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

Percentage Accurate: 40.7% → 89.5%

Time: 11.9s

Alternatives: 8

Speedup: 2.1×

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: 40.7% 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: 89.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* (pow re -0.5) (* im 0.5))
   (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 = pow(re, -0.5) * (im * 0.5);
	} 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 = Math.pow(re, -0.5) * (im * 0.5);
	} 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 = math.pow(re, -0.5) * (im * 0.5)
	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((re ^ -0.5) * Float64(im * 0.5));
	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 = (re ^ -0.5) * (im * 0.5);
	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[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $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 8.3%

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

      1. sub-neg 8.3%

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

      2. sqr-neg 8.3%

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

      3. sub-neg 8.3%

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

      4. sqr-neg 8.3%

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

      5. hypot-def 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in im around 0 93.9%

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

      1. associate-*r* 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-*r* 94.4%

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

      4. associate-*r* 94.4%

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

   6. Simplified 94.4%

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

      1. expm1-log1p-u 94.4%

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

      2. expm1-udef 10.9%

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

      3. associate-*r* 10.9%

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

      4. *-commutative 10.9%

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

      5. sqrt-div 10.9%

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

      6. metadata-eval 10.9%

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

      7. associate-*l/ 10.9%

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

      8. *-un-lft-identity 10.9%

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

      9. *-commutative 10.9%

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

      10. *-commutative 10.9%

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

      11. associate-*l* 10.9%

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

   8. Applied egg-rr 10.9%

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

      1. expm1-def 94.5%

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

      2. expm1-log1p 94.5%

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

      3. associate-*r/ 94.4%

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

      4. associate-*r* 93.9%

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

      5. *-commutative 93.9%

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

      6. associate-*l* 93.9%

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

      7. *-commutative 93.9%

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

      8. associate-*r* 94.4%

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

      9. associate-*l* 94.4%

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

   10. Simplified 94.4%

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

       1. clear-num 93.5%

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

       2. associate-/r/ 94.3%

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

       3. pow1/2 94.3%

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

       4. pow-flip 94.3%

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

       5. metadata-eval 94.3%

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

       6. associate-*l* 93.9%

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

       7. add-sqr-sqrt 93.9%

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

       8. sqrt-unprod 93.9%

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

       9. swap-sqr 93.9%

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

       10. rem-square-sqrt 93.9%

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

       11. metadata-eval 93.9%

           \[\leadsto {re}^{-0.5} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{2 \cdot \color{blue}{0.25}}\right)\right) \]

       12. metadata-eval 93.9%

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

       13. rem-square-sqrt 95.4%

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

   12. Applied egg-rr 95.4%

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

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

   1. Initial program 47.5%

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

      1. sub-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. sub-neg 47.5%

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

      4. sqr-neg 47.5%

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

      5. hypot-def 90.6%

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

   3. Simplified 90.6%

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

      1. add-sqr-sqrt 90.0%

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

      2. sqrt-unprod 90.6%

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

      3. *-commutative 90.6%

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

      4. *-commutative 90.6%

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

      5. swap-sqr 90.6%

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

      6. add-sqr-sqrt 90.6%

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

      7. metadata-eval 90.6%

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

   5. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. metadata-eval 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 91.4%

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

Alternative 2: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-74} \lor \neg \left(re \leq 8.5 \cdot 10^{+142}\right) \land re \leq 6.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.5e-10)
   (* 0.5 (sqrt (* re -4.0)))
   (if (or (<= re 1.55e-74) (and (not (<= re 8.5e+142)) (<= re 6.5e+172)))
     (sqrt (* 0.5 (- im re)))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.5e-10) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if ((re <= 1.55e-74) || (!(re <= 8.5e+142) && (re <= 6.5e+172))) {
		tmp = sqrt((0.5 * (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 <= (-9.5d-10)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if ((re <= 1.55d-74) .or. (.not. (re <= 8.5d+142)) .and. (re <= 6.5d+172)) then
        tmp = sqrt((0.5d0 * (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 <= -9.5e-10) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if ((re <= 1.55e-74) || (!(re <= 8.5e+142) && (re <= 6.5e+172))) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.5e-10:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif (re <= 1.55e-74) or (not (re <= 8.5e+142) and (re <= 6.5e+172)):
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.5e-10)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif ((re <= 1.55e-74) || (!(re <= 8.5e+142) && (re <= 6.5e+172)))
		tmp = sqrt(Float64(0.5 * 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 <= -9.5e-10)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif ((re <= 1.55e-74) || (~((re <= 8.5e+142)) && (re <= 6.5e+172)))
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.5e-10], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.55e-74], And[N[Not[LessEqual[re, 8.5e+142]], $MachinePrecision], LessEqual[re, 6.5e+172]]], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.50000000000000028e-10

