powComplex, real part

Percentage Accurate: 41.0% → 80.6%

Time: 19.3s

Alternatives: 9

Speedup: 2.7×

• 35.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.0% 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: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;x.im \leq -1020000000:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - \log \left(\frac{-1}{x.im}\right) \cdot y.re} \cdot \cos t_0\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot \sin t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (hypot x.im x.re)))))
   (if (<= x.im -1020000000.0)
     (*
      (exp (- (* (atan2 x.im x.re) (- y.im)) (* (log (/ -1.0 x.im)) y.re)))
      (cos t_0))
     (*
      (exp (- (* y.re (log (hypot x.re x.im))) (* (atan2 x.im x.re) y.im)))
      (- 1.0 (* (atan2 x.im x.re) (* y.re (sin 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_im * log(hypot(x_46_im, x_46_re));
	double tmp;
	if (x_46_im <= -1020000000.0) {
		tmp = exp(((atan2(x_46_im, x_46_re) * -y_46_im) - (log((-1.0 / x_46_im)) * y_46_re))) * cos(t_0);
	} else {
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - (atan2(x_46_im, x_46_re) * y_46_im))) * (1.0 - (atan2(x_46_im, x_46_re) * (y_46_re * sin(t_0))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double tmp;
	if (x_46_im <= -1020000000.0) {
		tmp = Math.exp(((Math.atan2(x_46_im, x_46_re) * -y_46_im) - (Math.log((-1.0 / x_46_im)) * y_46_re))) * Math.cos(t_0);
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im))) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * (1.0 - (Math.atan2(x_46_im, x_46_re) * (y_46_re * Math.sin(t_0))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	tmp = 0
	if x_46_im <= -1020000000.0:
		tmp = math.exp(((math.atan2(x_46_im, x_46_re) * -y_46_im) - (math.log((-1.0 / x_46_im)) * y_46_re))) * math.cos(t_0)
	else:
		tmp = math.exp(((y_46_re * math.log(math.hypot(x_46_re, x_46_im))) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * (1.0 - (math.atan2(x_46_im, x_46_re) * (y_46_re * math.sin(t_0))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -1020000000.0)
		tmp = Float64(exp(Float64(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)) - Float64(log(Float64(-1.0 / x_46_im)) * y_46_re))) * cos(t_0));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * Float64(1.0 - Float64(atan(x_46_im, x_46_re) * Float64(y_46_re * sin(t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	tmp = 0.0;
	if (x_46_im <= -1020000000.0)
		tmp = exp(((atan2(x_46_im, x_46_re) * -y_46_im) - (log((-1.0 / x_46_im)) * y_46_re))) * cos(t_0);
	else
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - (atan2(x_46_im, x_46_re) * y_46_im))) * (1.0 - (atan2(x_46_im, x_46_re) * (y_46_re * sin(t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1020000000.0], N[(N[Exp[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision] - N[(N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.02e9

   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

   3. Simplified 81.7%

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

   4. Taylor expanded in x.im around -inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. *-commutative 81.0%

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

      3. distribute-rgt-neg-in 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in y.re around 0 41.5%

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

      1. +-commutative 41.5%

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

      2. unpow2 41.5%

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

      3. unpow2 41.5%

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

      4. hypot-def 89.4%

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

      5. hypot-def 41.5%

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

      6. unpow2 41.5%

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

      7. unpow2 41.5%

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

      8. +-commutative 41.5%

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

      9. unpow2 41.5%

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

      10. unpow2 41.5%

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

      11. hypot-def 89.4%

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

   9. Simplified 89.4%

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

   if -1.02e9 < x.im

   1. Initial program 40.7%

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

   3. Simplified 78.6%

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

      1. log1p-expm1-u 74.9%

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

      2. *-commutative 74.9%

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

   5. Applied egg-rr 74.9%

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

   6. Taylor expanded in y.re around 0 40.2%

      \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \cdot \color{blue}{\left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + -1 \cdot \left(y.re \cdot \left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 40.2%

         \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + \color{blue}{\left(-y.re \cdot \left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right) \]

      2. unsub-neg 40.2%

         \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \mathsf{log1p}\left(\mathsf{expm1}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \cdot \color{blue}{\left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) - y.re \cdot \left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   8. Simplified 77.1%

