powComplex, real part

Percentage Accurate: 41.1% → 82.0%

Time: 28.9s

Alternatives: 12

Speedup: 4.1×

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ e^{t\_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(t\_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (cos (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    t_0 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    code = exp(((t_0 * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * cos(((t_0 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	return math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.cos(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 41.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ e^{t\_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(t\_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (cos (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    t_0 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    code = exp(((t_0 * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * cos(((t_0 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	return math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.cos(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.re \leq 10^{-300}:\\ \;\;\;\;e^{{\left(\sqrt[3]{y.im \cdot \left(t\_0 \cdot \frac{y.re}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(t\_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-64}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (<= y.re 1e-300)
     (*
      (exp
       (pow (cbrt (* y.im (- (* t_0 (/ y.re y.im)) (atan2 x.im x.re)))) 3.0))
      (cos (fma t_0 y.im (* y.re (atan2 x.im x.re)))))
     (if (<= y.re 3.9e-64)
       (exp (* y.im (- (atan2 x.im x.re))))
       (*
        (exp
         (*
          y.im
          (- (* y.re (/ (log (hypot x.im x.re)) y.im)) (atan2 x.im x.re))))
        (cos (* y.im t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= 1e-300) {
		tmp = exp(pow(cbrt((y_46_im * ((t_0 * (y_46_re / y_46_im)) - atan2(x_46_im, x_46_re)))), 3.0)) * cos(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	} else if (y_46_re <= 3.9e-64) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else {
		tmp = exp((y_46_im * ((y_46_re * (log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan2(x_46_im, x_46_re)))) * cos((y_46_im * t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= 1e-300)
		tmp = Float64(exp((cbrt(Float64(y_46_im * Float64(Float64(t_0 * Float64(y_46_re / y_46_im)) - atan(x_46_im, x_46_re)))) ^ 3.0)) * cos(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	elseif (y_46_re <= 3.9e-64)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	else
		tmp = Float64(exp(Float64(y_46_im * Float64(Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan(x_46_im, x_46_re)))) * cos(Float64(y_46_im * t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, 1e-300], N[(N[Exp[N[Power[N[Power[N[(y$46$im * N[(N[(t$95$0 * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.9e-64], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(y$46$im * N[(N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$im * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < 1.00000000000000003e-300

   1. Initial program 46.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 46.7%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 46.7%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 46.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 46.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 46.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 46.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 88.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 88.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 88.7%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 72.7%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 72.7%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 72.7%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 72.7%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 72.7%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 72.7%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 72.7%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 88.7%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   8. Step-by-step derivation

      1. add-cube-cbrt 88.7%

         \[\leadsto e^{\color{blue}{\left(\sqrt[3]{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sqrt[3]{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \cdot \sqrt[3]{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. pow3 88.7%

         \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. *-commutative 88.7%

         \[\leadsto e^{{\left(\sqrt[3]{y.im \cdot \left(\color{blue}{\frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} \cdot y.re} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-*l/ 88.7%

         \[\leadsto e^{{\left(\sqrt[3]{y.im \cdot \left(\color{blue}{\frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.re}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. associate-/l* 88.7%

         \[\leadsto e^{{\left(\sqrt[3]{y.im \cdot \left(\color{blue}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \frac{y.re}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-undefine 68.5%

         \[\leadsto e^{{\left(\sqrt[3]{y.im \cdot \left(\log \color{blue}{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \cdot \frac{y.re}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. +-commutative 68.5%

         \[\leadsto e^{{\left(\sqrt[3]{y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.re \cdot x.re + x.im \cdot x.im}}\right) \cdot \frac{y.re}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. hypot-define 88.7%

         \[\leadsto e^{{\left(\sqrt[3]{y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \frac{y.re}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Applied egg-rr 88.7%

      \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot \frac{y.re}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   if 1.00000000000000003e-300 < y.re < 3.8999999999999997e-64

   1. Initial program 39.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 52.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 52.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 29.4%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   6. Taylor expanded in y.re around 0 81.8%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
   7. Step-by-step derivation

      1. distribute-lft-neg-in 81.8%

         \[\leadsto e^{\color{blue}{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   8. Simplified 81.8%

      \[\leadsto \color{blue}{e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   if 3.8999999999999997e-64 < y.re

