powComplex, real part

Percentage Accurate: 39.9% → 80.3%

Time: 25.6s

Alternatives: 17

Speedup: 2.6×

• Could not converge on a ground truth

  1. reg0 = (bf "-8.602944290194490159204702323684667451404e129")
  2. reg1 = (bf 1294120883603318043844968765985771065704000)
  3. reg2 = (bf "-4.905109255320236596564917094753833690511e-278")
  4. reg3 = (bf "-5.389095823079168641200535483455230300987e244")

• 36.6% 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: 39.9% 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: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -1.06 \cdot 10^{+98} \lor \neg \left(y.re \leq 1.15 \cdot 10^{+54}\right):\\ \;\;\;\;e^{y.re \cdot \mathsf{log1p}\left(\mathsf{hypot}\left(x.re, x.im\right) + -1\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(y.re \cdot \left(t\_0 - y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right)\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re))))
   (if (or (<= y.re -1.06e+98) (not (<= y.re 1.15e+54)))
     (*
      (exp
       (-
        (* y.re (log1p (+ (hypot x.re x.im) -1.0)))
        (* (atan2 x.im x.re) y.im)))
      (cos (* y.im t_0)))
     (*
      (pow E (* y.re (- t_0 (* y.im (/ (atan2 x.im x.re) y.re)))))
      (cos (* y.re (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 t_0 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if ((y_46_re <= -1.06e+98) || !(y_46_re <= 1.15e+54)) {
		tmp = exp(((y_46_re * log1p((hypot(x_46_re, x_46_im) + -1.0))) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos((y_46_im * t_0));
	} else {
		tmp = pow(((double) M_E), (y_46_re * (t_0 - (y_46_im * (atan2(x_46_im, x_46_re) / y_46_re))))) * cos((y_46_re * 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 t_0 = Math.log(Math.hypot(x_46_im, x_46_re));
	double tmp;
	if ((y_46_re <= -1.06e+98) || !(y_46_re <= 1.15e+54)) {
		tmp = Math.exp(((y_46_re * Math.log1p((Math.hypot(x_46_re, x_46_im) + -1.0))) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos((y_46_im * t_0));
	} else {
		tmp = Math.pow(Math.E, (y_46_re * (t_0 - (y_46_im * (Math.atan2(x_46_im, x_46_re) / y_46_re))))) * Math.cos((y_46_re * 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):
	t_0 = math.log(math.hypot(x_46_im, x_46_re))
	tmp = 0
	if (y_46_re <= -1.06e+98) or not (y_46_re <= 1.15e+54):
		tmp = math.exp(((y_46_re * math.log1p((math.hypot(x_46_re, x_46_im) + -1.0))) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.cos((y_46_im * t_0))
	else:
		tmp = math.pow(math.e, (y_46_re * (t_0 - (y_46_im * (math.atan2(x_46_im, x_46_re) / y_46_re))))) * math.cos((y_46_re * 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)
	t_0 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_re <= -1.06e+98) || !(y_46_re <= 1.15e+54))
		tmp = Float64(exp(Float64(Float64(y_46_re * log1p(Float64(hypot(x_46_re, x_46_im) + -1.0))) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(y_46_im * t_0)));
	else
		tmp = Float64((exp(1) ^ Float64(y_46_re * Float64(t_0 - Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / y_46_re))))) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	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$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y$46$re, -1.06e+98], N[Not[LessEqual[y$46$re, 1.15e+54]], $MachinePrecision]], N[(N[Exp[N[(N[(y$46$re * N[Log[1 + N[(N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$im * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[E, N[(y$46$re * N[(t$95$0 - N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.0600000000000001e98 or 1.14999999999999997e54 < y.re

   1. Initial program 44.2%

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

   3. Taylor expanded in y.re around 0 49.5%

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

      1. *-commutative 49.5%

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

      2. unpow2 49.5%

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

      3. unpow2 49.5%

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

      4. hypot-undefine 83.2%

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

   5. Simplified 83.2%

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

      1. log1p-expm1-u 82.2%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

      2. hypot-define 84.2%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

      3. expm1-undefine 84.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)} - 1}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

      4. add-exp-log 84.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(x.re, x.im\right)} - 1\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

   7. Applied egg-rr 84.2%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\mathsf{hypot}\left(x.re, x.im\right) - 1\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

   if -1.0600000000000001e98 < y.re < 1.14999999999999997e54

   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. Step-by-step derivation

      1. cancel-sign-sub-inv 37.4%

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

      2. fma-define 37.4%

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

      3. hypot-define 37.4%

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

      4. distribute-lft-neg-in 37.4%

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

      5. distribute-rgt-neg-out 37.4%

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

      6. fma-define 37.4%

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

      7. hypot-define 83.1%

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

      8. *-commutative 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in y.re around -inf 50.8%

