powComplex, imaginary part

Percentage Accurate: 41.3% → 77.3%

Time: 22.1s

Alternatives: 16

Speedup: 3.7×

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)))
    (sin (+ (* 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))) * sin(((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))) * sin(((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.sin(((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.sin(((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))) * sin(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))) * sin(((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[Sin[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 41.3% accurate, 1.0× speedup

Math

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

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)))
    (sin (+ (* 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))) * sin(((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))) * sin(((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.sin(((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.sin(((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))) * sin(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))) * sin(((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[Sin[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: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := x.re \cdot x.re + x.im \cdot x.im\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \sin t\_2\\ t_4 := t\_3 + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos t\_2\right)\\ t_5 := e^{y.re \cdot \log \left(\sqrt{t\_1}\right) - t\_0} \cdot t\_4\\ \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-12}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;y.re \leq 3.1:\\ \;\;\;\;\frac{t\_4}{\frac{e^{t\_0}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+279}:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{{t\_1}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (+ (* x.re x.re) (* x.im x.im)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (sin t_2))
        (t_4 (+ t_3 (* y.im (* (log (hypot x.im x.re)) (cos t_2)))))
        (t_5 (* (exp (- (* y.re (log (sqrt t_1))) t_0)) t_4)))
   (if (<= y.re -1.4e-12)
     t_5
     (if (<= y.re 3.1)
       (/ t_4 (/ (exp t_0) (pow (hypot x.re x.im) y.re)))
       (if (<= y.re 5.5e+279) t_5 (/ t_3 (pow t_1 (/ y.re -2.0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = sin(t_2);
	double t_4 = t_3 + (y_46_im * (log(hypot(x_46_im, x_46_re)) * cos(t_2)));
	double t_5 = exp(((y_46_re * log(sqrt(t_1))) - t_0)) * t_4;
	double tmp;
	if (y_46_re <= -1.4e-12) {
		tmp = t_5;
	} else if (y_46_re <= 3.1) {
		tmp = t_4 / (exp(t_0) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else if (y_46_re <= 5.5e+279) {
		tmp = t_5;
	} else {
		tmp = t_3 / pow(t_1, (y_46_re / -2.0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = Math.sin(t_2);
	double t_4 = t_3 + (y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.cos(t_2)));
	double t_5 = Math.exp(((y_46_re * Math.log(Math.sqrt(t_1))) - t_0)) * t_4;
	double tmp;
	if (y_46_re <= -1.4e-12) {
		tmp = t_5;
	} else if (y_46_re <= 3.1) {
		tmp = t_4 / (Math.exp(t_0) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else if (y_46_re <= 5.5e+279) {
		tmp = t_5;
	} else {
		tmp = t_3 / Math.pow(t_1, (y_46_re / -2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = math.sin(t_2)
	t_4 = t_3 + (y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * math.cos(t_2)))
	t_5 = math.exp(((y_46_re * math.log(math.sqrt(t_1))) - t_0)) * t_4
	tmp = 0
	if y_46_re <= -1.4e-12:
		tmp = t_5
	elif y_46_re <= 3.1:
		tmp = t_4 / (math.exp(t_0) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	elif y_46_re <= 5.5e+279:
		tmp = t_5
	else:
		tmp = t_3 / math.pow(t_1, (y_46_re / -2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = sin(t_2)
	t_4 = Float64(t_3 + Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * cos(t_2))))
	t_5 = Float64(exp(Float64(Float64(y_46_re * log(sqrt(t_1))) - t_0)) * t_4)
	tmp = 0.0
	if (y_46_re <= -1.4e-12)
		tmp = t_5;
	elseif (y_46_re <= 3.1)
		tmp = Float64(t_4 / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	elseif (y_46_re <= 5.5e+279)
		tmp = t_5;
	else
		tmp = Float64(t_3 / (t_1 ^ Float64(y_46_re / -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = sin(t_2);
	t_4 = t_3 + (y_46_im * (log(hypot(x_46_im, x_46_re)) * cos(t_2)));
	t_5 = exp(((y_46_re * log(sqrt(t_1))) - t_0)) * t_4;
	tmp = 0.0;
	if (y_46_re <= -1.4e-12)
		tmp = t_5;
	elseif (y_46_re <= 3.1)
		tmp = t_4 / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	elseif (y_46_re <= 5.5e+279)
		tmp = t_5;
	else
		tmp = t_3 / (t_1 ^ (y_46_re / -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]}, If[LessEqual[y$46$re, -1.4e-12], t$95$5, If[LessEqual[y$46$re, 3.1], N[(t$95$4 / N[(N[Exp[t$95$0], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.5e+279], t$95$5, N[(t$95$3 / N[Power[t$95$1, N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.4000000000000001e-12 or 3.10000000000000009 < y.re < 5.50000000000000038e279