   1. Initial program 31.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 31.2%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 31.2%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 31.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 31.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 31.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 31.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 76.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 76.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 61.0%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 61.0%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 61.0%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 61.0%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 61.0%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 61.0%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 61.0%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 76.6%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 76.6%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 33.9%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 33.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 33.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 33.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 79.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 79.2%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 10^{-300}:\\ \;\;\;\;e^{{\left(\sqrt[3]{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot \frac{y.re}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-64}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.re \leq 6.1 \cdot 10^{-295}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(t\_1, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot t\_0\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-68}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \cos \left(y.im \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (*
           y.im
           (- (* y.re (/ (log (hypot x.im x.re)) y.im)) (atan2 x.im x.re)))))
        (t_1 (log (hypot x.re x.im))))
   (if (<= y.re 6.1e-295)
     (* (cos (fma t_1 y.im (* y.re (atan2 x.im x.re)))) t_0)
     (if (<= y.re 3.5e-68)
       (exp (* y.im (- (atan2 x.im x.re))))
       (* t_0 (cos (* y.im t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_im * ((y_46_re * (log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan2(x_46_im, x_46_re))));
	double t_1 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= 6.1e-295) {
		tmp = cos(fma(t_1, y_46_im, (y_46_re * atan2(x_46_im, x_46_re)))) * t_0;
	} else if (y_46_re <= 3.5e-68) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0 * cos((y_46_im * t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * Float64(Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan(x_46_im, x_46_re))))
	t_1 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= 6.1e-295)
		tmp = Float64(cos(fma(t_1, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))) * t_0);
	elseif (y_46_re <= 3.5e-68)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	else
		tmp = Float64(t_0 * cos(Float64(y_46_im * t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$im * N[(N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, 6.1e-295], N[(N[Cos[N[(t$95$1 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 3.5e-68], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[(t$95$0 * N[Cos[N[(y$46$im * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < 6.09999999999999974e-295

   1. Initial program 46.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 46.7%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 46.7%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 46.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 46.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 46.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 46.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 88.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 88.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 88.7%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 72.7%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 72.7%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 72.7%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 72.7%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 72.7%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 72.7%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 72.7%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 88.7%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   if 6.09999999999999974e-295 < y.re < 3.50000000000000013e-68

   1. Initial program 39.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 52.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 52.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 29.4%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   6. Taylor expanded in y.re around 0 81.8%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
   7. Step-by-step derivation

      1. distribute-lft-neg-in 81.8%

         \[\leadsto e^{\color{blue}{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   8. Simplified 81.8%

      \[\leadsto \color{blue}{e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   if 3.50000000000000013e-68 < y.re

   1. Initial program 31.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 31.2%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 31.2%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 31.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 31.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 31.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 31.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 76.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 76.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 61.0%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 61.0%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 61.0%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 61.0%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 61.0%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 61.0%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 61.0%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 76.6%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 76.6%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 33.9%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 33.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 33.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 33.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 79.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 79.2%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 6.1 \cdot 10^{-295}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-68}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im 1.5e-12)
   (*
    (exp
     (* y.im (- (* y.re (/ (log (hypot x.im x.re)) y.im)) (atan2 x.im x.re))))
    (cos (* y.im (log (hypot x.re x.im)))))
   (exp (- (* y.re (log x.im)) (* y.im (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= 1.5e-12) {
		tmp = exp((y_46_im * ((y_46_re * (log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan2(x_46_im, x_46_re)))) * cos((y_46_im * log(hypot(x_46_re, x_46_im))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - (y_46_im * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= 1.5e-12) {
		tmp = Math.exp((y_46_im * ((y_46_re * (Math.log(Math.hypot(x_46_im, x_46_re)) / y_46_im)) - Math.atan2(x_46_im, x_46_re)))) * Math.cos((y_46_im * Math.log(Math.hypot(x_46_re, x_46_im))));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= 1.5e-12:
		tmp = math.exp((y_46_im * ((y_46_re * (math.log(math.hypot(x_46_im, x_46_re)) / y_46_im)) - math.atan2(x_46_im, x_46_re)))) * math.cos((y_46_im * math.log(math.hypot(x_46_re, x_46_im))))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= 1.5e-12)
		tmp = Float64(exp(Float64(y_46_im * Float64(Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan(x_46_im, x_46_re)))) * cos(Float64(y_46_im * log(hypot(x_46_re, x_46_im)))));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - Float64(y_46_im * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.5e-12)
		tmp = exp((y_46_im * ((y_46_re * (log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan2(x_46_im, x_46_re)))) * cos((y_46_im * log(hypot(x_46_re, x_46_im))));
	else
		tmp = exp(((y_46_re * log(x_46_im)) - (y_46_im * atan2(x_46_im, x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, 1.5e-12], N[(N[Exp[N[(y$46$im * N[(N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.5000000000000001e-12