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

      1. mul-1-neg 50.8%

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

      2. fma-define 50.8%

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

      3. unpow2 50.8%

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

      4. unpow2 50.8%

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

      5. hypot-undefine 83.1%

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

      6. associate-/l* 83.1%

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

   7. Simplified 83.1%

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

      1. *-un-lft-identity 83.1%

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

      2. exp-prod 83.2%

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

      3. distribute-rgt-neg-in 83.2%

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

   9. Applied egg-rr 83.2%

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

       1. exp-1-e 83.2%

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

       2. hypot-undefine 50.8%

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

       3. unpow2 50.8%

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

       4. unpow2 50.8%

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

       5. associate-*r/ 50.8%

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

       6. fma-define 50.8%

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

       7. +-commutative 50.8%

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

       8. mul-1-neg 50.8%

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

       9. unsub-neg 50.8%

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

       10. associate-*r/ 50.8%

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

   11. Simplified 83.2%

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

   12. Taylor expanded in y.im around 0 84.1%

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

       1. *-commutative 84.1%

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

   14. Simplified 84.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.06 \cdot 10^{+98} \lor \neg \left(y.re \leq 1.15 \cdot 10^{+54}\right):\\ \;\;\;\;e^{y.re \cdot \mathsf{log1p}\left(\mathsf{hypot}\left(x.re, x.im\right) + -1\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right)\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{t\_1 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_3 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;t\_2 \cdot \cos \left(t\_1 \cdot y.im + t\_0\right) \leq -0.85:\\ \;\;\;\;t\_2 \cdot \left|\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot t\_3\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(y.re \cdot \left(t\_3 - y.im \cdot \frac{{\left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{3}\right)}^{3}}{y.re}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (exp (- (* t_1 y.re) (* (atan2 x.im x.re) y.im))))
        (t_3 (log (hypot x.im x.re))))
   (if (<= (* t_2 (cos (+ (* t_1 y.im) t_0))) -0.85)
     (* t_2 (fabs (cos (fma y.re (atan2 x.im x.re) (* y.im t_3)))))
     (*
      (pow
       E
       (*
        y.re
        (-
         t_3
         (*
          y.im
          (/ (pow (pow (cbrt (cbrt (atan2 x.im x.re))) 3.0) 3.0) y.re)))))
      (cos (fma (log (hypot x.re x.im)) 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = exp(((t_1 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_3 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if ((t_2 * cos(((t_1 * y_46_im) + t_0))) <= -0.85) {
		tmp = t_2 * fabs(cos(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * t_3))));
	} else {
		tmp = pow(((double) M_E), (y_46_re * (t_3 - (y_46_im * (pow(pow(cbrt(cbrt(atan2(x_46_im, x_46_re))), 3.0), 3.0) / y_46_re))))) * cos(fma(log(hypot(x_46_re, x_46_im)), 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(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = exp(Float64(Float64(t_1 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_3 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (Float64(t_2 * cos(Float64(Float64(t_1 * y_46_im) + t_0))) <= -0.85)
		tmp = Float64(t_2 * abs(cos(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * t_3)))));
	else
		tmp = Float64((exp(1) ^ Float64(y_46_re * Float64(t_3 - Float64(y_46_im * Float64(((cbrt(cbrt(atan(x_46_im, x_46_re))) ^ 3.0) ^ 3.0) / y_46_re))))) * cos(fma(log(hypot(x_46_re, x_46_im)), 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[Cos[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.85], N[(t$95$2 * N[Abs[N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[E, N[(y$46$re * N[(t$95$3 - N[(y$46$im * N[(N[Power[N[Power[N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 3.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.849999999999999978

   1. Initial program 30.6%

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

      1. fma-define 30.6%

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

      2. hypot-define 30.6%

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

      3. *-commutative 30.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 62.8%

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

      6. pow2 62.8%

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

   4. Applied egg-rr 62.8%

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

      1. unpow2 62.8%

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

      2. rem-sqrt-square 62.8%

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

      3. *-commutative 62.8%

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

      4. hypot-undefine 62.8%

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

      5. unpow2 62.8%

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

      6. unpow2 62.8%

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

      7. +-commutative 62.8%

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

      8. unpow2 62.8%

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

      9. unpow2 62.8%

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

      10. hypot-undefine 62.8%

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

   6. Simplified 62.8%

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

   if -0.849999999999999978 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 40.8%

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

      1. cancel-sign-sub-inv 40.8%

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

      2. fma-define 40.8%

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

      3. hypot-define 40.8%

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

      4. distribute-lft-neg-in 40.8%

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

      5. distribute-rgt-neg-out 40.8%

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

      6. fma-define 40.8%

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

      7. hypot-define 85.3%

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

      8. *-commutative 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y.re around -inf 62.1%

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

      1. mul-1-neg 62.1%

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

      2. fma-define 62.1%

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

      3. unpow2 62.1%

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

      4. unpow2 62.1%

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

      5. hypot-undefine 85.3%

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

      6. associate-/l* 85.3%

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

   7. Simplified 85.3%

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

      1. *-un-lft-identity 85.3%

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

      2. exp-prod 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

   9. Applied egg-rr 85.3%

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

       1. exp-1-e 85.3%

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

       2. hypot-undefine 62.1%

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

       3. unpow2 62.1%

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

       4. unpow2 62.1%

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

       5. associate-*r/ 62.1%

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

       6. fma-define 62.1%

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

       7. +-commutative 62.1%

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

       8. mul-1-neg 62.1%

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

       9. unsub-neg 62.1%

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

       10. associate-*r/ 62.1%

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

   11. Simplified 85.3%

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

       1. add-cube-cbrt 85.3%

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

       2. pow3 85.3%

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

   13. Applied egg-rr 85.3%

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

       1. add-cube-cbrt 85.4%

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

       2. pow3 85.4%

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

   15. Applied egg-rr 85.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.85:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left|\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - y.im \cdot \frac{{\left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{3}\right)}^{3}}{y.re}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{t\_1 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_3 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;t\_2 \cdot \cos \left(t\_1 \cdot y.im + t\_0\right) \leq -0.85:\\ \;\;\;\;t\_2 \cdot \left|\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot t\_3\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(y.re \cdot \left(t\_3 - y.im \cdot \frac{{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}{y.re}\right)\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_0\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (exp (- (* t_1 y.re) (* (atan2 x.im x.re) y.im))))
        (t_3 (log (hypot x.im x.re))))
   (if (<= (* t_2 (cos (+ (* t_1 y.im) t_0))) -0.85)
     (* t_2 (fabs (cos (fma y.re (atan2 x.im x.re) (* y.im t_3)))))
     (*
      (pow
       E
       (* y.re (- t_3 (* y.im (/ (pow (cbrt (atan2 x.im x.re)) 3.0) y.re)))))
      (expm1 (log1p (cos (fma (log (hypot x.re x.im)) 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = exp(((t_1 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_3 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if ((t_2 * cos(((t_1 * y_46_im) + t_0))) <= -0.85) {
		tmp = t_2 * fabs(cos(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * t_3))));
	} else {
		tmp = pow(((double) M_E), (y_46_re * (t_3 - (y_46_im * (pow(cbrt(atan2(x_46_im, x_46_re)), 3.0) / y_46_re))))) * expm1(log1p(cos(fma(log(hypot(x_46_re, x_46_im)), 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(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = exp(Float64(Float64(t_1 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_3 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (Float64(t_2 * cos(Float64(Float64(t_1 * y_46_im) + t_0))) <= -0.85)
		tmp = Float64(t_2 * abs(cos(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * t_3)))));
	else
		tmp = Float64((exp(1) ^ Float64(y_46_re * Float64(t_3 - Float64(y_46_im * Float64((cbrt(atan(x_46_im, x_46_re)) ^ 3.0) / y_46_re))))) * expm1(log1p(cos(fma(log(hypot(x_46_re, x_46_im)), 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[Cos[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.85], N[(t$95$2 * N[Abs[N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[E, N[(y$46$re * N[(t$95$3 - N[(y$46$im * N[(N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.849999999999999978

   1. Initial program 30.6%

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

      1. fma-define 30.6%

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

      2. hypot-define 30.6%

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

      3. *-commutative 30.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 62.8%

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

      6. pow2 62.8%

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

   4. Applied egg-rr 62.8%

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

      1. unpow2 62.8%

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

      2. rem-sqrt-square 62.8%

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

      3. *-commutative 62.8%

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

      4. hypot-undefine 62.8%

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

      5. unpow2 62.8%

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

      6. unpow2 62.8%

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

      7. +-commutative 62.8%

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

      8. unpow2 62.8%

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

      9. unpow2 62.8%

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

      10. hypot-undefine 62.8%

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

   6. Simplified 62.8%

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

   if -0.849999999999999978 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 40.8%

