powComplex, real part

Percentage Accurate: 40.7% → 79.8%

Time: 13.5s

Alternatives: 13

Speedup: 4.9×

• 35.1% 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: 40.7% 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: 79.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_4 := e^{t\_3 \cdot y.re - t\_1}\\ t_5 := \cos \left(\mathsf{fma}\left(t\_0, y.im, -t\_2\right)\right)\\ \mathbf{if}\;t\_4 \cdot \cos \left(t\_3 \cdot y.im + t\_2\right) \leq 5 \cdot 10^{+31}:\\ \;\;\;\;t\_4 \cdot \frac{1}{\frac{t\_5}{t\_5 \cdot \cos \left(\mathsf{fma}\left(t\_0, y.im, t\_2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - t\_1} \cdot \cos t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 0.5 (log (fma x.im x.im (* x.re x.re)))))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_4 (exp (- (* t_3 y.re) t_1)))
        (t_5 (cos (fma t_0 y.im (- t_2)))))
   (if (<= (* t_4 (cos (+ (* t_3 y.im) t_2))) 5e+31)
     (* t_4 (/ 1.0 (/ t_5 (* t_5 (cos (fma t_0 y.im t_2))))))
     (* (exp (- (* y.re (log (hypot x.re x.im))) t_1)) (cos t_2)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.5 * log(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_4 = exp(((t_3 * y_46_re) - t_1));
	double t_5 = cos(fma(t_0, y_46_im, -t_2));
	double tmp;
	if ((t_4 * cos(((t_3 * y_46_im) + t_2))) <= 5e+31) {
		tmp = t_4 * (1.0 / (t_5 / (t_5 * cos(fma(t_0, y_46_im, t_2)))));
	} else {
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - t_1)) * cos(t_2);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(0.5 * log(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_4 = exp(Float64(Float64(t_3 * y_46_re) - t_1))
	t_5 = cos(fma(t_0, y_46_im, Float64(-t_2)))
	tmp = 0.0
	if (Float64(t_4 * cos(Float64(Float64(t_3 * y_46_im) + t_2))) <= 5e+31)
		tmp = Float64(t_4 * Float64(1.0 / Float64(t_5 / Float64(t_5 * cos(fma(t_0, y_46_im, t_2))))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - t_1)) * cos(t_2));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(0.5 * N[Log[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(t$95$0 * y$46$im + (-t$95$2)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$4 * N[Cos[N[(N[(t$95$3 * y$46$im), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+31], N[(t$95$4 * N[(1.0 / N[(t$95$5 / N[(t$95$5 * N[Cos[N[(t$95$0 * y$46$im + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 5.00000000000000027e31

   1. Initial program 87.9%

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

   4. Applied egg-rr 87.9%

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

   if 5.00000000000000027e31 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 13.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 48.2

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

   5. Simplified 48.2%

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

      1. lower-hypot.f64 75.0

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

   7. Applied egg-rr 75.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 5 \cdot 10^{+31}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \frac{1}{\frac{\cos \left(\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right), y.im, -y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}{\cos \left(\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right), y.im, -y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \cos \left(\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t\_2 \cdot y.re - t\_0} \cdot \cos \left(t\_2 \cdot y.im + t\_1\right)\\ \mathbf{if}\;t\_3 \leq 0.9830716708561194:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - t\_0} \cdot \cos t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (* (exp (- (* t_2 y.re) t_0)) (cos (+ (* t_2 y.im) t_1)))))
   (if (<= t_3 0.9830716708561194)
     t_3
     (* (exp (- (* y.re (log (hypot x.re x.im))) t_0)) (cos t_1)))))