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 42.9%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 42.9%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 42.9%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 42.9%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 42.9%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 42.9%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 81.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 81.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 65.1%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 65.1%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 65.1%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 65.1%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 65.1%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 65.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 65.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 81.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 81.3%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 42.4%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 42.4%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 42.4%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 42.4%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 81.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 81.3%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   if 1.5000000000000001e-12 < x.im

   1. Initial program 33.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 53.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 61.4%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 85.7%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.46 \cdot 10^{+63}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq 30000:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.46e+63)
   (exp (* y.im (- (atan2 x.im x.re))))
   (if (<= y.im 30000.0)
     (*
      (exp (* y.re (log (hypot x.re x.im))))
      (+ 1.0 (* -0.5 (pow (* y.im (log (hypot x.im x.re))) 2.0))))
     (exp
      (-
       (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
       (* y.im (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.46e+63) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 30000.0) {
		tmp = exp((y_46_re * log(hypot(x_46_re, x_46_im)))) * (1.0 + (-0.5 * pow((y_46_im * log(hypot(x_46_im, x_46_re))), 2.0)));
	} else {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.46e+63) {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 30000.0) {
		tmp = Math.exp((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im)))) * (1.0 + (-0.5 * Math.pow((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))), 2.0)));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.46e+63:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	elif y_46_im <= 30000.0:
		tmp = math.exp((y_46_re * math.log(math.hypot(x_46_re, x_46_im)))) * (1.0 + (-0.5 * math.pow((y_46_im * math.log(math.hypot(x_46_im, x_46_re))), 2.0)))
	else:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.46e+63)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	elseif (y_46_im <= 30000.0)
		tmp = Float64(exp(Float64(y_46_re * log(hypot(x_46_re, x_46_im)))) * Float64(1.0 + Float64(-0.5 * (Float64(y_46_im * log(hypot(x_46_im, x_46_re))) ^ 2.0))));
	else
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(y_46_im * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.46e+63)
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	elseif (y_46_im <= 30000.0)
		tmp = exp((y_46_re * log(hypot(x_46_re, x_46_im)))) * (1.0 + (-0.5 * ((y_46_im * log(hypot(x_46_im, x_46_re))) ^ 2.0)));
	else
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.46e+63], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$im, 30000.0], N[(N[Exp[N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.4599999999999999e63

   1. Initial program 37.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 44.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 46.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 25.5%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   6. Taylor expanded in y.re around 0 65.5%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
   7. Step-by-step derivation

      1. distribute-lft-neg-in 65.5%

         \[\leadsto e^{\color{blue}{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   8. Simplified 65.5%

      \[\leadsto \color{blue}{e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   if -1.4599999999999999e63 < y.im < 3e4

   1. Initial program 42.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 42.3%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 42.3%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 42.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 42.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 42.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 42.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 87.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 87.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 87.6%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 64.2%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 64.2%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 64.2%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 64.2%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 64.2%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 64.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 64.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 87.6%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 87.6%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 43.8%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 43.8%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 43.8%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 43.8%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 89.8%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 89.8%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   11. Taylor expanded in y.im around 0 45.8%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.im}^{2} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.im \cdot y.im\right)} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right) \]

       2. +-commutative 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{2}\right)\right) \]

       3. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{2}\right)\right) \]

       4. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{2}\right)\right) \]

       5. hypot-undefine 91.0%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{2}\right)\right) \]

       6. unpow2 91.0%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}\right)\right) \]