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

      1. cancel-sign-sub-inv 40.8%

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

      2. fma-define 40.8%

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

      3. hypot-define 40.8%

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

      4. distribute-lft-neg-in 40.8%

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

      5. distribute-rgt-neg-out 40.8%

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

      6. fma-define 40.8%

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

      7. hypot-define 85.3%

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

      8. *-commutative 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y.re around -inf 62.1%

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

      1. mul-1-neg 62.1%

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

      2. fma-define 62.1%

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

      3. unpow2 62.1%

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

      4. unpow2 62.1%

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

      5. hypot-undefine 85.3%

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

      6. associate-/l* 85.3%

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

   7. Simplified 85.3%

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

      1. *-un-lft-identity 85.3%

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

      2. exp-prod 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

   9. Applied egg-rr 85.3%

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

       1. exp-1-e 85.3%

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

       2. hypot-undefine 62.1%

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

       3. unpow2 62.1%

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

       4. unpow2 62.1%

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

       5. associate-*r/ 62.1%

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

       6. fma-define 62.1%

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

       7. +-commutative 62.1%

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

       8. mul-1-neg 62.1%

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

       9. unsub-neg 62.1%

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

       10. associate-*r/ 62.1%

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

   11. Simplified 85.3%

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

       1. add-cube-cbrt 85.3%

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

       2. pow3 85.3%

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

   13. Applied egg-rr 85.3%

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

       1. expm1-log1p-u 85.4%

          \[\leadsto {e}^{\left(y.re \cdot \left(-\left(y.im \cdot \frac{{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}{y.re} - \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)} \]

   15. Applied egg-rr 85.4%

       \[\leadsto {e}^{\left(y.re \cdot \left(-\left(y.im \cdot \frac{{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}{y.re} - \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.85:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left|\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - y.im \cdot \frac{{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}{y.re}\right)\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{t\_1 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_3 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;t\_2 \cdot \cos \left(t\_1 \cdot y.im + t\_0\right) \leq -0.85:\\ \;\;\;\;t\_2 \cdot \left|\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot t\_3\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_0\right)\right) \cdot {e}^{\left(y.re \cdot \left(t\_3 - y.im \cdot \frac{{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}{y.re}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (exp (- (* t_1 y.re) (* (atan2 x.im x.re) y.im))))
        (t_3 (log (hypot x.im x.re))))
   (if (<= (* t_2 (cos (+ (* t_1 y.im) t_0))) -0.85)
     (* t_2 (fabs (cos (fma y.re (atan2 x.im x.re) (* y.im t_3)))))
     (*
      (cos (fma (log (hypot x.re x.im)) y.im t_0))
      (pow
       E
       (*
        y.re
        (- t_3 (* y.im (/ (pow (cbrt (atan2 x.im x.re)) 3.0) y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = exp(((t_1 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_3 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if ((t_2 * cos(((t_1 * y_46_im) + t_0))) <= -0.85) {
		tmp = t_2 * fabs(cos(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * t_3))));
	} else {
		tmp = cos(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0)) * pow(((double) M_E), (y_46_re * (t_3 - (y_46_im * (pow(cbrt(atan2(x_46_im, x_46_re)), 3.0) / y_46_re)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = exp(Float64(Float64(t_1 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_3 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (Float64(t_2 * cos(Float64(Float64(t_1 * y_46_im) + t_0))) <= -0.85)
		tmp = Float64(t_2 * abs(cos(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * t_3)))));
	else
		tmp = Float64(cos(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0)) * (exp(1) ^ Float64(y_46_re * Float64(t_3 - Float64(y_46_im * Float64((cbrt(atan(x_46_im, x_46_re)) ^ 3.0) / y_46_re))))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[Cos[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.85], N[(t$95$2 * N[Abs[N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[E, N[(y$46$re * N[(t$95$3 - N[(y$46$im * N[(N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.849999999999999978

   1. Initial program 30.6%

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

      1. fma-define 30.6%

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

      2. hypot-define 30.6%

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

      3. *-commutative 30.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 62.8%

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

      6. pow2 62.8%

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

   4. Applied egg-rr 62.8%

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

      1. unpow2 62.8%

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

      2. rem-sqrt-square 62.8%

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

      3. *-commutative 62.8%

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

      4. hypot-undefine 62.8%

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

      5. unpow2 62.8%

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

      6. unpow2 62.8%

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

      7. +-commutative 62.8%

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

      8. unpow2 62.8%

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

      9. unpow2 62.8%

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

      10. hypot-undefine 62.8%

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

   6. Simplified 62.8%

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

   if -0.849999999999999978 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 40.8%

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

      1. cancel-sign-sub-inv 40.8%

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

      2. fma-define 40.8%

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

      3. hypot-define 40.8%

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

      4. distribute-lft-neg-in 40.8%

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

      5. distribute-rgt-neg-out 40.8%

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

      6. fma-define 40.8%

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

      7. hypot-define 85.3%

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

      8. *-commutative 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y.re around -inf 62.1%

