. expm1-undefine 41.9%

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

   3. sqrt-unprod 41.9%

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

5. Applied egg-rr 41.9%

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

   1. log1p-undefine 41.9%

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

   2. rem-exp-log 44.8%

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

   3. +-commutative 44.8%

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

   4. associate--l+ 52.2%

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

   5. metadata-eval 52.2%

      \[\leadsto 0.5 \cdot \left(\sqrt{im \cdot 2} + \color{blue}{0}\right) \]

   6. +-rgt-identity 52.2%

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

7. Simplified 52.2%

   \[\leadsto 0.5 \cdot \color{blue}{\sqrt{im \cdot 2}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(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)))))