       7. swap-sqr 91.0%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)}\right) \]

       8. unpow2 91.0%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{2}}\right) \]

   13. Simplified 91.0%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right)} \]

   14. Taylor expanded in y.im around 0 65.4%

       \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}} \cdot \left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right) \]
   15. Step-by-step derivation

       1. +-commutative 65.4%

          \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)} \cdot \left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right) \]

       2. unpow2 65.4%

          \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)} \cdot \left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right) \]

       3. unpow2 65.4%

          \[\leadsto e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)} \cdot \left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right) \]

       4. hypot-undefine 88.0%

          \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right) \]

   16. Simplified 88.0%

       \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right) \]

   if 3e4 < y.im

   1. Initial program 39.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 66.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 70.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.46 \cdot 10^{+63}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq 30000:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-309}:\\ \;\;\;\;t\_0 \cdot \cos \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \cos \left(y.im \cdot \log x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (*
           y.im
           (- (* y.re (/ (log (hypot x.im x.re)) y.im)) (atan2 x.im x.re))))))
   (if (<= x.re -5e-309)
     (* t_0 (cos (* y.im (log (/ -1.0 x.re)))))
     (* t_0 (cos (* y.im (log x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_im * ((y_46_re * (log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan2(x_46_im, x_46_re))));
	double tmp;
	if (x_46_re <= -5e-309) {
		tmp = t_0 * cos((y_46_im * log((-1.0 / x_46_re))));
	} else {
		tmp = t_0 * cos((y_46_im * log(x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp((y_46_im * ((y_46_re * (Math.log(Math.hypot(x_46_im, x_46_re)) / y_46_im)) - Math.atan2(x_46_im, x_46_re))));
	double tmp;
	if (x_46_re <= -5e-309) {
		tmp = t_0 * Math.cos((y_46_im * Math.log((-1.0 / x_46_re))));
	} else {
		tmp = t_0 * Math.cos((y_46_im * Math.log(x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((y_46_im * ((y_46_re * (math.log(math.hypot(x_46_im, x_46_re)) / y_46_im)) - math.atan2(x_46_im, x_46_re))))
	tmp = 0
	if x_46_re <= -5e-309:
		tmp = t_0 * math.cos((y_46_im * math.log((-1.0 / x_46_re))))
	else:
		tmp = t_0 * math.cos((y_46_im * math.log(x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * Float64(Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (x_46_re <= -5e-309)
		tmp = Float64(t_0 * cos(Float64(y_46_im * log(Float64(-1.0 / x_46_re)))));
	else
		tmp = Float64(t_0 * cos(Float64(y_46_im * log(x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((y_46_im * ((y_46_re * (log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan2(x_46_im, x_46_re))));
	tmp = 0.0;
	if (x_46_re <= -5e-309)
		tmp = t_0 * cos((y_46_im * log((-1.0 / x_46_re))));
	else
		tmp = t_0 * cos((y_46_im * log(x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$im * N[(N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -5e-309], N[(t$95$0 * N[Cos[N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Cos[N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.9999999999999995e-309

   1. Initial program 46.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 46.0%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 46.0%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 46.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 46.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 46.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 46.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 83.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 83.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 66.3%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 66.3%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 66.3%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 66.3%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 66.3%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 66.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 66.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 83.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 83.3%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 46.8%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 46.8%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 46.8%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 46.8%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 84.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 84.1%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   11. Taylor expanded in x.re around -inf 84.4%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 84.4%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \color{blue}{\left(-y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)} \]

       2. cos-neg 84.4%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)} \]