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

      1. mul-1-neg 62.1%

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

      2. fma-define 62.1%

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

      3. unpow2 62.1%

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

      4. unpow2 62.1%

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

      5. hypot-undefine 85.3%

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

      6. associate-/l* 85.3%

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

   7. Simplified 85.3%

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

      1. *-un-lft-identity 85.3%

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

      2. exp-prod 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

   9. Applied egg-rr 85.3%

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

       1. exp-1-e 85.3%

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

       2. hypot-undefine 62.1%

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

       3. unpow2 62.1%

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

       4. unpow2 62.1%

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

       5. associate-*r/ 62.1%

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

       6. fma-define 62.1%

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

       7. +-commutative 62.1%

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

       8. mul-1-neg 62.1%

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

       9. unsub-neg 62.1%

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

       10. associate-*r/ 62.1%

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

   11. Simplified 85.3%

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

       1. add-cube-cbrt 85.3%

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

       2. pow3 85.3%

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

   13. Applied egg-rr 85.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.85:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left|\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {e}^{\left(y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - y.im \cdot \frac{{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}{y.re}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{t\_1 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_3 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;t\_2 \cdot \cos \left(t\_1 \cdot y.im + t\_0\right) \leq -0.85:\\ \;\;\;\;t\_2 \cdot \left|\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot t\_3\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_0\right)\right) \cdot {e}^{\left(y.re \cdot \left(t\_3 - y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (exp (- (* t_1 y.re) (* (atan2 x.im x.re) y.im))))
        (t_3 (log (hypot x.im x.re))))
   (if (<= (* t_2 (cos (+ (* t_1 y.im) t_0))) -0.85)
     (* t_2 (fabs (cos (fma y.re (atan2 x.im x.re) (* y.im t_3)))))
     (*
      (cos (fma (log (hypot x.re x.im)) y.im t_0))
      (pow E (* y.re (- t_3 (* 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = exp(((t_1 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_3 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if ((t_2 * cos(((t_1 * y_46_im) + t_0))) <= -0.85) {
		tmp = t_2 * fabs(cos(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * t_3))));
	} else {
		tmp = cos(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0)) * pow(((double) M_E), (y_46_re * (t_3 - (y_46_im * (atan2(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)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = exp(Float64(Float64(t_1 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_3 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (Float64(t_2 * cos(Float64(Float64(t_1 * y_46_im) + t_0))) <= -0.85)
		tmp = Float64(t_2 * abs(cos(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * t_3)))));
	else
		tmp = Float64(cos(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0)) * (exp(1) ^ Float64(y_46_re * Float64(t_3 - Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / y_46_re))))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[Cos[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.85], N[(t$95$2 * N[Abs[N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[E, N[(y$46$re * N[(t$95$3 - N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.849999999999999978

   1. Initial program 30.6%

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

      1. fma-define 30.6%

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

      2. hypot-define 30.6%

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

      3. *-commutative 30.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 62.8%

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

      6. pow2 62.8%

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

   4. Applied egg-rr 62.8%

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

      1. unpow2 62.8%

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

      2. rem-sqrt-square 62.8%

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

      3. *-commutative 62.8%

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

      4. hypot-undefine 62.8%

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

      5. unpow2 62.8%

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

      6. unpow2 62.8%

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

      7. +-commutative 62.8%

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

      8. unpow2 62.8%

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

      9. unpow2 62.8%

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

      10. hypot-undefine 62.8%

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

   6. Simplified 62.8%

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

   if -0.849999999999999978 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 40.8%

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

      1. cancel-sign-sub-inv 40.8%

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

      2. fma-define 40.8%

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

      3. hypot-define 40.8%

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

      4. distribute-lft-neg-in 40.8%

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

      5. distribute-rgt-neg-out 40.8%

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

      6. fma-define 40.8%

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

      7. hypot-define 85.3%

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

      8. *-commutative 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y.re around -inf 62.1%

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

      1. mul-1-neg 62.1%

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

      2. fma-define 62.1%

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

      3. unpow2 62.1%

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

      4. unpow2 62.1%

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

      5. hypot-undefine 85.3%

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

      6. associate-/l* 85.3%

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

   7. Simplified 85.3%

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

      1. *-un-lft-identity 85.3%

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

      2. exp-prod 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

   9. Applied egg-rr 85.3%

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

       1. exp-1-e 85.3%

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

       2. hypot-undefine 62.1%

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

       3. unpow2 62.1%

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

       4. unpow2 62.1%

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

       5. associate-*r/ 62.1%

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

       6. fma-define 62.1%

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

       7. +-commutative 62.1%

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

       8. mul-1-neg 62.1%

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

       9. unsub-neg 62.1%

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

       10. associate-*r/ 62.1%

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

   11. Simplified 85.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.85:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left|\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {e}^{\left(y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -0.000135:\\ \;\;\;\;t\_0 \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 220000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -0.000135)
     (*
      t_0
      (/ (pow (hypot x.re x.im) y.re) (+ 1.0 (* (atan2 x.im x.re) y.im))))
     (if (<= y.re 220000000.0)
       (*
        (cos (* y.im (log (hypot x.im x.re))))
        (exp (* (atan2 x.im x.re) (- y.im))))
       (pow
        (pow (* t_0 (pow (hypot x.im x.re) y.re)) 3.0)
        0.3333333333333333)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = cos((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -0.000135) {
		tmp = t_0 * (pow(hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 220000000.0) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = pow(pow((t_0 * pow(hypot(x_46_im, x_46_re), y_46_re)), 3.0), 0.3333333333333333);
	}
	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.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -0.000135) {
		tmp = t_0 * (Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 220000000.0) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.pow(Math.pow((t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re)), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_re <= -0.000135:
		tmp = t_0 * (math.pow(math.hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (math.atan2(x_46_im, x_46_re) * y_46_im)))
	elif y_46_re <= 220000000.0:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.pow(math.pow((t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -0.000135)
		tmp = Float64(t_0 * Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(1.0 + Float64(atan(x_46_im, x_46_re) * y_46_im))));
	elseif (y_46_re <= 220000000.0)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = (Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re)) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_re <= -0.000135)
		tmp = t_0 * ((hypot(x_46_re, x_46_im) ^ y_46_re) / (1.0 + (atan2(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 220000000.0)
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = ((t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re)) ^ 3.0) ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -0.000135], N[(t$95$0 * N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(1.0 + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 220000000.0], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.35000000000000002e-4

   1. Initial program 48.6%

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

      1. exp-diff 43.1%

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

      2. exp-to-pow 43.1%

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

      3. hypot-define 43.1%

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

      4. *-commutative 43.1%

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

      5. exp-prod 40.3%

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

      6. fma-define 40.3%

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

      7. hypot-define 75.0%

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

      8. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in y.im around 0 83.4%

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

   6. Taylor expanded in y.im around 0 86.2%

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

      1. *-commutative 88.9%

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

   8. Simplified 86.2%

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

   if -1.35000000000000002e-4 < y.re < 2.2e8

   1. Initial program 34.5%

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

      1. cancel-sign-sub-inv 34.5%

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

      2. fma-define 34.5%

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

      3. hypot-define 34.5%

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

      4. distribute-lft-neg-in 34.5%

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

      5. distribute-rgt-neg-out 34.5%

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

      6. fma-define 34.5%

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

      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.re around 0 33.9%

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

      1. *-commutative 33.9%

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

      2. unpow2 33.9%

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

      3. unpow2 33.9%

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

      4. hypot-undefine 78.9%

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

      5. neg-mul-1 78.9%

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

      6. distribute-rgt-neg-in 78.9%

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

   7. Simplified 78.9%

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

   if 2.2e8 < y.re

   1. Initial program 41.5%

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

      1. cancel-sign-sub-inv 41.5%

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

      2. fma-define 41.5%

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

      3. hypot-define 41.5%

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

      4. distribute-lft-neg-in 41.5%

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

      5. distribute-rgt-neg-out 41.5%

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

      6. fma-define 41.5%

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

      7. hypot-define 66.0%

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

      8. *-commutative 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in y.im around 0 60.5%