   1. Initial program 37.1%

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

   2. Taylor expanded in re around -inf 76.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   if -9.50000000000000028e-10 < re < 1.5500000000000001e-74 or 8.49999999999999955e142 < re < 6.4999999999999997e172

   1. Initial program 55.5%

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

      1. sub-neg 55.5%

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

      2. sqr-neg 55.5%

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

      3. sub-neg 55.5%

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

      4. sqr-neg 55.5%

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

      5. hypot-def 92.2%

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

   3. Simplified 92.2%

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

      1. add-sqr-sqrt 91.6%

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

      2. sqrt-unprod 92.2%

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

      3. *-commutative 92.2%

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

      4. *-commutative 92.2%

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

      5. swap-sqr 92.2%

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

      6. add-sqr-sqrt 92.2%

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

      7. metadata-eval 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. associate-*r* 92.2%

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

      3. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in re around 0 79.5%

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

      1. neg-mul-1 79.5%

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

      2. unsub-neg 79.5%

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

   10. Simplified 79.5%

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

   if 1.5500000000000001e-74 < re < 8.49999999999999955e142 or 6.4999999999999997e172 < re

   1. Initial program 20.4%

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

      1. sub-neg 20.4%

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

      2. sqr-neg 20.4%

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

      3. sub-neg 20.4%

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

      4. sqr-neg 20.4%

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

      5. hypot-def 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in im around 0 77.8%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*r* 78.2%

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

      4. associate-*r* 78.2%

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

   6. Simplified 78.2%

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

      1. expm1-log1p-u 77.7%

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

      2. expm1-udef 26.0%

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

      3. associate-*r* 26.0%

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

      4. *-commutative 26.0%

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

      5. sqrt-div 26.0%

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

      6. metadata-eval 26.0%

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

      7. associate-*l/ 26.0%

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

      8. *-un-lft-identity 26.0%

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

      9. *-commutative 26.0%

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

      10. *-commutative 26.0%

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

      11. associate-*l* 26.0%

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

   8. Applied egg-rr 26.0%

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

      1. expm1-def 77.8%

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

      2. expm1-log1p 78.3%

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

      3. associate-*r/ 78.3%

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

      4. associate-*r* 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*l* 77.9%

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

      7. *-commutative 77.9%

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

      8. associate-*r* 78.3%

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

      9. associate-*l* 78.3%

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

   10. Simplified 78.3%

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

       1. expm1-log1p-u 77.7%

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

       2. expm1-udef 26.0%

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

   12. Applied egg-rr 26.0%

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

       1. expm1-def 78.3%

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

       2. expm1-log1p 78.9%

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

       3. associate-/l* 78.9%

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

       4. associate-*r/ 78.8%

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

   14. Simplified 78.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-74} \lor \neg \left(re \leq 8.5 \cdot 10^{+142}\right) \land re \leq 6.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-76} \lor \neg \left(re \leq 8.5 \cdot 10^{+142}\right) \land re \leq 6.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-9)
   (* 0.5 (sqrt (* re -4.0)))
   (if (or (<= re 1.3e-76) (and (not (<= re 8.5e+142)) (<= re 6.5e+172)))
     (sqrt (* 0.5 (- im re)))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-9) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if ((re <= 1.3e-76) || (!(re <= 8.5e+142) && (re <= 6.5e+172))) {
		tmp = sqrt((0.5 * (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 <= (-4.5d-9)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if ((re <= 1.3d-76) .or. (.not. (re <= 8.5d+142)) .and. (re <= 6.5d+172)) then
        tmp = sqrt((0.5d0 * (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 <= -4.5e-9) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if ((re <= 1.3e-76) || (!(re <= 8.5e+142) && (re <= 6.5e+172))) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e-9:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif (re <= 1.3e-76) or (not (re <= 8.5e+142) and (re <= 6.5e+172)):
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-9)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif ((re <= 1.3e-76) || (!(re <= 8.5e+142) && (re <= 6.5e+172)))
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e-9)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif ((re <= 1.3e-76) || (~((re <= 8.5e+142)) && (re <= 6.5e+172)))
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-9], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.3e-76], And[N[Not[LessEqual[re, 8.5e+142]], $MachinePrecision], LessEqual[re, 6.5e+172]]], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.49999999999999976e-9