   13. Simplified 84.4%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)} \]

   if -4.9999999999999995e-309 < x.re

   1. Initial program 35.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 35.1%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 35.1%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 35.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 35.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 35.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 35.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 78.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 78.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 78.3%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 58.9%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 58.9%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 58.9%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 58.9%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 58.9%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 58.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 58.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 78.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 78.3%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 34.3%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 34.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 34.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 34.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 77.5%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 77.5%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   11. Taylor expanded in x.im around 0 80.6%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-309}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -5e-310)
   (exp (- (* y.re (log (- x.re))) (* y.im (atan2 x.im x.re))))
   (*
    (exp
     (* y.im (- (* y.re (/ (log (hypot x.im x.re)) y.im)) (atan2 x.im x.re))))
    (cos (* y.im (log x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -5e-310) {
		tmp = exp(((y_46_re * log(-x_46_re)) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = exp((y_46_im * ((y_46_re * (log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan2(x_46_im, x_46_re)))) * cos((y_46_im * log(x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -5e-310) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_re)) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = Math.exp((y_46_im * ((y_46_re * (Math.log(Math.hypot(x_46_im, x_46_re)) / y_46_im)) - Math.atan2(x_46_im, x_46_re)))) * Math.cos((y_46_im * Math.log(x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -5e-310:
		tmp = math.exp(((y_46_re * math.log(-x_46_re)) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	else:
		tmp = math.exp((y_46_im * ((y_46_re * (math.log(math.hypot(x_46_im, x_46_re)) / y_46_im)) - math.atan2(x_46_im, x_46_re)))) * math.cos((y_46_im * math.log(x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -5e-310)
		tmp = exp(Float64(Float64(y_46_re * log(Float64(-x_46_re))) - Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(exp(Float64(y_46_im * Float64(Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan(x_46_im, x_46_re)))) * cos(Float64(y_46_im * log(x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -5e-310)
		tmp = exp(((y_46_re * log(-x_46_re)) - (y_46_im * atan2(x_46_im, x_46_re))));
	else
		tmp = exp((y_46_im * ((y_46_re * (log(hypot(x_46_im, x_46_re)) / y_46_im)) - atan2(x_46_im, x_46_re)))) * cos((y_46_im * log(x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -5e-310], N[Exp[N[(N[(y$46$re * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(y$46$im * N[(N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.999999999999985e-310

   1. Initial program 46.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 64.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 67.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around -inf 77.4%

      \[\leadsto e^{\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]
   6. Step-by-step derivation

      1. mul-1-neg 77.4%

         \[\leadsto e^{\log \color{blue}{\left(-x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   7. Simplified 77.4%

      \[\leadsto e^{\log \color{blue}{\left(-x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   if -4.999999999999985e-310 < x.re

   1. Initial program 35.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 35.1%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 35.1%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 35.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 35.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 35.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 35.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 78.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 78.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 78.3%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 58.9%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 58.9%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 58.9%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 58.9%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 58.9%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 58.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 58.9%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 78.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 78.3%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 34.3%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 34.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 34.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 34.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 77.5%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 77.5%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   11. Taylor expanded in x.im around 0 80.6%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+27}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq 62000000000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.2e+27)
   (exp (* y.im (- (atan2 x.im x.re))))
   (if (<= y.im 62000000000000.0)
     (* (cos (* y.im (log (hypot x.re x.im)))) (pow (hypot x.re x.im) y.re))
     (exp
      (-
       (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
       (* y.im (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+27) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 62000000000000.0) {
		tmp = cos((y_46_im * log(hypot(x_46_re, x_46_im)))) * pow(hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+27) {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 62000000000000.0) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_re, x_46_im)))) * Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.2e+27:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	elif y_46_im <= 62000000000000.0:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_re, x_46_im)))) * math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	else:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e+27)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	elseif (y_46_im <= 62000000000000.0)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_re, x_46_im)))) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(y_46_im * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.2e+27)
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	elseif (y_46_im <= 62000000000000.0)
		tmp = cos((y_46_im * log(hypot(x_46_re, x_46_im)))) * (hypot(x_46_re, x_46_im) ^ y_46_re);
	else
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.2e+27], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$im, 62000000000000.0], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.19999999999999999e27

   1. Initial program 32.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 48.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 48.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 22.7%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   6. Taylor expanded in y.re around 0 63.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
   7. Step-by-step derivation

      1. distribute-lft-neg-in 63.6%

         \[\leadsto e^{\color{blue}{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   if -1.19999999999999999e27 < y.im < 6.2e13

   1. Initial program 44.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 44.5%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 44.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 44.5%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 44.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 44.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 44.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 88.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 88.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 86.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{1}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Taylor expanded in y.re around 0 43.5%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 46.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 46.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 46.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 92.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   8. Simplified 89.8%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   if 6.2e13 < y.im