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

   9. Taylor expanded in y.im around 0 79.0%

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

       1. log1p-expm1-u 82.7%

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

       2. *-commutative 82.7%

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

   11. Applied egg-rr 82.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1020000000:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - \log \left(\frac{-1}{x.im}\right) \cdot y.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{x.im}\right) \cdot y.re\\ t_1 := \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - t_0} \cdot t_1\\ \mathbf{if}\;x.im \leq -2.6 \cdot 10^{+26}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq -2.1 \cdot 10^{-153}:\\ \;\;\;\;t_1 \cdot e^{\left(\frac{y.re \cdot \left(x.re \cdot \left(x.re \cdot 0.5\right)\right)}{x.im \cdot x.im} - t_0\right) - t_2}\\ \mathbf{elif}\;x.im \leq -1.3 \cdot 10^{-295}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{+72}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - t_2}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log x.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (log (/ -1.0 x.im)) y.re))
        (t_1 (cos (* y.im (log (hypot x.im x.re)))))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (* (exp (- (* (atan2 x.im x.re) (- y.im)) t_0)) t_1)))
   (if (<= x.im -2.6e+26)
     t_3
     (if (<= x.im -2.1e-153)
       (*
        t_1
        (exp (- (- (/ (* y.re (* x.re (* x.re 0.5))) (* x.im x.im)) t_0) t_2)))
       (if (<= x.im -1.3e-295)
         t_3
         (if (<= x.im 6e+72)
           (exp (- (* y.re (log (sqrt (+ (* x.im x.im) (* x.re x.re))))) t_2))
           (*
            (cos (+ (* y.im (log x.im)) (* y.re (atan2 x.im x.re))))
            (exp (- (* y.re (log x.im)) t_2)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log((-1.0 / x_46_im)) * y_46_re;
	double t_1 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = exp(((atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * t_1;
	double tmp;
	if (x_46_im <= -2.6e+26) {
		tmp = t_3;
	} else if (x_46_im <= -2.1e-153) {
		tmp = t_1 * exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_2));
	} else if (x_46_im <= -1.3e-295) {
		tmp = t_3;
	} else if (x_46_im <= 6e+72) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_2));
	} else {
		tmp = cos(((y_46_im * log(x_46_im)) + (y_46_re * atan2(x_46_im, x_46_re)))) * exp(((y_46_re * log(x_46_im)) - t_2));
	}
	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((-1.0 / x_46_im)) * y_46_re;
	double t_1 = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = Math.exp(((Math.atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * t_1;
	double tmp;
	if (x_46_im <= -2.6e+26) {
		tmp = t_3;
	} else if (x_46_im <= -2.1e-153) {
		tmp = t_1 * Math.exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_2));
	} else if (x_46_im <= -1.3e-295) {
		tmp = t_3;
	} else if (x_46_im <= 6e+72) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_2));
	} else {
		tmp = Math.cos(((y_46_im * Math.log(x_46_im)) + (y_46_re * Math.atan2(x_46_im, x_46_re)))) * Math.exp(((y_46_re * Math.log(x_46_im)) - t_2));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log((-1.0 / x_46_im)) * y_46_re
	t_1 = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	t_2 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_3 = math.exp(((math.atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * t_1
	tmp = 0
	if x_46_im <= -2.6e+26:
		tmp = t_3
	elif x_46_im <= -2.1e-153:
		tmp = t_1 * math.exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_2))
	elif x_46_im <= -1.3e-295:
		tmp = t_3
	elif x_46_im <= 6e+72:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_2))
	else:
		tmp = math.cos(((y_46_im * math.log(x_46_im)) + (y_46_re * math.atan2(x_46_im, x_46_re)))) * math.exp(((y_46_re * math.log(x_46_im)) - t_2))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(log(Float64(-1.0 / x_46_im)) * y_46_re)
	t_1 = cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_3 = Float64(exp(Float64(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)) - t_0)) * t_1)
	tmp = 0.0
	if (x_46_im <= -2.6e+26)
		tmp = t_3;
	elseif (x_46_im <= -2.1e-153)
		tmp = Float64(t_1 * exp(Float64(Float64(Float64(Float64(y_46_re * Float64(x_46_re * Float64(x_46_re * 0.5))) / Float64(x_46_im * x_46_im)) - t_0) - t_2)));
	elseif (x_46_im <= -1.3e-295)
		tmp = t_3;
	elseif (x_46_im <= 6e+72)
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))) - t_2));
	else
		tmp = Float64(cos(Float64(Float64(y_46_im * log(x_46_im)) + Float64(y_46_re * atan(x_46_im, x_46_re)))) * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log((-1.0 / x_46_im)) * y_46_re;
	t_1 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	t_3 = exp(((atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * t_1;
	tmp = 0.0;
	if (x_46_im <= -2.6e+26)
		tmp = t_3;
	elseif (x_46_im <= -2.1e-153)
		tmp = t_1 * exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_2));
	elseif (x_46_im <= -1.3e-295)
		tmp = t_3;
	elseif (x_46_im <= 6e+72)
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_2));
	else
		tmp = cos(((y_46_im * log(x_46_im)) + (y_46_re * atan2(x_46_im, x_46_re)))) * exp(((y_46_re * log(x_46_im)) - t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x$46$im, -2.6e+26], t$95$3, If[LessEqual[x$46$im, -2.1e-153], N[(t$95$1 * N[Exp[N[(N[(N[(N[(y$46$re * N[(x$46$re * N[(x$46$re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -1.3e-295], t$95$3, If[LessEqual[x$46$im, 6e+72], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -2.60000000000000002e26 or -2.10000000000000004e-153 < x.im < -1.29999999999999993e-295