   1. Initial program 37.1%

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

   2. Taylor expanded in re around -inf 76.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   if -4.49999999999999976e-9 < re < 1.3e-76 or 8.49999999999999955e142 < re < 6.4999999999999997e172

   1. Initial program 55.5%

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

      1. sub-neg 55.5%

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

      2. sqr-neg 55.5%

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

      3. sub-neg 55.5%

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

      4. sqr-neg 55.5%

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

      5. hypot-def 92.2%

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

   3. Simplified 92.2%

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

      1. add-sqr-sqrt 91.6%

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

      2. sqrt-unprod 92.2%

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

      3. *-commutative 92.2%

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

      4. *-commutative 92.2%

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

      5. swap-sqr 92.2%

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

      6. add-sqr-sqrt 92.2%

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

      7. metadata-eval 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. associate-*r* 92.2%

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

      3. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in re around 0 79.5%

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

      1. neg-mul-1 79.5%

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

      2. unsub-neg 79.5%

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

   10. Simplified 79.5%

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

   if 1.3e-76 < re < 8.49999999999999955e142 or 6.4999999999999997e172 < re

   1. Initial program 20.4%

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

      1. sub-neg 20.4%

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

      2. sqr-neg 20.4%

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

      3. sub-neg 20.4%

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

      4. sqr-neg 20.4%

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

      5. hypot-def 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in im around 0 77.8%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-*r* 78.2%

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

      4. associate-*r* 78.2%

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

   6. Simplified 78.2%

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

      1. expm1-log1p-u 77.7%

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

      2. expm1-udef 26.0%

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

      3. associate-*r* 26.0%

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

      4. *-commutative 26.0%

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

      5. sqrt-div 26.0%

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

      6. metadata-eval 26.0%

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

      7. associate-*l/ 26.0%

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

      8. *-un-lft-identity 26.0%

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

      9. *-commutative 26.0%

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

      10. *-commutative 26.0%

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

      11. associate-*l* 26.0%

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

   8. Applied egg-rr 26.0%

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

      1. expm1-def 77.8%

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

      2. expm1-log1p 78.3%

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

      3. associate-*r/ 78.3%

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

      4. associate-*r* 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*l* 77.9%

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

      7. *-commutative 77.9%

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

      8. associate-*r* 78.3%

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

      9. associate-*l* 78.3%

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

   10. Simplified 78.3%

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

       1. associate-*l* 77.9%

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

       2. add-sqr-sqrt 77.9%

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

       3. sqrt-unprod 77.9%

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

       4. swap-sqr 77.9%

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

       5. rem-square-sqrt 77.9%

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

       6. metadata-eval 77.9%

          \[\leadsto \frac{im \cdot \left(\sqrt{0.5} \cdot \sqrt{2 \cdot \color{blue}{0.25}}\right)}{\sqrt{re}} \]

       7. metadata-eval 77.9%

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

       8. rem-square-sqrt 78.9%

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

       9. expm1-log1p-u 77.2%

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

       10. expm1-udef 35.7%

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

   12. Applied egg-rr 35.7%

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

       1. expm1-def 77.2%

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

       2. expm1-log1p 78.9%

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

   14. Simplified 78.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-76} \lor \neg \left(re \leq 8.5 \cdot 10^{+142}\right) \land re \leq 6.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+172}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (* 0.5 (- im re)))))
   (if (<= re -8.5e-9)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re 1.45e-74)
       t_0
       (if (<= re 8.5e+142)
         (* 0.5 (* im (pow re -0.5)))
         (if (<= re 6.5e+172) t_0 (/ (* im 0.5) (sqrt re))))))))