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 66.4%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 71.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+27}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq 62000000000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+26}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq 62000000000000:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.4e+26)
   (exp (* y.im (- (atan2 x.im x.re))))
   (if (<= y.im 62000000000000.0)
     (pow (hypot x.im x.re) y.re)
     (exp
      (-
       (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
       (* y.im (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.4e+26) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 62000000000000.0) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.4e+26) {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 62000000000000.0) {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5.4e+26:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	elif y_46_im <= 62000000000000.0:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.4e+26)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	elseif (y_46_im <= 62000000000000.0)
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	else
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(y_46_im * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -5.4e+26)
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	elseif (y_46_im <= 62000000000000.0)
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	else
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.4e+26], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$im, 62000000000000.0], N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.4e26

   1. Initial program 32.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 48.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 48.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 22.7%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   6. Taylor expanded in y.re around 0 63.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
   7. Step-by-step derivation

      1. distribute-lft-neg-in 63.6%

         \[\leadsto e^{\color{blue}{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   if -5.4e26 < y.im < 6.2e13

   1. Initial program 44.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 44.5%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 44.5%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 44.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 44.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 44.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 44.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 89.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 89.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 64.1%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 64.1%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 64.1%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 64.1%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 64.1%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 64.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 64.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 89.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 89.2%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 46.1%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 46.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 46.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 46.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 92.3%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 92.3%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   11. Taylor expanded in y.im around 0 44.4%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.im}^{2} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. unpow2 44.4%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.im \cdot y.im\right)} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right) \]

       2. +-commutative 44.4%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{2}\right)\right) \]

       3. unpow2 44.4%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{2}\right)\right) \]

       4. unpow2 44.4%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{2}\right)\right) \]

       5. hypot-undefine 89.6%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{2}\right)\right) \]

       6. unpow2 89.6%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}\right)\right) \]

       7. swap-sqr 89.6%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)}\right) \]

       8. unpow2 89.6%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{2}}\right) \]

   13. Simplified 89.6%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right)} \]

   14. Taylor expanded in y.im around 0 64.1%

       \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   15. Step-by-step derivation

       1. unpow2 64.1%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

       2. unpow2 64.1%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

       3. hypot-undefine 89.1%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   16. Simplified 89.1%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   if 6.2e13 < y.im

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 66.4%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 71.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+26}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq 62000000000000:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+26}:\\ \;\;\;\;e^{y.im \cdot t\_0}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+57}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{y.im}\right)}^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (atan2 x.im x.re))))
   (if (<= y.im -5.4e+26)
     (exp (* y.im t_0))
     (if (<= y.im 4.2e+57)
       (pow (hypot x.im x.re) y.re)
       (pow (exp y.im) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= -5.4e+26) {
		tmp = exp((y_46_im * t_0));
	} else if (y_46_im <= 4.2e+57) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = pow(exp(y_46_im), t_0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= -5.4e+26) {
		tmp = Math.exp((y_46_im * t_0));
	} else if (y_46_im <= 4.2e+57) {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.pow(Math.exp(y_46_im), t_0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_im <= -5.4e+26:
		tmp = math.exp((y_46_im * t_0))
	elif y_46_im <= 4.2e+57:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.pow(math.exp(y_46_im), t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(-atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= -5.4e+26)
		tmp = exp(Float64(y_46_im * t_0));
	elseif (y_46_im <= 4.2e+57)
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	else
		tmp = exp(y_46_im) ^ t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_im <= -5.4e+26)
		tmp = exp((y_46_im * t_0));
	elseif (y_46_im <= 4.2e+57)
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	else
		tmp = exp(y_46_im) ^ t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])}, If[LessEqual[y$46$im, -5.4e+26], N[Exp[N[(y$46$im * t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$im, 4.2e+57], N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision], N[Power[N[Exp[y$46$im], $MachinePrecision], t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.4e26

   1. Initial program 32.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 48.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 48.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 22.7%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   6. Taylor expanded in y.re around 0 63.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
   7. Step-by-step derivation

      1. distribute-lft-neg-in 63.6%

         \[\leadsto e^{\color{blue}{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   if -5.4e26 < y.im < 4.19999999999999982e57