   1. Initial program 34.7%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in x.im around -inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. *-commutative 80.3%

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

      3. distribute-rgt-neg-in 80.3%

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

   6. Simplified 80.3%

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

   7. Taylor expanded in y.re around 0 35.1%

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

      1. +-commutative 35.1%

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

      2. unpow2 35.1%

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

      3. unpow2 35.1%

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

      4. hypot-def 87.7%

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

      5. hypot-def 35.1%

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

      6. unpow2 35.1%

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

      7. unpow2 35.1%

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

      8. +-commutative 35.1%

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

      9. unpow2 35.1%

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

      10. unpow2 35.1%

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

      11. hypot-def 87.7%

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

   9. Simplified 87.7%

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

   if -2.60000000000000002e26 < x.im < -2.10000000000000004e-153

   1. Initial program 61.8%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in x.im around -inf 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. associate-*r/ 73.9%

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

      5. unpow2 73.9%

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

      6. associate-*r* 73.9%

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

      7. unpow2 73.9%

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

      8. associate-*r* 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in y.re around 0 61.8%

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

      1. +-commutative 47.3%

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

      2. unpow2 47.3%

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

      3. unpow2 47.3%

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

      4. hypot-def 59.2%

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

      5. hypot-def 47.3%

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

      6. unpow2 47.3%

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

      7. unpow2 47.3%

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

      8. +-commutative 47.3%

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

      9. unpow2 47.3%

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

      10. unpow2 47.3%

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

      11. hypot-def 59.2%

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

   9. Simplified 76.8%

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

   if -1.29999999999999993e-295 < x.im < 6.00000000000000006e72

   1. Initial program 53.2%

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

   2. Taylor expanded in y.im around 0 71.0%

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

   3. Taylor expanded in y.re around 0 75.3%

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

   if 6.00000000000000006e72 < x.im

   1. Initial program 22.8%

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

   2. Taylor expanded in x.re around 0 67.3%

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

   3. Taylor expanded in x.re around 0 86.0%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.6 \cdot 10^{+26}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - \log \left(\frac{-1}{x.im}\right) \cdot y.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq -2.1 \cdot 10^{-153}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\left(\frac{y.re \cdot \left(x.re \cdot \left(x.re \cdot 0.5\right)\right)}{x.im \cdot x.im} - \log \left(\frac{-1}{x.im}\right) \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq -1.3 \cdot 10^{-295}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - \log \left(\frac{-1}{x.im}\right) \cdot y.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{+72}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log x.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 3: 72.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{x.im}\right) \cdot y.re\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - t_0} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq -2.8 \cdot 10^{-47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq -4 \cdot 10^{-159}:\\ \;\;\;\;e^{\left(\frac{y.re \cdot \left(x.re \cdot \left(x.re \cdot 0.5\right)\right)}{x.im \cdot x.im} - t_0\right) - t_1} \cdot \cos t_3\\ \mathbf{elif}\;x.im \leq -4.6 \cdot 10^{-302}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+69}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log x.im + t_3\right) \cdot e^{y.re \cdot \log x.im - t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (log (/ -1.0 x.im)) y.re))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2
         (*
          (exp (- (* (atan2 x.im x.re) (- y.im)) t_0))
          (cos (* y.im (log (hypot x.im x.re))))))
        (t_3 (* y.re (atan2 x.im x.re))))
   (if (<= x.im -2.8e-47)
     t_2
     (if (<= x.im -4e-159)
       (*
        (exp (- (- (/ (* y.re (* x.re (* x.re 0.5))) (* x.im x.im)) t_0) t_1))
        (cos t_3))
       (if (<= x.im -4.6e-302)
         t_2
         (if (<= x.im 4e+69)
           (exp (- (* y.re (log (sqrt (+ (* x.im x.im) (* x.re x.re))))) t_1))
           (*
            (cos (+ (* y.im (log x.im)) t_3))