C

double code(double re, double im) {
	double t_0 = sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -8.5e-9) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.45e-74) {
		tmp = t_0;
	} else if (re <= 8.5e+142) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 6.5e+172) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (im - re)))
    if (re <= (-8.5d-9)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.45d-74) then
        tmp = t_0
    else if (re <= 8.5d+142) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 6.5d+172) then
        tmp = t_0
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -8.5e-9) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.45e-74) {
		tmp = t_0;
	} else if (re <= 8.5e+142) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 6.5e+172) {
		tmp = t_0;
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt((0.5 * (im - re)))
	tmp = 0
	if re <= -8.5e-9:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.45e-74:
		tmp = t_0
	elif re <= 8.5e+142:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 6.5e+172:
		tmp = t_0
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(0.5 * Float64(im - re)))
	tmp = 0.0
	if (re <= -8.5e-9)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.45e-74)
		tmp = t_0;
	elseif (re <= 8.5e+142)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 6.5e+172)
		tmp = t_0;
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt((0.5 * (im - re)));
	tmp = 0.0;
	if (re <= -8.5e-9)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.45e-74)
		tmp = t_0;
	elseif (re <= 8.5e+142)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 6.5e+172)
		tmp = t_0;
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -8.5e-9], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.45e-74], t$95$0, If[LessEqual[re, 8.5e+142], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+172], t$95$0, N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.5e-9

   1. Initial program 37.1%

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

   2. Taylor expanded in re around -inf 76.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   if -8.5e-9 < re < 1.45e-74 or 8.49999999999999955e142 < re < 6.4999999999999997e172

   1. Initial program 55.5%

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

      1. sub-neg 55.5%

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

      2. sqr-neg 55.5%

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

      3. sub-neg 55.5%

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

      4. sqr-neg 55.5%

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

      5. hypot-def 92.2%

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

   3. Simplified 92.2%

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

      1. add-sqr-sqrt 91.6%

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

      2. sqrt-unprod 92.2%

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

      3. *-commutative 92.2%

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

      4. *-commutative 92.2%

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

      5. swap-sqr 92.2%

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

      6. add-sqr-sqrt 92.2%

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

      7. metadata-eval 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. associate-*r* 92.2%

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

      3. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in re around 0 79.5%

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

      1. neg-mul-1 79.5%

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

      2. unsub-neg 79.5%

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

   10. Simplified 79.5%

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

   if 1.45e-74 < re < 8.49999999999999955e142

   1. Initial program 32.9%

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

      1. sub-neg 32.9%

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

      2. sqr-neg 32.9%

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

      3. sub-neg 32.9%

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

      4. sqr-neg 32.9%

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

      5. hypot-def 35.5%

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

   3. Simplified 35.5%

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

      1. add-sqr-sqrt 35.4%

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

      2. sqrt-unprod 35.5%

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

      3. *-commutative 35.5%

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

      4. *-commutative 35.5%

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

      5. swap-sqr 35.5%

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

      6. add-sqr-sqrt 35.5%

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

      7. metadata-eval 35.5%

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

   5. Applied egg-rr 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-*r* 35.5%

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

      3. metadata-eval 35.5%

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

   7. Simplified 35.5%

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

   8. Taylor expanded in im around 0 73.7%

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

      1. *-commutative 73.7%

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

      2. unpow2 73.7%

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

      3. rem-square-sqrt 74.9%

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

      4. associate-*l* 74.9%

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

      5. unpow1/2 74.9%

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

      6. exp-to-pow 70.8%

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

      7. *-commutative 70.8%

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

      8. log-rec 70.8%

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

      9. neg-mul-1 70.8%

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

      10. associate-*r* 70.8%

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

      11. metadata-eval 70.8%

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

      12. log-pow 70.8%

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

      13. rem-exp-log 74.9%

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

   10. Simplified 74.9%

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

   if 6.4999999999999997e172 < re

   1. Initial program 3.0%

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

      1. sub-neg 3.0%

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

      2. sqr-neg 3.0%

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

      3. sub-neg 3.0%

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

      4. sqr-neg 3.0%

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

      5. hypot-def 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in im around 0 83.5%