   1. Initial program 46.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 46.6%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 46.6%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 46.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 46.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 46.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 46.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 89.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 89.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 65.2%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 65.2%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 65.2%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 65.2%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 65.2%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 65.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 65.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 89.0%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 89.0%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 48.1%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 48.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 48.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 48.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 92.0%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 92.0%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   11. Taylor expanded in y.im around 0 45.8%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.im}^{2} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.im \cdot y.im\right)} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right) \]

       2. +-commutative 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{2}\right)\right) \]

       3. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{2}\right)\right) \]

       4. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{2}\right)\right) \]

       5. hypot-undefine 88.7%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{2}\right)\right) \]

       6. unpow2 88.7%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}\right)\right) \]

       7. swap-sqr 88.7%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)}\right) \]

       8. unpow2 88.7%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{2}}\right) \]

   13. Simplified 88.7%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right)} \]

   14. Taylor expanded in y.im around 0 64.5%

       \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   15. Step-by-step derivation

       1. unpow2 64.5%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

       2. unpow2 64.5%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

       3. hypot-undefine 88.2%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   16. Simplified 88.2%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   if 4.19999999999999982e57 < y.im

   1. Initial program 34.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 64.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 69.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 39.7%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   6. Taylor expanded in y.re around 0 66.1%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
   7. Step-by-step derivation

      1. distribute-rgt-neg-in 66.1%

         \[\leadsto e^{\color{blue}{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]

      2. exp-prod 67.7%

         \[\leadsto \color{blue}{{\left(e^{y.im}\right)}^{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]

   8. Simplified 67.7%

      \[\leadsto \color{blue}{{\left(e^{y.im}\right)}^{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+26}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+57}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{y.im}\right)}^{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.6 \cdot 10^{+27} \lor \neg \left(y.im \leq 1.05 \cdot 10^{+58}\right):\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9.6e+27) (not (<= y.im 1.05e+58)))
   (exp (* y.im (- (atan2 x.im x.re))))
   (pow (hypot x.im x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.6e+27) || !(y_46_im <= 1.05e+58)) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.6e+27) || !(y_46_im <= 1.05e+58)) {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -9.6e+27) or not (y_46_im <= 1.05e+58):
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	else:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -9.6e+27) || !(y_46_im <= 1.05e+58))
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	else
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -9.6e+27) || ~((y_46_im <= 1.05e+58)))
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	else
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9.6e+27], N[Not[LessEqual[y$46$im, 1.05e+58]], $MachinePrecision]], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.59999999999999991e27 or 1.05000000000000006e58 < y.im

   1. Initial program 33.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0 55.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Taylor expanded in y.re around 0 58.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

   5. Taylor expanded in x.re around 0 30.8%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   6. Taylor expanded in y.re around 0 64.8%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
   7. Step-by-step derivation

      1. distribute-lft-neg-in 64.8%

         \[\leadsto e^{\color{blue}{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   8. Simplified 64.8%

      \[\leadsto \color{blue}{e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   if -9.59999999999999991e27 < y.im < 1.05000000000000006e58

   1. Initial program 46.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 46.6%

         \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. fma-define 46.6%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-define 46.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. distribute-lft-neg-in 46.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. distribute-rgt-neg-out 46.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. fma-define 46.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      7. hypot-define 89.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. *-commutative 89.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around inf 65.2%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 65.2%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. mul-1-neg 65.2%

         \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. unsub-neg 65.2%

         \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. associate-/l* 65.2%

         \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 65.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. unpow2 65.2%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-undefine 89.0%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Simplified 89.0%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Taylor expanded in y.re around 0 48.1%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 48.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

      2. unpow2 48.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

      3. unpow2 48.1%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

      4. hypot-undefine 92.0%

         \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

   10. Simplified 92.0%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

   11. Taylor expanded in y.im around 0 45.8%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.im}^{2} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.im \cdot y.im\right)} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right) \]

       2. +-commutative 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{2}\right)\right) \]

       3. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{2}\right)\right) \]

       4. unpow2 45.8%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{2}\right)\right) \]

       5. hypot-undefine 88.7%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{2}\right)\right) \]

       6. unpow2 88.7%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}\right)\right) \]

       7. swap-sqr 88.7%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)}\right) \]