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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0)) * cos(((t_2 * y_46_im) + t_1));
	double tmp;
	if (t_3 <= 0.9830716708561194) {
		tmp = t_3;
	} else {
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - t_0)) * cos(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.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = Math.exp(((t_2 * y_46_re) - t_0)) * Math.cos(((t_2 * y_46_im) + t_1));
	double tmp;
	if (t_3 <= 0.9830716708561194) {
		tmp = t_3;
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im))) - t_0)) * Math.cos(t_1);
	}
	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 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	t_3 = math.exp(((t_2 * y_46_re) - t_0)) * math.cos(((t_2 * y_46_im) + t_1))
	tmp = 0
	if t_3 <= 0.9830716708561194:
		tmp = t_3
	else:
		tmp = math.exp(((y_46_re * math.log(math.hypot(x_46_re, x_46_im))) - t_0)) * math.cos(t_1)
	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 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * cos(Float64(Float64(t_2 * y_46_im) + t_1)))
	tmp = 0.0
	if (t_3 <= 0.9830716708561194)
		tmp = t_3;
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - t_0)) * cos(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 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	t_3 = exp(((t_2 * y_46_re) - t_0)) * cos(((t_2 * y_46_im) + t_1));
	tmp = 0.0;
	if (t_3 <= 0.9830716708561194)
		tmp = t_3;
	else
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - t_0)) * cos(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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.9830716708561194], t$95$3, N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 0.98307167085611935

   1. Initial program 82.6%

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

   if 0.98307167085611935 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 28.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 57.1

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

   5. Simplified 57.1%

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

      1. lower-hypot.f64 79.3

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

   7. Applied egg-rr 79.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 0.9830716708561194:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -6.2 \cdot 10^{+118}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - t\_0} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -6.2e+118)
     (exp (- (* y.re (log (- x.im))) t_0))
     (*
      (exp (- (* y.re (log (hypot x.re x.im))) t_0))
      (cos (* y.re (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -6.2e+118) {
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	} else {
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -6.2e+118) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im))) - t_0)) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -6.2e+118:
		tmp = math.exp(((y_46_re * math.log(-x_46_im)) - t_0))
	else:
		tmp = math.exp(((y_46_re * math.log(math.hypot(x_46_re, x_46_im))) - t_0)) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -6.2e+118)
		tmp = exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - t_0)) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -6.2e+118)
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	else
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -6.2e+118], N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - 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] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -6.19999999999999973e118

   1. Initial program 17.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 57.2

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

   5. Simplified 57.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 70.0%

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

   9. Taylor expanded in x.im around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 85.0

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

   11. Simplified 85.0%

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

   if -6.19999999999999973e118 < x.im

   1. Initial program 48.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.7

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

   5. Simplified 62.7%

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

      1. lower-hypot.f64 78.8

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

   7. Applied egg-rr 78.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.2 \cdot 10^{+118}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -1.1 \cdot 10^{-295}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - t\_0}\\ \mathbf{elif}\;x.im \leq 4.6 \cdot 10^{-257}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \mathsf{fma}\left({\tan^{-1}_* \frac{x.im}{x.re}}^{2}, -0.5 \cdot \left(y.re \cdot y.re\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - t\_0}\\ \end{array} \end{array} \]

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 -1.1e-295)
     (exp (- (* y.re (log (- x.im))) t_0))
     (if (<= x.im 4.6e-257)
       (*
        (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re)
        (fma (pow (atan2 x.im x.re) 2.0) (* -0.5 (* y.re y.re)) 1.0))
       (*
        (cos (* y.re (atan2 x.im x.re)))
        (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 <= -1.1e-295) {
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	} else if (x_46_im <= 4.6e-257) {
		tmp = pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re) * fma(pow(atan2(x_46_im, x_46_re), 2.0), (-0.5 * (y_46_re * y_46_re)), 1.0);
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * exp(((y_46_re * 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 <= -1.1e-295)
		tmp = exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0));
	elseif (x_46_im <= 4.6e-257)
		tmp = Float64((sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re) * fma((atan(x_46_im, x_46_re) ^ 2.0), Float64(-0.5 * Float64(y_46_re * y_46_re)), 1.0));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -1.1e-295], N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[x$46$im, 4.6e-257], N[(N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision] * N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.1000000000000001e-295