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

      1. associate-*r* 83.5%

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

      2. *-commutative 83.5%

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

      3. associate-*r* 84.0%

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

      4. associate-*r* 84.0%

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

   6. Simplified 84.0%

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

      1. expm1-log1p-u 82.8%

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

      2. expm1-udef 44.2%

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

      3. associate-*r* 44.2%

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

      4. *-commutative 44.2%

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

      5. sqrt-div 44.2%

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

      6. metadata-eval 44.2%

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

      7. associate-*l/ 44.2%

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

      8. *-un-lft-identity 44.2%

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

      9. *-commutative 44.2%

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

      10. *-commutative 44.2%

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

      11. associate-*l* 44.2%

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

   8. Applied egg-rr 44.2%

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

      1. expm1-def 82.8%

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

      2. expm1-log1p 84.1%

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

      3. associate-*r/ 84.1%

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

      4. associate-*r* 83.6%

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

      5. *-commutative 83.6%

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

      6. associate-*l* 83.6%

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

      7. *-commutative 83.6%

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

      8. associate-*r* 84.1%

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

      9. associate-*l* 84.1%

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

   10. Simplified 84.1%

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

       1. associate-*l* 83.6%

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

       2. add-sqr-sqrt 83.6%

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

       3. sqrt-unprod 83.6%

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

       4. swap-sqr 83.6%

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

       5. rem-square-sqrt 83.6%

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

       6. metadata-eval 83.6%

          \[\leadsto \frac{im \cdot \left(\sqrt{0.5} \cdot \sqrt{2 \cdot \color{blue}{0.25}}\right)}{\sqrt{re}} \]

       7. metadata-eval 83.6%

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

       8. rem-square-sqrt 84.7%

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

       9. expm1-log1p-u 80.9%

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

       10. expm1-udef 57.6%

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

   12. Applied egg-rr 57.6%

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

       1. expm1-def 80.9%

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

       2. expm1-log1p 84.7%

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

   14. Simplified 84.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+142}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+172}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (* 0.5 (- im re)))))
   (if (<= re -4.4e-11)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re 8.4e-75)
       t_0
       (if (<= re 8.5e+142)
         (* (pow re -0.5) (* im 0.5))
         (if (<= re 6.5e+172) t_0 (/ (* im 0.5) (sqrt re))))))))

C

double code(double re, double im) {
	double t_0 = sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -4.4e-11) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 8.4e-75) {
		tmp = t_0;
	} else if (re <= 8.5e+142) {
		tmp = pow(re, -0.5) * (im * 0.5);
	} else if (re <= 6.5e+172) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (im - re)))
    if (re <= (-4.4d-11)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 8.4d-75) then
        tmp = t_0
    else if (re <= 8.5d+142) then
        tmp = (re ** (-0.5d0)) * (im * 0.5d0)
    else if (re <= 6.5d+172) then
        tmp = t_0
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -4.4e-11) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 8.4e-75) {
		tmp = t_0;
	} else if (re <= 8.5e+142) {
		tmp = Math.pow(re, -0.5) * (im * 0.5);
	} else if (re <= 6.5e+172) {
		tmp = t_0;
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt((0.5 * (im - re)))
	tmp = 0
	if re <= -4.4e-11:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 8.4e-75:
		tmp = t_0
	elif re <= 8.5e+142:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	elif re <= 6.5e+172:
		tmp = t_0
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(0.5 * Float64(im - re)))
	tmp = 0.0
	if (re <= -4.4e-11)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 8.4e-75)
		tmp = t_0;
	elseif (re <= 8.5e+142)
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	elseif (re <= 6.5e+172)
		tmp = t_0;
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt((0.5 * (im - re)));
	tmp = 0.0;
	if (re <= -4.4e-11)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 8.4e-75)
		tmp = t_0;
	elseif (re <= 8.5e+142)
		tmp = (re ^ -0.5) * (im * 0.5);
	elseif (re <= 6.5e+172)
		tmp = t_0;
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -4.4e-11], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.4e-75], t$95$0, If[LessEqual[re, 8.5e+142], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+172], t$95$0, N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.4000000000000003e-11