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

      1. unpow2 60.5%

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

      2. unpow2 60.5%

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

      3. hypot-undefine 60.5%

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

   7. Simplified 60.5%

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

      1. add-cbrt-cube 60.5%

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

      2. pow1/3 64.3%

         \[\leadsto \color{blue}{{\left(\left(\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right) \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right) \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 64.3%

         \[\leadsto {\color{blue}{\left({\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}^{3}\right)}}^{0.3333333333333333} \]

   9. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{{\left({\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}^{3}\right)}^{0.3333333333333333}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.000135:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 220000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.00013:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 220000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(y.re \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -0.00013)
   (*
    (cos (* y.re (atan2 x.im x.re)))
    (/ (pow (hypot x.re x.im) y.re) (+ 1.0 (* (atan2 x.im x.re) y.im))))
   (if (<= y.re 220000000.0)
     (*
      (cos (* y.im (log (hypot x.im x.re))))
      (exp (* (atan2 x.im x.re) (- y.im))))
     (*
      (pow (hypot x.im x.re) y.re)
      (cos (* y.re (log1p (expm1 (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_re <= -0.00013) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (pow(hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 220000000.0) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) * cos((y_46_re * log1p(expm1(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_re <= -0.00013) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * (Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 220000000.0) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re) * Math.cos((y_46_re * Math.log1p(Math.expm1(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_re <= -0.00013:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * (math.pow(math.hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (math.atan2(x_46_im, x_46_re) * y_46_im)))
	elif y_46_re <= 220000000.0:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re) * math.cos((y_46_re * math.log1p(math.expm1(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_re <= -0.00013)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(1.0 + Float64(atan(x_46_im, x_46_re) * y_46_im))));
	elseif (y_46_re <= 220000000.0)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) * cos(Float64(y_46_re * log1p(expm1(atan(x_46_im, x_46_re))))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -0.00013], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(1.0 + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 220000000.0], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Cos[N[(y$46$re * N[Log[1 + N[(Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.29999999999999989e-4

   1. Initial program 48.6%

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

      1. exp-diff 43.1%

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

      2. exp-to-pow 43.1%

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

      3. hypot-define 43.1%

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

      4. *-commutative 43.1%

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

      5. exp-prod 40.3%

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

      6. fma-define 40.3%

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

      7. hypot-define 75.0%

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

      8. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in y.im around 0 83.4%

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

   6. Taylor expanded in y.im around 0 86.2%

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

      1. *-commutative 88.9%

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

   8. Simplified 86.2%

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

   if -1.29999999999999989e-4 < y.re < 2.2e8

   1. Initial program 34.5%

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

      1. cancel-sign-sub-inv 34.5%

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

      2. fma-define 34.5%

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

      3. hypot-define 34.5%

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

      4. distribute-lft-neg-in 34.5%

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

      5. distribute-rgt-neg-out 34.5%

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

      6. fma-define 34.5%

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

      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.re around 0 33.9%

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

      1. *-commutative 33.9%

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

      2. unpow2 33.9%

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

      3. unpow2 33.9%

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

      4. hypot-undefine 78.9%

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

      5. neg-mul-1 78.9%

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

      6. distribute-rgt-neg-in 78.9%

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

   7. Simplified 78.9%

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

   if 2.2e8 < y.re

   1. Initial program 41.5%

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

      1. cancel-sign-sub-inv 41.5%

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

      2. fma-define 41.5%

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

      3. hypot-define 41.5%

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

      4. distribute-lft-neg-in 41.5%

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

      5. distribute-rgt-neg-out 41.5%

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

      6. fma-define 41.5%

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

      7. hypot-define 66.0%

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

      8. *-commutative 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in y.im around 0 60.5%

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

      1. unpow2 60.5%

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

      2. unpow2 60.5%

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

      3. hypot-undefine 60.5%

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

   7. Simplified 60.5%

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

      1. log1p-expm1-u 62.4%

         \[\leadsto \cos \left(y.re \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   9. Applied egg-rr 62.4%

      \[\leadsto \cos \left(y.re \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00013:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 220000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(y.re \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (pow
   E
   (* y.re (- (log (hypot x.im x.re)) (* y.im (/ (atan2 x.im x.re) y.re)))))
  (cos (* y.re (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return pow(((double) M_E), (y_46_re * (log(hypot(x_46_im, x_46_re)) - (y_46_im * (atan2(x_46_im, x_46_re) / y_46_re))))) * cos((y_46_re * atan2(x_46_im, x_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.E, (y_46_re * (Math.log(Math.hypot(x_46_im, x_46_re)) - (y_46_im * (Math.atan2(x_46_im, x_46_re) / y_46_re))))) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.pow(math.e, (y_46_re * (math.log(math.hypot(x_46_im, x_46_re)) - (y_46_im * (math.atan2(x_46_im, x_46_re) / y_46_re))))) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64((exp(1) ^ Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) - Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / y_46_re))))) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (2.71828182845904523536 ^ (y_46_re * (log(hypot(x_46_im, x_46_re)) - (y_46_im * (atan2(x_46_im, x_46_re) / y_46_re))))) * cos((y_46_re * atan2(x_46_im, x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Power[E, N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] - N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

   1. cancel-sign-sub-inv 39.9%

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

   2. fma-define 39.9%

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

   3. hypot-define 39.9%

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

   4. distribute-lft-neg-in 39.9%

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

   5. distribute-rgt-neg-out 39.9%

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

   6. fma-define 39.9%

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

   7. hypot-define 80.4%

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

   8. *-commutative 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in y.re around -inf 59.3%