            (exp (- (* y.re (log x.im)) t_1)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log((-1.0 / x_46_im)) * y_46_re;
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = exp(((atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	double t_3 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= -2.8e-47) {
		tmp = t_2;
	} else if (x_46_im <= -4e-159) {
		tmp = exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_1)) * cos(t_3);
	} else if (x_46_im <= -4.6e-302) {
		tmp = t_2;
	} else if (x_46_im <= 4e+69) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_1));
	} else {
		tmp = cos(((y_46_im * log(x_46_im)) + t_3)) * exp(((y_46_re * log(x_46_im)) - t_1));
	}
	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((-1.0 / x_46_im)) * y_46_re;
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = Math.exp(((Math.atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double t_3 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= -2.8e-47) {
		tmp = t_2;
	} else if (x_46_im <= -4e-159) {
		tmp = Math.exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_1)) * Math.cos(t_3);
	} else if (x_46_im <= -4.6e-302) {
		tmp = t_2;
	} else if (x_46_im <= 4e+69) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_1));
	} else {
		tmp = Math.cos(((y_46_im * Math.log(x_46_im)) + t_3)) * Math.exp(((y_46_re * Math.log(x_46_im)) - t_1));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log((-1.0 / x_46_im)) * y_46_re
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_2 = math.exp(((math.atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	t_3 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if x_46_im <= -2.8e-47:
		tmp = t_2
	elif x_46_im <= -4e-159:
		tmp = math.exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_1)) * math.cos(t_3)
	elif x_46_im <= -4.6e-302:
		tmp = t_2
	elif x_46_im <= 4e+69:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_1))
	else:
		tmp = math.cos(((y_46_im * math.log(x_46_im)) + t_3)) * math.exp(((y_46_re * math.log(x_46_im)) - t_1))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(log(Float64(-1.0 / x_46_im)) * y_46_re)
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = Float64(exp(Float64(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)) - t_0)) * cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))
	t_3 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_im <= -2.8e-47)
		tmp = t_2;
	elseif (x_46_im <= -4e-159)
		tmp = Float64(exp(Float64(Float64(Float64(Float64(y_46_re * Float64(x_46_re * Float64(x_46_re * 0.5))) / Float64(x_46_im * x_46_im)) - t_0) - t_1)) * cos(t_3));
	elseif (x_46_im <= -4.6e-302)
		tmp = t_2;
	elseif (x_46_im <= 4e+69)
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))) - t_1));
	else
		tmp = Float64(cos(Float64(Float64(y_46_im * log(x_46_im)) + t_3)) * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log((-1.0 / x_46_im)) * y_46_re;
	t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	t_2 = exp(((atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	t_3 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (x_46_im <= -2.8e-47)
		tmp = t_2;
	elseif (x_46_im <= -4e-159)
		tmp = exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_1)) * cos(t_3);
	elseif (x_46_im <= -4.6e-302)
		tmp = t_2;
	elseif (x_46_im <= 4e+69)
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_1));
	else
		tmp = cos(((y_46_im * log(x_46_im)) + t_3)) * exp(((y_46_re * log(x_46_im)) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -2.8e-47], t$95$2, If[LessEqual[x$46$im, -4e-159], N[(N[Exp[N[(N[(N[(N[(y$46$re * N[(x$46$re * N[(x$46$re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -4.6e-302], t$95$2, If[LessEqual[x$46$im, 4e+69], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -2.79999999999999993e-47 or -3.99999999999999995e-159 < x.im < -4.60000000000000004e-302

   1. Initial program 39.4%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in x.im around -inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. *-commutative 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in y.re around 0 39.7%

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

      1. +-commutative 39.7%

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

      2. unpow2 39.7%

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

      3. unpow2 39.7%

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

      4. hypot-def 88.8%

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

      5. hypot-def 39.7%

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

      6. unpow2 39.7%

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

      7. unpow2 39.7%

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

      8. +-commutative 39.7%

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

      9. unpow2 39.7%

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

      10. unpow2 39.7%

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

      11. hypot-def 88.8%

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

   9. Simplified 88.8%

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

   if -2.79999999999999993e-47 < x.im < -3.99999999999999995e-159

   1. Initial program 52.0%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x.im around -inf 68.5%

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

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. associate-*r/ 68.5%

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

      5. unpow2 68.5%

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

      6. associate-*r* 68.5%

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

      7. unpow2 68.5%

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

      8. associate-*r* 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in y.im around 0 64.6%