   1. Initial program 37.1%

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

   2. Taylor expanded in re around -inf 76.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   if -4.4000000000000003e-11 < re < 8.4000000000000004e-75 or 8.49999999999999955e142 < re < 6.4999999999999997e172

   1. Initial program 55.5%

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

      1. sub-neg 55.5%

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

      2. sqr-neg 55.5%

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

      3. sub-neg 55.5%

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

      4. sqr-neg 55.5%

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

      5. hypot-def 92.2%

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

   3. Simplified 92.2%

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

      1. add-sqr-sqrt 91.6%

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

      2. sqrt-unprod 92.2%

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

      3. *-commutative 92.2%

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

      4. *-commutative 92.2%

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

      5. swap-sqr 92.2%

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

      6. add-sqr-sqrt 92.2%

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

      7. metadata-eval 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. associate-*r* 92.2%

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

      3. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in re around 0 79.5%

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

      1. neg-mul-1 79.5%

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

      2. unsub-neg 79.5%

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

   10. Simplified 79.5%

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

   if 8.4000000000000004e-75 < re < 8.49999999999999955e142

   1. Initial program 32.9%

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

      1. sub-neg 32.9%

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

      2. sqr-neg 32.9%

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

      3. sub-neg 32.9%

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

      4. sqr-neg 32.9%

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

      5. hypot-def 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in im around 0 73.7%

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

      1. associate-*r* 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*r* 74.1%

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

      4. associate-*r* 74.1%

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

   6. Simplified 74.1%

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

      1. expm1-log1p-u 74.1%

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

      2. expm1-udef 12.9%

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

      3. associate-*r* 12.9%

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

      4. *-commutative 12.9%

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

      5. sqrt-div 12.9%

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

      6. metadata-eval 12.9%

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

      7. associate-*l/ 12.9%

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

      8. *-un-lft-identity 12.9%

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

      9. *-commutative 12.9%

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

      10. *-commutative 12.9%

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

      11. associate-*l* 12.9%

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

   8. Applied egg-rr 12.9%

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

      1. expm1-def 74.2%

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

      2. expm1-log1p 74.2%

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

      3. associate-*r/ 74.1%

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

      4. associate-*r* 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*l* 73.7%

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

      7. *-commutative 73.7%

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

      8. associate-*r* 74.1%

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

      9. associate-*l* 74.1%

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

   10. Simplified 74.1%

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

       1. clear-num 73.2%

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

       2. associate-/r/ 74.0%

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

       3. pow1/2 74.0%

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

       4. pow-flip 74.1%

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

       5. metadata-eval 74.1%

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

       6. associate-*l* 73.8%

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

       7. add-sqr-sqrt 73.8%

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

       8. sqrt-unprod 73.8%

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

       9. swap-sqr 73.8%

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

       10. rem-square-sqrt 73.8%

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

       11. metadata-eval 73.8%

           \[\leadsto {re}^{-0.5} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{2 \cdot \color{blue}{0.25}}\right)\right) \]

       12. metadata-eval 73.8%

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

       13. rem-square-sqrt 74.9%

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

   12. Applied egg-rr 74.9%

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

   if 6.4999999999999997e172 < re

   1. Initial program 3.0%

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

      1. sub-neg 3.0%

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

      2. sqr-neg 3.0%

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

      3. sub-neg 3.0%

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

      4. sqr-neg 3.0%

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

      5. hypot-def 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in im around 0 83.5%