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

   1. mul-1-neg 59.3%

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

   2. fma-define 59.3%

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

   3. unpow2 59.3%

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

   4. unpow2 59.3%

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

   5. hypot-undefine 80.4%

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

   6. associate-/l* 80.4%

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

7. Simplified 80.4%

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

   1. *-un-lft-identity 80.4%

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

   2. exp-prod 80.4%

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

   3. distribute-rgt-neg-in 80.4%

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

9. Applied egg-rr 80.4%

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

    1. exp-1-e 80.4%

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

    2. hypot-undefine 59.3%

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

    3. unpow2 59.3%

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

    4. unpow2 59.3%

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

    5. associate-*r/ 59.3%

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

    6. fma-define 59.3%

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

    7. +-commutative 59.3%

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

    8. mul-1-neg 59.3%

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

    9. unsub-neg 59.3%

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

    10. associate-*r/ 59.3%

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

11. Simplified 80.4%

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

12. Taylor expanded in y.im around 0 80.2%

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

    1. *-commutative 80.2%

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

14. Simplified 80.2%

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

15. Final simplification 80.2%

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

Alternative 9: 74.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.00013 \lor \neg \left(y.re \leq 220000000\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -0.00013) (not (<= y.re 220000000.0)))
   (*
    (cos (* y.re (atan2 x.im x.re)))
    (/ (pow (hypot x.re x.im) y.re) (+ 1.0 (* (atan2 x.im x.re) y.im))))
   (*
    (cos (* y.im (log (hypot x.im x.re))))
    (exp (* (atan2 x.im x.re) (- y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -0.00013) || !(y_46_re <= 220000000.0)) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (pow(hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	}
	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_re <= -0.00013) || !(y_46_re <= 220000000.0)) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * (Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -0.00013) or not (y_46_re <= 220000000.0):
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * (math.pow(math.hypot(x_46_re, x_46_im), y_46_re) / (1.0 + (math.atan2(x_46_im, x_46_re) * y_46_im)))
	else:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -0.00013) || !(y_46_re <= 220000000.0))
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(1.0 + Float64(atan(x_46_im, x_46_re) * y_46_im))));
	else
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	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_re <= -0.00013) || ~((y_46_re <= 220000000.0)))
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * ((hypot(x_46_re, x_46_im) ^ y_46_re) / (1.0 + (atan2(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -0.00013], N[Not[LessEqual[y$46$re, 220000000.0]], $MachinePrecision]], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(1.0 + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.29999999999999989e-4 or 2.2e8 < y.re

   1. Initial program 45.6%

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

      1. exp-diff 36.8%

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

      2. exp-to-pow 36.8%

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

      3. hypot-define 36.8%

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

      4. *-commutative 36.8%

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

      5. exp-prod 35.2%

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

      6. fma-define 35.2%

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

      7. hypot-define 63.2%

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

      8. *-commutative 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in y.im around 0 73.7%

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

   6. Taylor expanded in y.im around 0 76.1%

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

      1. *-commutative 80.0%

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

   8. Simplified 76.1%

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

   if -1.29999999999999989e-4 < y.re < 2.2e8

   1. Initial program 34.5%

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

      1. cancel-sign-sub-inv 34.5%

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

      2. fma-define 34.5%

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

      3. hypot-define 34.5%

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

      4. distribute-lft-neg-in 34.5%

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

      5. distribute-rgt-neg-out 34.5%

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

      6. fma-define 34.5%

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

      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.re around 0 33.9%

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

      1. *-commutative 33.9%

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

      2. unpow2 33.9%

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

      3. unpow2 33.9%

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

      4. hypot-undefine 78.9%

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

      5. neg-mul-1 78.9%

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

      6. distribute-rgt-neg-in 78.9%

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

   7. Simplified 78.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00013 \lor \neg \left(y.re \leq 220000000\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.00012 \lor \neg \left(y.re \leq 220000000\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -0.00012) (not (<= y.re 220000000.0)))
   (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
   (*
    (cos (* y.im (log (hypot x.im x.re))))
    (exp (* (atan2 x.im x.re) (- y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -0.00012) || !(y_46_re <= 220000000.0)) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	}
	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_re <= -0.00012) || !(y_46_re <= 220000000.0)) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -0.00012) or not (y_46_re <= 220000000.0):
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -0.00012) || !(y_46_re <= 220000000.0))
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	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_re <= -0.00012) || ~((y_46_re <= 220000000.0)))
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -0.00012], N[Not[LessEqual[y$46$re, 220000000.0]], $MachinePrecision]], N[(N[Cos[N[(y$46$re * 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]), $MachinePrecision], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.20000000000000003e-4 or 2.2e8 < y.re

   1. Initial program 45.6%

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

      1. cancel-sign-sub-inv 45.6%

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

      2. fma-define 45.6%

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

      3. hypot-define 45.6%

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

      4. distribute-lft-neg-in 45.6%

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

      5. distribute-rgt-neg-out 45.6%

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

      6. fma-define 45.6%

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

      7. hypot-define 80.0%

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

      8. *-commutative 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in y.im around 0 73.7%

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

      1. unpow2 73.7%

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

      2. unpow2 73.7%

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

      3. hypot-undefine 73.7%

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

   7. Simplified 73.7%

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

   if -1.20000000000000003e-4 < y.re < 2.2e8

   1. Initial program 34.5%

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

      1. cancel-sign-sub-inv 34.5%

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

      2. fma-define 34.5%

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

      3. hypot-define 34.5%

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

      4. distribute-lft-neg-in 34.5%

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

      5. distribute-rgt-neg-out 34.5%

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

      6. fma-define 34.5%

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

      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.re around 0 33.9%

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

      1. *-commutative 33.9%

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

      2. unpow2 33.9%

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

      3. unpow2 33.9%

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

      4. hypot-undefine 78.9%

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

      5. neg-mul-1 78.9%

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

      6. distribute-rgt-neg-in 78.9%

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

   7. Simplified 78.9%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00012 \lor \neg \left(y.re \leq 220000000\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{-7} \lor \neg \left(y.re \leq 1.1 \cdot 10^{-32}\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3e-7) (not (<= y.re 1.1e-32)))
   (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
   (- 1.0 (log1p (expm1 (* (atan2 x.im x.re) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3e-7) || !(y_46_re <= 1.1e-32)) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = 1.0 - log1p(expm1((atan2(x_46_im, x_46_re) * y_46_im)));
	}
	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_re <= -3e-7) || !(y_46_re <= 1.1e-32)) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = 1.0 - Math.log1p(Math.expm1((Math.atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3e-7) or not (y_46_re <= 1.1e-32):
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = 1.0 - math.log1p(math.expm1((math.atan2(x_46_im, x_46_re) * y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3e-7) || !(y_46_re <= 1.1e-32))
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(1.0 - log1p(expm1(Float64(atan(x_46_im, x_46_re) * y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3e-7], N[Not[LessEqual[y$46$re, 1.1e-32]], $MachinePrecision]], N[(N[Cos[N[(y$46$re * 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]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.9999999999999999e-7 or 1.1e-32 < y.re

   1. Initial program 45.8%

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

      1. cancel-sign-sub-inv 45.8%

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

      2. fma-define 45.8%

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

      3. hypot-define 45.8%

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

      4. distribute-lft-neg-in 45.8%

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

      5. distribute-rgt-neg-out 45.8%

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

      6. fma-define 45.8%

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

      7. hypot-define 79.6%

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

      8. *-commutative 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in y.im around 0 70.9%