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

   if -4.60000000000000004e-302 < x.im < 4.0000000000000003e69

   1. Initial program 53.2%

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

   2. Taylor expanded in y.im around 0 71.0%

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

   3. Taylor expanded in y.re around 0 75.3%

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

   if 4.0000000000000003e69 < x.im

   1. Initial program 22.8%

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

   2. Taylor expanded in x.re around 0 67.3%

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

   3. Taylor expanded in x.re around 0 86.0%

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.8 \cdot 10^{-47}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - \log \left(\frac{-1}{x.im}\right) \cdot y.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq -4 \cdot 10^{-159}:\\ \;\;\;\;e^{\left(\frac{y.re \cdot \left(x.re \cdot \left(x.re \cdot 0.5\right)\right)}{x.im \cdot x.im} - \log \left(\frac{-1}{x.im}\right) \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -4.6 \cdot 10^{-302}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - \log \left(\frac{-1}{x.im}\right) \cdot y.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+69}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log x.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 4: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{x.im}\right) \cdot y.re\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - t_0} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;x.im \leq -6.8 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq -2.7 \cdot 10^{-157}:\\ \;\;\;\;e^{\left(\frac{y.re \cdot \left(x.re \cdot \left(x.re \cdot 0.5\right)\right)}{x.im \cdot x.im} - t_0\right) - t_1} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -6 \cdot 10^{-309}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq 0.00125:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (log (/ -1.0 x.im)) y.re))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2
         (*
          (exp (- (* (atan2 x.im x.re) (- y.im)) t_0))
          (cos (* y.im (log (hypot x.im x.re)))))))
   (if (<= x.im -6.8e-37)
     t_2
     (if (<= x.im -2.7e-157)
       (*
        (exp (- (- (/ (* y.re (* x.re (* x.re 0.5))) (* x.im x.im)) t_0) t_1))
        (cos (* y.re (atan2 x.im x.re))))
       (if (<= x.im -6e-309)
         t_2
         (if (<= x.im 0.00125)
           (exp (- (* y.re (log (sqrt (+ (* x.im x.im) (* x.re x.re))))) t_1))
           (exp (- (* y.re (log x.im)) t_1))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log((-1.0 / x_46_im)) * y_46_re;
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = exp(((atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (x_46_im <= -6.8e-37) {
		tmp = t_2;
	} else if (x_46_im <= -2.7e-157) {
		tmp = exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_1)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (x_46_im <= -6e-309) {
		tmp = t_2;
	} else if (x_46_im <= 0.00125) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_1));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_1));
	}
	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((-1.0 / x_46_im)) * y_46_re;
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = Math.exp(((Math.atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (x_46_im <= -6.8e-37) {
		tmp = t_2;
	} else if (x_46_im <= -2.7e-157) {
		tmp = Math.exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_1)) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (x_46_im <= -6e-309) {
		tmp = t_2;
	} else if (x_46_im <= 0.00125) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_1));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_1));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log((-1.0 / x_46_im)) * y_46_re
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_2 = math.exp(((math.atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if x_46_im <= -6.8e-37:
		tmp = t_2
	elif x_46_im <= -2.7e-157:
		tmp = math.exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_1)) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif x_46_im <= -6e-309:
		tmp = t_2
	elif x_46_im <= 0.00125:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_1))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_1))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(log(Float64(-1.0 / x_46_im)) * y_46_re)
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = Float64(exp(Float64(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)) - t_0)) * cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))
	tmp = 0.0
	if (x_46_im <= -6.8e-37)
		tmp = t_2;
	elseif (x_46_im <= -2.7e-157)
		tmp = Float64(exp(Float64(Float64(Float64(Float64(y_46_re * Float64(x_46_re * Float64(x_46_re * 0.5))) / Float64(x_46_im * x_46_im)) - t_0) - t_1)) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (x_46_im <= -6e-309)
		tmp = t_2;
	elseif (x_46_im <= 0.00125)
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))) - t_1));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log((-1.0 / x_46_im)) * y_46_re;
	t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	t_2 = exp(((atan2(x_46_im, x_46_re) * -y_46_im) - t_0)) * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (x_46_im <= -6.8e-37)
		tmp = t_2;
	elseif (x_46_im <= -2.7e-157)
		tmp = exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - t_0) - t_1)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (x_46_im <= -6e-309)
		tmp = t_2;
	elseif (x_46_im <= 0.00125)
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_1));
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -6.8e-37], t$95$2, If[LessEqual[x$46$im, -2.7e-157], N[(N[Exp[N[(N[(N[(N[(y$46$re * N[(x$46$re * N[(x$46$re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -6e-309], t$95$2, If[LessEqual[x$46$im, 0.00125], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -6.80000000000000037e-37 or -2.7e-157 < x.im < -6.000000000000001e-309

   1. Initial program 39.4%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in x.im around -inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. *-commutative 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in y.re around 0 39.7%