   1. Initial program 43.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.8

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

   5. Simplified 62.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 65.3%

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

   9. Taylor expanded in x.im around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 71.8

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

   11. Simplified 71.8%

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

   if -1.1000000000000001e-295 < x.im < 4.6e-257

   1. Initial program 44.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 63.8

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

   5. Simplified 63.8%

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

      1. lower-hypot.f64 69.8

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

   7. Applied egg-rr 69.8%

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

   8. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.1

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

   10. Simplified 76.1%

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

   11. Taylor expanded in y.im around 0

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

       1. lower-pow.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 70.0

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

   13. Simplified 70.0%

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

   if 4.6e-257 < x.im

   1. Initial program 43.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 60.5

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

   5. Simplified 60.5%

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

      1. lower-hypot.f64 82.5

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

   7. Applied egg-rr 82.5%

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

   8. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 77.3

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

   10. Simplified 77.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.1 \cdot 10^{-295}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.6 \cdot 10^{-257}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \mathsf{fma}\left({\tan^{-1}_* \frac{x.im}{x.re}}^{2}, -0.5 \cdot \left(y.re \cdot y.re\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.re - t\_0}\\ \end{array} \end{array} \]

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 -5e-310)
     (exp (- (* y.re (log (- x.re))) t_0))
     (* (cos (* y.re (atan2 x.im x.re))) (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 <= -5e-310) {
		tmp = exp(((y_46_re * log(-x_46_re)) - t_0));
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * 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 <= (-5d-310)) then
        tmp = exp(((y_46re * log(-x_46re)) - t_0))
    else
        tmp = cos((y_46re * atan2(x_46im, x_46re))) * 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 <= -5e-310) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_re)) - t_0));
	} else {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * 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 <= -5e-310:
		tmp = math.exp(((y_46_re * math.log(-x_46_re)) - t_0))
	else:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * 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 <= -5e-310)
		tmp = exp(Float64(Float64(y_46_re * log(Float64(-x_46_re))) - t_0));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * 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 <= -5e-310)
		tmp = exp(((y_46_re * log(-x_46_re)) - t_0));
	else
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * 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, -5e-310], N[Exp[N[(N[(y$46$re * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 42.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 58.3

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

   5. Simplified 58.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 60.7%

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

   9. Taylor expanded in x.re around -inf

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 73.3

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

   11. Simplified 73.3%

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

   if -4.999999999999985e-310 < x.re

   1. Initial program 44.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 65.9

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

   5. Simplified 65.9%

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

      1. lower-hypot.f64 80.8

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

   7. Applied egg-rr 80.8%

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

   8. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 76.8

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

   10. Simplified 76.8%

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

4. Final simplification 74.9%

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

Alternative 6: 70.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\\ \mathbf{if}\;x.re \leq -7.2 \cdot 10^{-174}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq 1.25 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left({\tan^{-1}_* \frac{x.im}{x.re}}^{2}, -0.5 \cdot \left(y.re \cdot y.re\right), 1\right) \cdot e^{t\_0}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(y.re, \log x.re, t\_0\right)}\\ \end{array} \end{array} \]

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 -7.2e-174)
     (exp (- (* y.re (log (- x.re))) (* (atan2 x.im x.re) y.im)))
     (if (<= x.re 1.25e-155)
       (*
        (fma (pow (atan2 x.im x.re) 2.0) (* -0.5 (* y.re y.re)) 1.0)
        (exp t_0))
       (exp (fma 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 <= -7.2e-174) {
		tmp = exp(((y_46_re * log(-x_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (x_46_re <= 1.25e-155) {
		tmp = fma(pow(atan2(x_46_im, x_46_re), 2.0), (-0.5 * (y_46_re * y_46_re)), 1.0) * exp(t_0);
	} else {
		tmp = exp(fma(y_46_re, 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) * Float64(-y_46_im))
	tmp = 0.0
	if (x_46_re <= -7.2e-174)
		tmp = exp(Float64(Float64(y_46_re * log(Float64(-x_46_re))) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	elseif (x_46_re <= 1.25e-155)
		tmp = Float64(fma((atan(x_46_im, x_46_re) ^ 2.0), Float64(-0.5 * Float64(y_46_re * y_46_re)), 1.0) * exp(t_0));
	else
		tmp = exp(fma(y_46_re, log(x_46_re), t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]}, If[LessEqual[x$46$re, -7.2e-174], N[Exp[N[(N[(y$46$re * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x$46$re, 1.25e-155], N[(N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], N[Exp[N[(y$46$re * N[Log[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -7.19999999999999997e-174