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

      1. unpow2 70.9%

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

      2. unpow2 70.9%

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

      3. hypot-undefine 71.0%

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

   7. Simplified 71.0%

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

   if -2.9999999999999999e-7 < y.re < 1.1e-32

   1. Initial program 33.5%

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

      1. cancel-sign-sub-inv 33.5%

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

      2. fma-define 33.5%

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

      3. hypot-define 33.5%

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

      4. distribute-lft-neg-in 33.5%

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

      5. distribute-rgt-neg-out 33.5%

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

      6. fma-define 33.5%

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

      7. hypot-define 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.re around 0 33.5%

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

      1. *-commutative 33.5%

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

      2. unpow2 33.5%

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

      3. unpow2 33.5%

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

      4. hypot-undefine 81.3%

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

      5. neg-mul-1 81.3%

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

      6. distribute-rgt-neg-in 81.3%

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

      7. mul-1-neg 81.3%

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

      8. exp-prod 80.4%

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

      9. mul-1-neg 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in y.im around 0 51.6%

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

      1. neg-mul-1 51.6%

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

      2. unsub-neg 51.6%

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

   10. Simplified 51.6%

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

       1. log1p-expm1-u 64.5%

          \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Applied egg-rr 64.5%

       \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{-7} \lor \neg \left(y.re \leq 1.1 \cdot 10^{-32}\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -32000000 \lor \neg \left(y.re \leq 5.8 \cdot 10^{-11}\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -32000000.0) (not (<= y.re 5.8e-11)))
   (* (cos (* y.re (atan2 x.im x.re))) (pow x.im y.re))
   (- 1.0 (log1p (expm1 (* (atan2 x.im x.re) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -32000000.0) || !(y_46_re <= 5.8e-11)) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(x_46_im, y_46_re);
	} else {
		tmp = 1.0 - log1p(expm1((atan2(x_46_im, x_46_re) * y_46_im)));
	}
	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_re <= -32000000.0) || !(y_46_re <= 5.8e-11)) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = 1.0 - Math.log1p(Math.expm1((Math.atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -32000000.0) or not (y_46_re <= 5.8e-11):
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(x_46_im, y_46_re)
	else:
		tmp = 1.0 - math.log1p(math.expm1((math.atan2(x_46_im, x_46_re) * y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -32000000.0) || !(y_46_re <= 5.8e-11))
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (x_46_im ^ y_46_re));
	else
		tmp = Float64(1.0 - log1p(expm1(Float64(atan(x_46_im, x_46_re) * y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -32000000.0], N[Not[LessEqual[y$46$re, 5.8e-11]], $MachinePrecision]], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.2e7 or 5.8e-11 < y.re

   1. Initial program 46.1%

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

      1. cancel-sign-sub-inv 46.1%

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

      2. fma-define 46.1%

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

      3. hypot-define 46.1%

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

      4. distribute-lft-neg-in 46.1%

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

      5. distribute-rgt-neg-out 46.1%

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

      6. fma-define 46.1%

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

      7. hypot-define 79.7%

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

      8. *-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in y.im around 0 72.0%

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

      1. unpow2 72.0%

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

      2. unpow2 72.0%

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

      3. hypot-undefine 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in x.re around 0 51.3%

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

      1. *-commutative 51.3%

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

   10. Simplified 51.3%

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

   if -3.2e7 < y.re < 5.8e-11

   1. Initial program 33.8%

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

      1. cancel-sign-sub-inv 33.8%

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

      2. fma-define 33.8%

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

      3. hypot-define 33.8%

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

      4. distribute-lft-neg-in 33.8%

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

      5. distribute-rgt-neg-out 33.8%

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

      6. fma-define 33.8%

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

      7. hypot-define 81.2%

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

      8. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in y.re around 0 33.2%

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

      1. *-commutative 33.2%

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

      2. unpow2 33.2%

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

      3. unpow2 33.2%

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

      4. hypot-undefine 79.3%

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

      5. neg-mul-1 79.3%

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

      6. distribute-rgt-neg-in 79.3%

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

      7. mul-1-neg 79.3%

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

      8. exp-prod 78.4%

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

      9. mul-1-neg 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in y.im around 0 50.0%

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

      1. neg-mul-1 50.0%

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

      2. unsub-neg 50.0%

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

   10. Simplified 50.0%

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

       1. log1p-expm1-u 62.3%

          \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Applied egg-rr 62.3%

       \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -32000000 \lor \neg \left(y.re \leq 5.8 \cdot 10^{-11}\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -32000000:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.3:\\ \;\;\;\;1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -32000000.0)
     (* t_0 (pow x.im y.re))
     (if (<= y.re 2.3)
       (- 1.0 (log1p (expm1 (* (atan2 x.im x.re) y.im))))
       (* t_0 (pow 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 = cos((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -32000000.0) {
		tmp = t_0 * pow(x_46_im, y_46_re);
	} else if (y_46_re <= 2.3) {
		tmp = 1.0 - log1p(expm1((atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = t_0 * pow(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 t_0 = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -32000000.0) {
		tmp = t_0 * Math.pow(x_46_im, y_46_re);
	} else if (y_46_re <= 2.3) {
		tmp = 1.0 - Math.log1p(Math.expm1((Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = t_0 * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_re <= -32000000.0:
		tmp = t_0 * math.pow(x_46_im, y_46_re)
	elif y_46_re <= 2.3:
		tmp = 1.0 - math.log1p(math.expm1((math.atan2(x_46_im, x_46_re) * y_46_im)))
	else:
		tmp = t_0 * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -32000000.0)
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	elseif (y_46_re <= 2.3)
		tmp = Float64(1.0 - log1p(expm1(Float64(atan(x_46_im, x_46_re) * y_46_im))));
	else
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -32000000.0], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.3], N[(1.0 - N[Log[1 + N[(Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.2e7

   1. Initial program 48.6%

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

      1. cancel-sign-sub-inv 48.6%

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

      2. fma-define 48.6%

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

      3. hypot-define 48.6%

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

      4. distribute-lft-neg-in 48.6%

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

      5. distribute-rgt-neg-out 48.6%

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

      6. fma-define 48.6%

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

      7. hypot-define 90.0%

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

      8. *-commutative 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y.im around 0 83.0%

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

      1. unpow2 83.0%

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

      2. unpow2 83.0%

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

      3. hypot-undefine 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in x.re around 0 64.7%

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

      1. *-commutative 64.7%

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

   10. Simplified 64.7%

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

   if -3.2e7 < y.re < 2.2999999999999998

   1. Initial program 34.2%

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

      1. cancel-sign-sub-inv 34.2%

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

      2. fma-define 34.2%

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

      3. hypot-define 34.2%

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

      4. distribute-lft-neg-in 34.2%

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

      5. distribute-rgt-neg-out 34.2%

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

      6. fma-define 34.2%

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

      7. hypot-define 81.0%

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

      8. *-commutative 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in y.re around 0 33.0%