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

      1. associate-*r* 83.5%

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

      2. *-commutative 83.5%

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

      3. associate-*r* 84.0%

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

      4. associate-*r* 84.0%

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

   6. Simplified 84.0%

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

      1. expm1-log1p-u 82.8%

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

      2. expm1-udef 44.2%

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

      3. associate-*r* 44.2%

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

      4. *-commutative 44.2%

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

      5. sqrt-div 44.2%

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

      6. metadata-eval 44.2%

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

      7. associate-*l/ 44.2%

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

      8. *-un-lft-identity 44.2%

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

      9. *-commutative 44.2%

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

      10. *-commutative 44.2%

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

      11. associate-*l* 44.2%

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

   8. Applied egg-rr 44.2%

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

      1. expm1-def 82.8%

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

      2. expm1-log1p 84.1%

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

      3. associate-*r/ 84.1%

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

      4. associate-*r* 83.6%

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

      5. *-commutative 83.6%

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

      6. associate-*l* 83.6%

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

      7. *-commutative 83.6%

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

      8. associate-*r* 84.1%

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

      9. associate-*l* 84.1%

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

   10. Simplified 84.1%

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

       1. associate-*l* 83.6%

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

       2. add-sqr-sqrt 83.6%

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

       3. sqrt-unprod 83.6%

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

       4. swap-sqr 83.6%

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

       5. rem-square-sqrt 83.6%

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

       6. metadata-eval 83.6%

          \[\leadsto \frac{im \cdot \left(\sqrt{0.5} \cdot \sqrt{2 \cdot \color{blue}{0.25}}\right)}{\sqrt{re}} \]

       7. metadata-eval 83.6%

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

       8. rem-square-sqrt 84.7%

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

       9. expm1-log1p-u 80.9%

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

       10. expm1-udef 57.6%

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

   12. Applied egg-rr 57.6%

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

       1. expm1-def 80.9%

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

       2. expm1-log1p 84.7%

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

   14. Simplified 84.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+142}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 6: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.95 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.95e-13) (* 0.5 (sqrt (* re -4.0))) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.95e-13) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.95d-13)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.95e-13) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.95e-13:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.95e-13)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.95e-13)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.95e-13], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.94999999999999983e-13

   1. Initial program 37.1%

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

   2. Taylor expanded in re around -inf 76.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   if -3.94999999999999983e-13 < re

   1. Initial program 43.0%

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

      1. sub-neg 43.0%

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

      2. sqr-neg 43.0%

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

      3. sub-neg 43.0%

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

      4. sqr-neg 43.0%

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

      5. hypot-def 71.6%

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

   3. Simplified 71.6%

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

      1. add-sqr-sqrt 71.2%

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

      2. sqrt-unprod 71.6%

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

      3. *-commutative 71.6%

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

      4. *-commutative 71.6%

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

      5. swap-sqr 71.6%

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

      6. add-sqr-sqrt 71.6%

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

      7. metadata-eval 71.6%

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

   5. Applied egg-rr 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-*r* 71.6%

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

      3. metadata-eval 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in re around 0 58.5%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.95 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 7: 53.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

   1. sub-neg 41.5%

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

   2. sqr-neg 41.5%

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

   3. sub-neg 41.5%

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

   4. sqr-neg 41.5%

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

   5. hypot-def 79.1%

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

3. Simplified 79.1%

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

   1. add-sqr-sqrt 78.5%

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

   2. sqrt-unprod 79.1%

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

   3. *-commutative 79.1%

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

   4. *-commutative 79.1%

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

   5. swap-sqr 79.1%

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

   6. add-sqr-sqrt 79.1%

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

   7. metadata-eval 79.1%

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

5. Applied egg-rr 79.1%

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

   1. *-commutative 79.1%

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

   2. associate-*r* 79.1%

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

   3. metadata-eval 79.1%

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

7. Simplified 79.1%

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

8. Taylor expanded in re around 0 53.6%

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

   1. neg-mul-1 53.6%

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

   2. unsub-neg 53.6%

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

10. Simplified 53.6%

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

11. Final simplification 53.6%

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

Alternative 8: 51.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

   1. sub-neg 41.5%

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

   2. sqr-neg 41.5%

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

   3. sub-neg 41.5%

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

   4. sqr-neg 41.5%

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

   5. hypot-def 79.1%

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

3. Simplified 79.1%

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

   1. add-sqr-sqrt 78.5%

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

   2. sqrt-unprod 79.1%

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

   3. *-commutative 79.1%

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

   4. *-commutative 79.1%

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

   5. swap-sqr 79.1%

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

   6. add-sqr-sqrt 79.1%

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

   7. metadata-eval 79.1%

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

5. Applied egg-rr 79.1%

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

   1. *-commutative 79.1%

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

   2. associate-*r* 79.1%

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

   3. metadata-eval 79.1%

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

7. Simplified 79.1%

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

8. Taylor expanded in re around 0 50.9%

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

9. Final simplification 50.9%

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

Reproduce

herbie shell --seed 2023322 
(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)))))