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

      1. +-commutative 39.7%

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

      2. unpow2 39.7%

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

      3. unpow2 39.7%

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

      4. hypot-def 88.8%

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

      5. hypot-def 39.7%

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

      6. unpow2 39.7%

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

      7. unpow2 39.7%

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

      8. +-commutative 39.7%

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

      9. unpow2 39.7%

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

      10. unpow2 39.7%

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

      11. hypot-def 88.8%

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

   9. Simplified 88.8%

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

   if -6.80000000000000037e-37 < x.im < -2.7e-157

   1. Initial program 52.0%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x.im around -inf 68.5%

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

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. associate-*r/ 68.5%

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

      5. unpow2 68.5%

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

      6. associate-*r* 68.5%

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

      7. unpow2 68.5%

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

      8. associate-*r* 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in y.im around 0 64.6%

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

   if -6.000000000000001e-309 < x.im < 0.00125000000000000003

   1. Initial program 51.2%

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

   2. Taylor expanded in y.im around 0 69.0%

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

   3. Taylor expanded in y.re around 0 70.7%

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

   if 0.00125000000000000003 < x.im

   1. Initial program 29.4%

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

   2. Taylor expanded in y.im around 0 63.8%

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

   3. Taylor expanded in y.re around 0 68.2%

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

   4. Taylor expanded in x.re around 0 86.8%

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

      1. *-commutative 86.8%

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

   6. Simplified 86.8%

      \[\leadsto e^{\color{blue}{\log x.im \cdot y.re} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.8 \cdot 10^{-37}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - \log \left(\frac{-1}{x.im}\right) \cdot y.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq -2.7 \cdot 10^{-157}:\\ \;\;\;\;e^{\left(\frac{y.re \cdot \left(x.re \cdot \left(x.re \cdot 0.5\right)\right)}{x.im \cdot x.im} - \log \left(\frac{-1}{x.im}\right) \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -6 \cdot 10^{-309}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right) - \log \left(\frac{-1}{x.im}\right) \cdot y.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 0.00125:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 5: 72.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{if}\;x.im \leq -7 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -1.06 \cdot 10^{-157}:\\ \;\;\;\;e^{\left(\frac{y.re \cdot \left(x.re \cdot \left(x.re \cdot 0.5\right)\right)}{x.im \cdot x.im} - \log \left(\frac{-1}{x.im}\right) \cdot y.re\right) - t_0} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 0.002:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (exp (- (* y.re (log (- x.im))) t_0))))
   (if (<= x.im -7e+56)
     t_1
     (if (<= x.im -1.06e-157)
       (*
        (exp
         (-
          (-
           (/ (* y.re (* x.re (* x.re 0.5))) (* x.im x.im))
           (* (log (/ -1.0 x.im)) y.re))
          t_0))
        (cos (* y.re (atan2 x.im x.re))))
       (if (<= x.im -2e-308)
         t_1
         (if (<= x.im 0.002)
           (exp (- (* y.re (log (sqrt (+ (* x.im x.im) (* x.re x.re))))) t_0))
           (exp (- (* y.re (log x.im)) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = exp(((y_46_re * log(-x_46_im)) - t_0));
	double tmp;
	if (x_46_im <= -7e+56) {
		tmp = t_1;
	} else if (x_46_im <= -1.06e-157) {
		tmp = exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - (log((-1.0 / x_46_im)) * y_46_re)) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (x_46_im <= -2e-308) {
		tmp = t_1;
	} else if (x_46_im <= 0.002) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_0));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = exp(((y_46re * log(-x_46im)) - t_0))
    if (x_46im <= (-7d+56)) then
        tmp = t_1
    else if (x_46im <= (-1.06d-157)) then
        tmp = exp(((((y_46re * (x_46re * (x_46re * 0.5d0))) / (x_46im * x_46im)) - (log(((-1.0d0) / x_46im)) * y_46re)) - t_0)) * cos((y_46re * atan2(x_46im, x_46re)))
    else if (x_46im <= (-2d-308)) then
        tmp = t_1
    else if (x_46im <= 0.002d0) then
        tmp = exp(((y_46re * log(sqrt(((x_46im * x_46im) + (x_46re * x_46re))))) - t_0))
    else
        tmp = exp(((y_46re * log(x_46im)) - t_0))
    end if
    code = tmp
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.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0));
	double tmp;
	if (x_46_im <= -7e+56) {
		tmp = t_1;
	} else if (x_46_im <= -1.06e-157) {
		tmp = Math.exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - (Math.log((-1.0 / x_46_im)) * y_46_re)) - t_0)) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (x_46_im <= -2e-308) {
		tmp = t_1;
	} else if (x_46_im <= 0.002) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_0));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.exp(((y_46_re * math.log(-x_46_im)) - t_0))
	tmp = 0
	if x_46_im <= -7e+56:
		tmp = t_1
	elif x_46_im <= -1.06e-157:
		tmp = math.exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - (math.log((-1.0 / x_46_im)) * y_46_re)) - t_0)) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif x_46_im <= -2e-308:
		tmp = t_1
	elif x_46_im <= 0.002:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_0))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0))
	tmp = 0.0
	if (x_46_im <= -7e+56)
		tmp = t_1;
	elseif (x_46_im <= -1.06e-157)
		tmp = Float64(exp(Float64(Float64(Float64(Float64(y_46_re * Float64(x_46_re * Float64(x_46_re * 0.5))) / Float64(x_46_im * x_46_im)) - Float64(log(Float64(-1.0 / x_46_im)) * y_46_re)) - t_0)) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (x_46_im <= -2e-308)
		tmp = t_1;
	elseif (x_46_im <= 0.002)
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))) - t_0));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = exp(((y_46_re * log(-x_46_im)) - t_0));
	tmp = 0.0;
	if (x_46_im <= -7e+56)
		tmp = t_1;
	elseif (x_46_im <= -1.06e-157)
		tmp = exp(((((y_46_re * (x_46_re * (x_46_re * 0.5))) / (x_46_im * x_46_im)) - (log((-1.0 / x_46_im)) * y_46_re)) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (x_46_im <= -2e-308)
		tmp = t_1;
	elseif (x_46_im <= 0.002)
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_0));
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -7e+56], t$95$1, If[LessEqual[x$46$im, -1.06e-157], N[(N[Exp[N[(N[(N[(N[(y$46$re * N[(x$46$re * N[(x$46$re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -2e-308], t$95$1, If[LessEqual[x$46$im, 0.002], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -6.99999999999999999e56 or -1.06e-157 < x.im < -1.9999999999999998e-308