   1. Initial program 44.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 58.5

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

   5. Simplified 58.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 63.5%

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

   9. Taylor expanded in x.re around -inf

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 78.7

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

   11. Simplified 78.7%

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

   if -7.19999999999999997e-174 < x.re < 1.25e-155

   1. Initial program 33.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 56.3

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

   5. Simplified 56.3%

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

      1. lower-hypot.f64 73.9

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

   7. Applied egg-rr 73.9%

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

   8. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.1

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

   10. Simplified 76.1%

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

   11. Taylor expanded in y.re around 0

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

       1. lower-exp.f64 N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 67.9

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

   13. Simplified 67.9%

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

   if 1.25e-155 < x.re

   1. Initial program 48.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 69.7

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

   5. Simplified 69.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 64.0%

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

   9. Taylor expanded in x.re around inf

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. log-rec N/A

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

       4. mul-1-neg N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-log.f64 N/A

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

       10. *-commutative N/A

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

       11. distribute-rgt-neg-in N/A

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

       12. neg-mul-1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-atan2.f64 N/A

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

       15. neg-mul-1 N/A

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

       16. lower-neg.f64 72.0

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

   11. Simplified 72.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7.2 \cdot 10^{-174}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq 1.25 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left({\tan^{-1}_* \frac{x.im}{x.re}}^{2}, -0.5 \cdot \left(y.re \cdot y.re\right), 1\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(y.re, \log x.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 42.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 58.3

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

   5. Simplified 58.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 60.7%

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

   9. Taylor expanded in x.re around -inf

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 73.3

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

   11. Simplified 73.3%

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

   if -4.999999999999985e-310 < x.re

   1. Initial program 44.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 56.9%

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

   9. Taylor expanded in x.re around inf

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. log-rec N/A

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

       4. mul-1-neg N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-log.f64 N/A

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

       10. *-commutative N/A

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

       11. distribute-rgt-neg-in N/A

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

       12. neg-mul-1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-atan2.f64 N/A

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

       15. neg-mul-1 N/A

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

       16. lower-neg.f64 68.4

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

   11. Simplified 68.4%

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

4. Final simplification 71.0%

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

Alternative 8: 71.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+31}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.6e+31)
   (* (cos (* y.re (atan2 x.im x.re))) (pow x.re y.re))
   (if (<= y.re 6.2e-7)
     (exp (* (atan2 x.im x.re) (- y.im)))
     (exp (* y.re (log (sqrt (fma x.im x.im (* x.re x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.6e+31) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(x_46_re, y_46_re);
	} else if (y_46_re <= 6.2e-7) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = exp((y_46_re * log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5.6e+31)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (x_46_re ^ y_46_re));
	elseif (y_46_re <= 6.2e-7)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = exp(Float64(y_46_re * log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.6e+31], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-7], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.60000000000000034e31