   1. Initial program 43.7%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)}\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(\color{blue}{y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right) \]

      15. atan2-lowering-atan2.f64 81.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 81.0%

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

   if -1.4000000000000001e-12 < y.re < 3.10000000000000009

   1. Initial program 39.2%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 83.7%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{atan2.f64}\left(x.im, x.re\right)}, y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      15. atan2-lowering-atan2.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 88.2%

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

   if 5.50000000000000038e279 < y.re

   1. Initial program 30.0%

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

   4. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 80.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 80.0%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-12}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 3.1:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+279}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.0% accurate, 0.5× speedup

Math

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

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 (- (* y.re t_2) t_0)) (sin (+ t_1 (* t_2 y.im))))))
   (if (<= t_3 INFINITY)
     t_3
     (/
      (sin (+ t_1 (* y.im (log (hypot x.re x.im)))))
      (/ (exp t_0) (pow (hypot x.re x.im) y.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 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(((y_46_re * t_2) - t_0)) * sin((t_1 + (t_2 * y_46_im)));
	double tmp;
	if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = sin((t_1 + (y_46_im * log(hypot(x_46_re, x_46_im))))) / (exp(t_0) / pow(hypot(x_46_re, x_46_im), y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.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(((y_46_re * t_2) - t_0)) * Math.sin((t_1 + (t_2 * y_46_im)));
	double tmp;
	if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = Math.sin((t_1 + (y_46_im * Math.log(Math.hypot(x_46_re, x_46_im))))) / (Math.exp(t_0) / Math.pow(Math.hypot(x_46_re, x_46_im), y_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
	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(((y_46_re * t_2) - t_0)) * math.sin((t_1 + (t_2 * y_46_im)))
	tmp = 0
	if t_3 <= math.inf:
		tmp = t_3
	else:
		tmp = math.sin((t_1 + (y_46_im * math.log(math.hypot(x_46_re, x_46_im))))) / (math.exp(t_0) / math.pow(math.hypot(x_46_re, x_46_im), y_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)
	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(y_46_re * t_2) - t_0)) * sin(Float64(t_1 + Float64(t_2 * y_46_im))))
	tmp = 0.0
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(sin(Float64(t_1 + Float64(y_46_im * log(hypot(x_46_re, x_46_im))))) / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_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;
	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(((y_46_re * t_2) - t_0)) * sin((t_1 + (t_2 * y_46_im)));
	tmp = 0.0;
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = sin((t_1 + (y_46_im * log(hypot(x_46_re, x_46_im))))) / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_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]}, 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[(y$46$re * t$95$2), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 + N[(t$95$2 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, Infinity], t$95$3, N[(N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[t$95$0], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.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)))) < +inf.0

   1. Initial program 80.8%

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

   if +inf.0 < (*.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))) (sin.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 0.0%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 73.3%