   1. Initial program 31.0%

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

   2. Taylor expanded in y.im around 0 48.4%

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

   3. Taylor expanded in y.re around 0 56.8%

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

   4. Taylor expanded in x.im around -inf 82.5%

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

      1. mul-1-neg 82.5%

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

   6. Simplified 82.5%

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

   if -6.99999999999999999e56 < x.im < -1.06e-157

   1. Initial program 62.2%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in x.im around -inf 73.7%

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

      1. +-commutative 73.7%

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

      2. mul-1-neg 73.7%

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

      3. unsub-neg 73.7%

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

      4. associate-*r/ 73.7%

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

      5. unpow2 73.7%

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

      6. associate-*r* 73.7%

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

      7. unpow2 73.7%

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

      8. associate-*r* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in y.im around 0 73.8%

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

   if -1.9999999999999998e-308 < x.im < 2e-3

   1. Initial program 51.2%

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

   2. Taylor expanded in y.im around 0 69.0%

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

   3. Taylor expanded in y.re around 0 70.7%

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

   if 2e-3 < x.im

   1. Initial program 29.4%

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

   2. Taylor expanded in y.im around 0 63.8%

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

   3. Taylor expanded in y.re around 0 68.2%

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

   4. Taylor expanded in x.re around 0 86.8%

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

      1. *-commutative 86.8%

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

   6. Simplified 86.8%

      \[\leadsto e^{\color{blue}{\log x.im \cdot y.re} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -7 \cdot 10^{+56}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq -1.06 \cdot 10^{-157}:\\ \;\;\;\;e^{\left(\frac{y.re \cdot \left(x.re \cdot \left(x.re \cdot 0.5\right)\right)}{x.im \cdot x.im} - \log \left(\frac{-1}{x.im}\right) \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -2 \cdot 10^{-308}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 0.002:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 6: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -3.2 \cdot 10^{-300}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{elif}\;x.im \leq 0.00017:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -3.2e-300)
     (exp (- (* y.re (log (- x.im))) t_0))
     (if (<= x.im 0.00017)
       (exp (- (* y.re (log (sqrt (+ (* x.im x.im) (* x.re x.re))))) t_0))
       (exp (- (* y.re (log x.im)) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -3.2e-300) {
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	} else if (x_46_im <= 0.00017) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_0));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46im <= (-3.2d-300)) then
        tmp = exp(((y_46re * log(-x_46im)) - t_0))
    else if (x_46im <= 0.00017d0) then
        tmp = exp(((y_46re * log(sqrt(((x_46im * x_46im) + (x_46re * x_46re))))) - t_0))
    else
        tmp = exp(((y_46re * log(x_46im)) - t_0))
    end if
    code = tmp
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.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -3.2e-300) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0));
	} else if (x_46_im <= 0.00017) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_0));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -3.2e-300:
		tmp = math.exp(((y_46_re * math.log(-x_46_im)) - t_0))
	elif x_46_im <= 0.00017:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_0))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -3.2e-300)
		tmp = exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0));
	elseif (x_46_im <= 0.00017)
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))) - t_0));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -3.2e-300)
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	elseif (x_46_im <= 0.00017)
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_0));
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -3.2e-300], N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[x$46$im, 0.00017], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.20000000000000021e-300