   1. Initial program 41.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 80.5

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

   5. Simplified 80.5%

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

      1. lower-hypot.f64 82.4

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

   7. Applied egg-rr 82.4%

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

   8. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 45.2

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

   10. Simplified 45.2%

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

   11. Taylor expanded in y.im around 0

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

       1. lower-pow.f64 68.9

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

   13. Simplified 68.9%

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

   if -5.60000000000000034e31 < y.re < 6.1999999999999999e-7

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 49.8

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

   5. Simplified 49.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 50.2%

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

   9. Taylor expanded in y.re around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-atan2.f64 N/A

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

       7. neg-mul-1 N/A

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

       8. lower-neg.f64 75.9

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

   11. Simplified 75.9%

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

       1. lift-atan2.f64 N/A

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

       2. rem-log-exp N/A

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

       3. lift-neg.f64 N/A

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

       4. rem-log-exp N/A

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

       5. lift-*.f64 N/A

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

       6. lift-exp.f64 N/A

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

       7. *-rgt-identity 75.9

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

   13. Applied egg-rr 75.9%

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

   if 6.1999999999999999e-7 < y.re

   1. Initial program 49.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 70.3

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

   5. Simplified 70.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 69.1%

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

   9. Taylor expanded in y.re around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 65.1

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

   11. Simplified 65.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+31}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}\\ \mathbf{if}\;y.re \leq -0.00019:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.re (log (sqrt (fma x.im x.im (* x.re x.re))))))))
   (if (<= y.re -0.00019)
     t_0
     (if (<= y.re 6.2e-7) (exp (* (atan2 x.im x.re) (- y.im))) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_re * log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))))));
	double tmp;
	if (y_46_re <= -0.00019) {
		tmp = t_0;
	} else if (y_46_re <= 6.2e-7) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_re * log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))))))
	tmp = 0.0
	if (y_46_re <= -0.00019)
		tmp = t_0;
	elseif (y_46_re <= 6.2e-7)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -0.00019], t$95$0, If[LessEqual[y$46$re, 6.2e-7], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.9000000000000001e-4 or 6.1999999999999999e-7 < y.re

   1. Initial program 47.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 73.5

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

   5. Simplified 73.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 68.0%

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

   9. Taylor expanded in y.re around inf

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

       1. lower-*.f64 N/A

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

       2. lower-log.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 64.3

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

   11. Simplified 64.3%

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

   if -1.9000000000000001e-4 < y.re < 6.1999999999999999e-7

   1. Initial program 39.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 49.4

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

   5. Simplified 49.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 49.4%

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

   9. Taylor expanded in y.re around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-atan2.f64 N/A

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

       7. neg-mul-1 N/A

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

       8. lower-neg.f64 76.8

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

   11. Simplified 76.8%

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

       1. lift-atan2.f64 N/A

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

       2. rem-log-exp N/A

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

       3. lift-neg.f64 N/A

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

       4. rem-log-exp N/A

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

       5. lift-*.f64 N/A

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

       6. lift-exp.f64 N/A

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

       7. *-rgt-identity 76.8

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

   13. Applied egg-rr 76.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00019:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{\left(x.re \cdot x.re\right) \cdot -0.5}{x.im} - x.im\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -5.4 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (- (/ (* (* x.re x.re) -0.5) x.im) x.im) y.re)))
   (if (<= y.re -5.4e+31)
     t_0
     (if (<= y.re 6.2e-7) (exp (* (atan2 x.im x.re) (- y.im))) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(((((x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -5.4e+31) {
		tmp = t_0;
	} else if (y_46_re <= 6.2e-7) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = 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 = ((((x_46re * x_46re) * (-0.5d0)) / x_46im) - x_46im) ** y_46re
    if (y_46re <= (-5.4d+31)) then
        tmp = t_0
    else if (y_46re <= 6.2d-7) then
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    else
        tmp = 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.pow(((((x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -5.4e+31) {
		tmp = t_0;
	} else if (y_46_re <= 6.2e-7) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(((((x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -5.4e+31:
		tmp = t_0
	elif y_46_re <= 6.2e-7:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -5.4e+31)
		tmp = t_0;
	elseif (y_46_re <= 6.2e-7)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = 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 = ((((x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -5.4e+31)
		tmp = t_0;
	elseif (y_46_re <= 6.2e-7)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = 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[Power[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * -0.5), $MachinePrecision] / x$46$im), $MachinePrecision] - x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -5.4e+31], t$95$0, If[LessEqual[y$46$re, 6.2e-7], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.39999999999999971e31 or 6.1999999999999999e-7 < y.re

   1. Initial program 46.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 74.4

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

   5. Simplified 74.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 68.1%

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

   9. Taylor expanded in x.im around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 21.8

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

   11. Simplified 21.8%

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

   12. Taylor expanded in y.im around 0

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(\left(x.im + \frac{1}{2} \cdot \frac{{x.re}^{2}}{x.im}\right)\right)\right)}^{y.re}} \cdot 1 \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-neg-in N/A

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

       4. distribute-lft-neg-in N/A

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

       5. metadata-eval N/A

          \[\leadsto {\left(\color{blue}{\frac{-1}{2}} \cdot \frac{{x.re}^{2}}{x.im} + \left(\mathsf{neg}\left(x.im\right)\right)\right)}^{y.re} \cdot 1 \]

       6. sub-neg N/A

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

       7. lower--.f64 N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

          \[\leadsto {\left(\frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {x.re}^{2}}{x.im} - x.im\right)}^{y.re} \cdot 1 \]