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

      1. *-commutative 33.0%

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

      2. unpow2 33.0%

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

      3. unpow2 33.0%

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

      4. hypot-undefine 77.7%

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

      5. neg-mul-1 77.7%

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

      6. distribute-rgt-neg-in 77.7%

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

      7. mul-1-neg 77.7%

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

      8. exp-prod 76.9%

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

      9. mul-1-neg 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in y.im around 0 48.6%

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

      1. neg-mul-1 48.6%

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

      2. unsub-neg 48.6%

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

   10. Simplified 48.6%

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

       1. log1p-expm1-u 60.6%

          \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Applied egg-rr 60.6%

       \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   if 2.2999999999999998 < y.re

   1. Initial program 42.6%

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

      1. cancel-sign-sub-inv 42.6%

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

      2. fma-define 42.6%

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

      3. hypot-define 42.6%

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

      4. distribute-lft-neg-in 42.6%

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

      5. distribute-rgt-neg-out 42.6%

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

      6. fma-define 42.6%

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

      7. hypot-define 66.7%

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

      8. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in y.im around 0 59.4%

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

      1. unpow2 59.4%

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

      2. unpow2 59.4%

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

      3. hypot-undefine 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in x.im around 0 50.5%

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

      1. *-commutative 50.5%

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

   10. Simplified 50.5%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -32000000:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.3:\\ \;\;\;\;1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- 1.0 (log1p (expm1 (* (atan2 x.im x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 - log1p(expm1((atan2(x_46_im, x_46_re) * y_46_im)));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 - Math.log1p(Math.expm1((Math.atan2(x_46_im, x_46_re) * y_46_im)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 - math.log1p(math.expm1((math.atan2(x_46_im, x_46_re) * y_46_im)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 - log1p(expm1(Float64(atan(x_46_im, x_46_re) * y_46_im))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 - N[Log[1 + N[(Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

   1. cancel-sign-sub-inv 39.9%

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

   2. fma-define 39.9%

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

   3. hypot-define 39.9%

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

   4. distribute-lft-neg-in 39.9%

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

   5. distribute-rgt-neg-out 39.9%

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

   6. fma-define 39.9%

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

   7. hypot-define 80.4%

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

   8. *-commutative 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in y.re around 0 25.7%

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

   1. *-commutative 25.7%

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

   2. unpow2 25.7%

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

   3. unpow2 25.7%

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

   4. hypot-undefine 55.8%

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

   5. neg-mul-1 55.8%

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

   6. distribute-rgt-neg-in 55.8%

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

   7. mul-1-neg 55.8%

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

   8. exp-prod 56.8%

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

   9. mul-1-neg 56.8%

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

7. Simplified 56.8%

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

8. Taylor expanded in y.im around 0 26.4%

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

   1. neg-mul-1 26.4%

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

   2. unsub-neg 26.4%

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

10. Simplified 26.4%

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

    1. log1p-expm1-u 36.9%

       \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

12. Applied egg-rr 36.9%

    \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

13. Final simplification 36.9%

    \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right) \]
14. Add Preprocessing

Alternative 15: 26.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ 1 + y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ 1.0 (* y.re (log (hypot x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 + (y_46_re * log(hypot(x_46_im, x_46_re)));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 + (y_46_re * Math.log(Math.hypot(x_46_im, x_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 + (y_46_re * math.log(math.hypot(x_46_im, x_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 + Float64(y_46_re * log(hypot(x_46_im, x_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 + (y_46_re * log(hypot(x_46_im, x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 + N[(y$46$re * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

   1. cancel-sign-sub-inv 39.9%

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

   2. fma-define 39.9%

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

   3. hypot-define 39.9%

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

   4. distribute-lft-neg-in 39.9%

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

   5. distribute-rgt-neg-out 39.9%

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

   6. fma-define 39.9%

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

   7. hypot-define 80.4%

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

   8. *-commutative 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in y.im around 0 50.0%

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

   1. unpow2 50.0%

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

   2. unpow2 50.0%

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

   3. hypot-undefine 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in y.re around 0 19.1%

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

   1. +-commutative 19.1%

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

   2. unpow2 19.1%

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

   3. unpow2 19.1%

      \[\leadsto y.re \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) + 1 \]

   4. hypot-undefine 26.5%

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

10. Simplified 26.5%

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

11. Final simplification 26.5%

    \[\leadsto 1 + y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \]
12. Add Preprocessing

Alternative 16: 25.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ 1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- 1.0 (* (atan2 x.im x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 - (atan2(x_46_im, x_46_re) * y_46_im);
}

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 = 1.0d0 - (atan2(x_46im, x_46re) * y_46im)
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 - (Math.atan2(x_46_im, x_46_re) * y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 - (math.atan2(x_46_im, x_46_re) * y_46_im)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 - Float64(atan(x_46_im, x_46_re) * y_46_im))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 - (atan2(x_46_im, x_46_re) * y_46_im);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

   1. cancel-sign-sub-inv 39.9%

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

   2. fma-define 39.9%

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

   3. hypot-define 39.9%

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

   4. distribute-lft-neg-in 39.9%

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

   5. distribute-rgt-neg-out 39.9%

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

   6. fma-define 39.9%

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

   7. hypot-define 80.4%

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

   8. *-commutative 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in y.re around 0 25.7%

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

   1. *-commutative 25.7%

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

   2. unpow2 25.7%

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

   3. unpow2 25.7%

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

   4. hypot-undefine 55.8%

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

   5. neg-mul-1 55.8%

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

   6. distribute-rgt-neg-in 55.8%

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

   7. mul-1-neg 55.8%

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

   8. exp-prod 56.8%

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

   9. mul-1-neg 56.8%

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

7. Simplified 56.8%

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

8. Taylor expanded in y.im around 0 26.4%

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

   1. neg-mul-1 26.4%

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

   2. unsub-neg 26.4%

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

10. Simplified 26.4%

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

11. Final simplification 26.4%

    \[\leadsto 1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im \]
12. Add Preprocessing

Alternative 17: 25.6% accurate, 829.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 39.9%

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

   1. cancel-sign-sub-inv 39.9%

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

   2. fma-define 39.9%

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

   3. hypot-define 39.9%

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

   4. distribute-lft-neg-in 39.9%

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

   5. distribute-rgt-neg-out 39.9%

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

   6. fma-define 39.9%

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

   7. hypot-define 80.4%

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

   8. *-commutative 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in y.im around 0 50.0%

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

   1. unpow2 50.0%

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

   2. unpow2 50.0%

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

   3. hypot-undefine 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in y.re around 0 26.1%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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)))))