   1. Initial program 41.9%

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

   2. Taylor expanded in y.im around 0 57.3%

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

   3. Taylor expanded in y.re around 0 60.4%

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

   4. Taylor expanded in x.im around -inf 75.6%

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

      1. mul-1-neg 75.6%

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

   6. Simplified 75.6%

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

   if -3.20000000000000021e-300 < x.im < 1.7e-4

   1. Initial program 51.2%

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

   2. Taylor expanded in y.im around 0 69.0%

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

   3. Taylor expanded in y.re around 0 70.7%

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

   if 1.7e-4 < x.im

   1. Initial program 29.4%

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

   2. Taylor expanded in y.im around 0 63.8%

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

   3. Taylor expanded in y.re around 0 68.2%

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

   4. Taylor expanded in x.re around 0 86.8%

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

      1. *-commutative 86.8%

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

   6. Simplified 86.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.2 \cdot 10^{-300}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 0.00017:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 7: 71.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -5e-310)
     (exp (- (* y.re (log (- x.im))) t_0))
     (exp (- (* y.re (log x.im)) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -5e-310) {
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46im <= (-5d-310)) then
        tmp = exp(((y_46re * log(-x_46im)) - t_0))
    else
        tmp = exp(((y_46re * log(x_46im)) - t_0))
    end if
    code = tmp
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.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -5e-310) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -5e-310:
		tmp = math.exp(((y_46_re * math.log(-x_46_im)) - t_0))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -5e-310)
		tmp = exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -5e-310)
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -5e-310], N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.9%

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

   2. Taylor expanded in y.im around 0 57.3%

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

   3. Taylor expanded in y.re around 0 60.4%

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

   4. Taylor expanded in x.im around -inf 75.6%

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

      1. mul-1-neg 75.6%

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

   6. Simplified 75.6%

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

   if -4.999999999999985e-310 < x.im

   1. Initial program 39.5%

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

   2. Taylor expanded in y.im around 0 66.2%

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

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

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

   4. Taylor expanded in x.re around 0 74.1%

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

      1. *-commutative 74.1%

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

   6. Simplified 74.1%

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

4. Final simplification 74.8%

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

Alternative 8: 53.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -9 \cdot 10^{-309}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -9e-309)
     (exp (- (* y.re (log x.im)) t_0))
     (exp (- (* y.re (log x.re)) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -9e-309) {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	} else {
		tmp = exp(((y_46_re * log(x_46_re)) - t_0));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46re <= (-9d-309)) then
        tmp = exp(((y_46re * log(x_46im)) - t_0))
    else
        tmp = exp(((y_46re * log(x_46re)) - t_0))
    end if
    code = tmp
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.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -9e-309) {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_0));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_re)) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_re <= -9e-309:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_0))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_re)) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -9e-309)
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_re <= -9e-309)
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	else
		tmp = exp(((y_46_re * log(x_46_re)) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -9e-309], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -9.0000000000000021e-309

   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. Taylor expanded in y.im around 0 63.7%

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

   3. Taylor expanded in y.re around 0 65.1%

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

   4. Taylor expanded in x.re around 0 40.5%

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

      1. *-commutative 40.5%

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

   6. Simplified 40.5%

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

   if -9.0000000000000021e-309 < x.re

   1. Initial program 34.1%

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

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

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

   3. Taylor expanded in y.re around 0 64.4%

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

   4. Taylor expanded in x.im around 0 67.1%

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

      1. *-commutative 67.1%

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

   6. Simplified 67.1%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -9 \cdot 10^{-309}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 9: 35.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (exp (- (* y.re (log x.im)) (* (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 exp(((y_46_re * log(x_46_im)) - (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 = exp(((y_46re * log(x_46im)) - (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 Math.exp(((y_46_re * Math.log(x_46_im)) - (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 math.exp(((y_46_re * math.log(x_46_im)) - (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 exp(Float64(Float64(y_46_re * log(x_46_im)) - 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 = exp(((y_46_re * log(x_46_im)) - (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[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 40.7%

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

2. Taylor expanded in y.im around 0 61.7%

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

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

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

4. Taylor expanded in x.re around 0 36.8%

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

   1. *-commutative 36.8%

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

6. Simplified 36.8%

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

7. Final simplification 36.8%

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

Reproduce

herbie shell --seed 2023268 
(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)))))