       8. unpow2 88.7%

          \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{2}}\right) \]

   13. Simplified 88.7%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right)} \]

   14. Taylor expanded in y.im around 0 64.5%

       \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   15. Step-by-step derivation

       1. unpow2 64.5%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

       2. unpow2 64.5%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

       3. hypot-undefine 88.2%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   16. Simplified 88.2%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.6 \cdot 10^{+27} \lor \neg \left(y.im \leq 1.05 \cdot 10^{+58}\right):\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (pow (hypot x.im x.re) y.re))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return pow(hypot(x_46_im, x_46_re), y_46_re);
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.pow(math.hypot(x_46_im, x_46_re), y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return hypot(x_46_im, x_46_re) ^ y_46_re
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]

Derivation

1. Initial program 40.5%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 40.5%

      \[\leadsto e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. fma-define 40.5%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. hypot-define 40.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. distribute-lft-neg-in 40.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. distribute-rgt-neg-out 40.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. fma-define 40.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   7. hypot-define 80.8%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   8. *-commutative 80.8%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

3. Simplified 80.8%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in y.im around inf 62.5%

   \[\leadsto e^{\color{blue}{y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
6. Step-by-step derivation

   1. +-commutative 62.5%

      \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + -1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   2. mul-1-neg 62.5%

      \[\leadsto e^{y.im \cdot \left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} + \color{blue}{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   3. unsub-neg 62.5%

      \[\leadsto e^{y.im \cdot \color{blue}{\left(\frac{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   4. associate-/l* 62.5%

      \[\leadsto e^{y.im \cdot \left(\color{blue}{y.re \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.im}} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   5. unpow2 62.5%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. unpow2 62.5%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. hypot-undefine 80.8%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

7. Simplified 80.8%

   \[\leadsto e^{\color{blue}{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

8. Taylor expanded in y.re around 0 40.5%

   \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 40.5%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right) \]

   2. unpow2 40.5%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right) \]

   3. unpow2 40.5%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right) \]

   4. hypot-undefine 80.8%

      \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \]

10. Simplified 80.8%

    \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)} \]

11. Taylor expanded in y.im around 0 37.1%

    \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.im}^{2} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right)} \]
12. Step-by-step derivation

    1. unpow2 37.1%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.im \cdot y.im\right)} \cdot {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2}\right)\right) \]

    2. +-commutative 37.1%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{2}\right)\right) \]

    3. unpow2 37.1%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{2}\right)\right) \]

    4. unpow2 37.1%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{2}\right)\right) \]

    5. hypot-undefine 64.8%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot {\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{2}\right)\right) \]

    6. unpow2 64.8%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \left(\left(y.im \cdot y.im\right) \cdot \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}\right)\right) \]

    7. swap-sqr 64.8%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)}\right) \]

    8. unpow2 64.8%

       \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(1 + -0.5 \cdot \color{blue}{{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{2}}\right) \]

13. Simplified 64.8%

    \[\leadsto e^{y.im \cdot \left(y.re \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.im} - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot {\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}^{2}\right)} \]

14. Taylor expanded in y.im around 0 51.0%

    \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
15. Step-by-step derivation

    1. unpow2 51.0%

       \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

    2. unpow2 51.0%

       \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

    3. hypot-undefine 58.2%

       \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

16. Simplified 58.2%

    \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

17. Final simplification 58.2%

    \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]
18. Add Preprocessing

Alternative 12: 39.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ {x.im}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (pow x.im y.re))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return pow(x_46_im, y_46_re);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im ** y_46re
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.pow(x_46_im, y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.pow(x_46_im, y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return x_46_im ^ y_46_re
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = x_46_im ^ y_46_re;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Power[x$46$im, y$46$re], $MachinePrecision]

Derivation

1. Initial program 40.5%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Add Preprocessing

3. Taylor expanded in y.im around 0 59.2%

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

4. Taylor expanded in y.re around 0 62.0%

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]

5. Taylor expanded in x.re around 0 33.7%

   \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

6. Taylor expanded in y.im around 0 36.5%

   \[\leadsto \color{blue}{{x.im}^{y.re}} \cdot 1 \]

7. Final simplification 36.5%

   \[\leadsto {x.im}^{y.re} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))