       10. distribute-lft-neg-in N/A

           \[\leadsto {\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2} \cdot {x.re}^{2}\right)}}{x.im} - x.im\right)}^{y.re} \cdot 1 \]

       11. lower-/.f64 N/A

           \[\leadsto {\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {x.re}^{2}\right)}{x.im}} - x.im\right)}^{y.re} \cdot 1 \]

       12. *-commutative N/A

           \[\leadsto {\left(\frac{\mathsf{neg}\left(\color{blue}{{x.re}^{2} \cdot \frac{1}{2}}\right)}{x.im} - x.im\right)}^{y.re} \cdot 1 \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto {\left(\frac{\color{blue}{{x.re}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x.im} - x.im\right)}^{y.re} \cdot 1 \]

       14. metadata-eval N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 54.8

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

   14. Simplified 54.8%

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

   if -5.39999999999999971e31 < y.re < 6.1999999999999999e-7

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 49.8

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

   5. Simplified 49.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 50.2%

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

   9. Taylor expanded in y.re around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-atan2.f64 N/A

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

       7. neg-mul-1 N/A

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

       8. lower-neg.f64 75.9

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

   11. Simplified 75.9%

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

       1. lift-atan2.f64 N/A

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

       2. rem-log-exp N/A

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

       3. lift-neg.f64 N/A

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

       4. rem-log-exp N/A

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

       5. lift-*.f64 N/A

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

       6. lift-exp.f64 N/A

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

       7. *-rgt-identity 75.9

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

   13. Applied egg-rr 75.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.4 \cdot 10^{+31}:\\ \;\;\;\;{\left(\frac{\left(x.re \cdot x.re\right) \cdot -0.5}{x.im} - x.im\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\left(x.re \cdot x.re\right) \cdot -0.5}{x.im} - x.im\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{\left(x.re \cdot x.re\right) \cdot -0.5}{x.im} - x.im\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -3.9 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.45 \cdot 10^{-8}:\\ \;\;\;\;1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (- (/ (* (* x.re x.re) -0.5) x.im) x.im) y.re)))
   (if (<= y.re -3.9e-47)
     t_0
     (if (<= y.re 2.45e-8) (- 1.0 (* (atan2 x.im x.re) y.im)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(((((x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -3.9e-47) {
		tmp = t_0;
	} else if (y_46_re <= 2.45e-8) {
		tmp = 1.0 - (atan2(x_46_im, x_46_re) * y_46_im);
	} else {
		tmp = 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 = ((((x_46re * x_46re) * (-0.5d0)) / x_46im) - x_46im) ** y_46re
    if (y_46re <= (-3.9d-47)) then
        tmp = t_0
    else if (y_46re <= 2.45d-8) then
        tmp = 1.0d0 - (atan2(x_46im, x_46re) * y_46im)
    else
        tmp = 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.pow(((((x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -3.9e-47) {
		tmp = t_0;
	} else if (y_46_re <= 2.45e-8) {
		tmp = 1.0 - (Math.atan2(x_46_im, x_46_re) * y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(((((x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -3.9e-47:
		tmp = t_0
	elif y_46_re <= 2.45e-8:
		tmp = 1.0 - (math.atan2(x_46_im, x_46_re) * y_46_im)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -3.9e-47)
		tmp = t_0;
	elseif (y_46_re <= 2.45e-8)
		tmp = Float64(1.0 - Float64(atan(x_46_im, x_46_re) * y_46_im));
	else
		tmp = 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 = ((((x_46_re * x_46_re) * -0.5) / x_46_im) - x_46_im) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -3.9e-47)
		tmp = t_0;
	elseif (y_46_re <= 2.45e-8)
		tmp = 1.0 - (atan2(x_46_im, x_46_re) * y_46_im);
	else
		tmp = 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[Power[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * -0.5), $MachinePrecision] / x$46$im), $MachinePrecision] - x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -3.9e-47], t$95$0, If[LessEqual[y$46$re, 2.45e-8], N[(1.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.89999999999999978e-47 or 2.4500000000000001e-8 < y.re

   1. Initial program 45.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 71.6

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

   5. Simplified 71.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 66.4%

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

   9. Taylor expanded in x.im around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 22.4

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

   11. Simplified 22.4%

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

   12. Taylor expanded in y.im around 0

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(\left(x.im + \frac{1}{2} \cdot \frac{{x.re}^{2}}{x.im}\right)\right)\right)}^{y.re}} \cdot 1 \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-neg-in N/A

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

       4. distribute-lft-neg-in N/A

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

       5. metadata-eval N/A

          \[\leadsto {\left(\color{blue}{\frac{-1}{2}} \cdot \frac{{x.re}^{2}}{x.im} + \left(\mathsf{neg}\left(x.im\right)\right)\right)}^{y.re} \cdot 1 \]

       6. sub-neg N/A

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

       7. lower--.f64 N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

          \[\leadsto {\left(\frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {x.re}^{2}}{x.im} - x.im\right)}^{y.re} \cdot 1 \]

       10. distribute-lft-neg-in N/A

           \[\leadsto {\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2} \cdot {x.re}^{2}\right)}}{x.im} - x.im\right)}^{y.re} \cdot 1 \]

       11. lower-/.f64 N/A

           \[\leadsto {\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {x.re}^{2}\right)}{x.im}} - x.im\right)}^{y.re} \cdot 1 \]

       12. *-commutative N/A

           \[\leadsto {\left(\frac{\mathsf{neg}\left(\color{blue}{{x.re}^{2} \cdot \frac{1}{2}}\right)}{x.im} - x.im\right)}^{y.re} \cdot 1 \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto {\left(\frac{\color{blue}{{x.re}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x.im} - x.im\right)}^{y.re} \cdot 1 \]

       14. metadata-eval N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 52.7

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

   14. Simplified 52.7%

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

   if -3.89999999999999978e-47 < y.re < 2.4500000000000001e-8

   1. Initial program 41.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 50.0

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

   5. Simplified 50.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 50.0%

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

   9. Taylor expanded in y.re around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-atan2.f64 N/A

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

       7. neg-mul-1 N/A

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

       8. lower-neg.f64 78.5

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

   11. Simplified 78.5%

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

   12. Taylor expanded in y.im around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-atan2.f64 47.4

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

   14. Simplified 47.4%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{-47}:\\ \;\;\;\;{\left(\frac{\left(x.re \cdot x.re\right) \cdot -0.5}{x.im} - x.im\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.45 \cdot 10^{-8}:\\ \;\;\;\;1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\left(x.re \cdot x.re\right) \cdot -0.5}{x.im} - x.im\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.9% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0 - (atan2(x_46im, x_46re) * y_46im)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 - (atan2(x_46_im, x_46_re) * y_46_im);
end

Wolfram

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

Derivation

1. Initial program 43.5%

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

3. Taylor expanded in y.im around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-atan2.f64 61.8

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

5. Simplified 61.8%

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

6. Taylor expanded in y.re around 0

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

8. Simplified 59.0%

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

9. Taylor expanded in y.re around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-rgt-neg-in N/A

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

    4. neg-mul-1 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-atan2.f64 N/A

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

    7. neg-mul-1 N/A

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

    8. lower-neg.f64 49.3

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

11. Simplified 49.3%

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

12. Taylor expanded in y.im around 0

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

    1. neg-mul-1 N/A

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

    2. unsub-neg N/A

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

    3. lower--.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-atan2.f64 23.2

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

14. Simplified 23.2%

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

15. Final simplification 23.2%

    \[\leadsto 1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im \]
16. Add Preprocessing

Alternative 13: 25.8% accurate, 680.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 43.5%

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

3. Taylor expanded in y.im around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-atan2.f64 61.8

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

5. Simplified 61.8%

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

6. Taylor expanded in y.re around 0

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

8. Simplified 59.0%

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

9. Taylor expanded in y.re around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-rgt-neg-in N/A

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

    4. neg-mul-1 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-atan2.f64 N/A

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

    7. neg-mul-1 N/A

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

    8. lower-neg.f64 49.3

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

11. Simplified 49.3%

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

12. Taylor expanded in y.im around 0

    \[\leadsto \color{blue}{1} \]
13. Step-by-step derivation

14. Simplified 23.0%

    \[\leadsto \color{blue}{1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))))