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

4. Final simplification 77.1%

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

Alternative 3: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+201}:\\ \;\;\;\;\frac{\sin t\_0 + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos t\_0\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* t_0 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -3.3e+103)
     t_1
     (if (<= y.re 3.4e+201)
       (/
        (+ (sin t_0) (* y.im (* (log (hypot x.im x.re)) (cos t_0))))
        (/ (exp (* (atan2 x.im x.re) y.im)) (pow (hypot x.re x.im) y.re)))
       t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -3.3e+103) {
		tmp = t_1;
	} else if (y_46_re <= 3.4e+201) {
		tmp = (sin(t_0) + (y_46_im * (log(hypot(x_46_im, x_46_re)) * cos(t_0)))) / (exp((atan2(x_46_im, x_46_re) * y_46_im)) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = 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 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -3.3e+103) {
		tmp = t_1;
	} else if (y_46_re <= 3.4e+201) {
		tmp = (Math.sin(t_0) + (y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.cos(t_0)))) / (Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im)) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -3.3e+103:
		tmp = t_1
	elif y_46_re <= 3.4e+201:
		tmp = (math.sin(t_0) + (y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * math.cos(t_0)))) / (math.exp((math.atan2(x_46_im, x_46_re) * y_46_im)) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.3e+103)
		tmp = t_1;
	elseif (y_46_re <= 3.4e+201)
		tmp = Float64(Float64(sin(t_0) + Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * cos(t_0)))) / Float64(exp(Float64(atan(x_46_im, x_46_re) * y_46_im)) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = 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 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.3e+103)
		tmp = t_1;
	elseif (y_46_re <= 3.4e+201)
		tmp = (sin(t_0) + (y_46_im * (log(hypot(x_46_im, x_46_re)) * cos(t_0)))) / (exp((atan2(x_46_im, x_46_re) * y_46_im)) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.3e+103], t$95$1, If[LessEqual[y$46$re, 3.4e+201], N[(N[(N[Sin[t$95$0], $MachinePrecision] + N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.30000000000000009e103 or 3.4e201 < y.re

   1. Initial program 40.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 70.4%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 74.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 74.4%

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

   if -3.30000000000000009e103 < y.re < 3.4e201

   1. Initial program 41.3%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 79.4%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{atan2.f64}\left(x.im, x.re\right)}, y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      15. atan2-lowering-atan2.f64 83.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 83.4%

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

Alternative 4: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+103}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 550000000000:\\ \;\;\;\;\frac{\sin \left(t\_0 + y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin t\_0}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -3.3e+103)
     (* t_0 (pow (hypot x.im x.re) y.re))
     (if (<= y.re 550000000000.0)
       (/
        (sin (+ t_0 (* y.im (log (hypot x.re x.im)))))
        (/ (exp (* (atan2 x.im x.re) y.im)) (pow (hypot x.re x.im) y.re)))
       (/ (sin t_0) (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re -2.0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -3.3e+103) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 550000000000.0) {
		tmp = sin((t_0 + (y_46_im * log(hypot(x_46_re, x_46_im))))) / (exp((atan2(x_46_im, x_46_re) * y_46_im)) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = sin(t_0) / pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -3.3e+103) {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 550000000000.0) {
		tmp = Math.sin((t_0 + (y_46_im * Math.log(Math.hypot(x_46_re, x_46_im))))) / (Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im)) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = Math.sin(t_0) / Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_re <= -3.3e+103:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 550000000000.0:
		tmp = math.sin((t_0 + (y_46_im * math.log(math.hypot(x_46_re, x_46_im))))) / (math.exp((math.atan2(x_46_im, x_46_re) * y_46_im)) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = math.sin(t_0) / math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -3.3e+103)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 550000000000.0)
		tmp = Float64(sin(Float64(t_0 + Float64(y_46_im * log(hypot(x_46_re, x_46_im))))) / Float64(exp(Float64(atan(x_46_im, x_46_re) * y_46_im)) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = Float64(sin(t_0) / (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.3e+103)
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 550000000000.0)
		tmp = sin((t_0 + (y_46_im * log(hypot(x_46_re, x_46_im))))) / (exp((atan2(x_46_im, x_46_re) * y_46_im)) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = sin(t_0) / (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.3e+103], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 550000000000.0], N[(N[Sin[N[(t$95$0 + N[(y$46$im * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[t$95$0], $MachinePrecision] / N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.30000000000000009e103

   1. Initial program 43.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 82.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 82.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 84.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 84.8%

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

   if -3.30000000000000009e103 < y.re < 5.5e11

   1. Initial program 41.9%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 85.4%

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

   if 5.5e11 < y.re

   1. Initial program 38.2%

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

   4. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 60.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+103}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 550000000000:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-25}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -1.75e-25)
     (* t_0 (pow (hypot x.im x.re) y.re))
     (if (<= y.re 7.8e+68)
       (/ (* y.im (log (hypot x.im x.re))) (exp (* (atan2 x.im x.re) y.im)))
       (/ t_0 (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re -2.0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -1.75e-25) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 7.8e+68) {
		tmp = (y_46_im * log(hypot(x_46_im, x_46_re))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = t_0 / pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -1.75e-25) {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 7.8e+68) {
		tmp = (y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = t_0 / Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_re <= -1.75e-25:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 7.8e+68:
		tmp = (y_46_im * math.log(math.hypot(x_46_im, x_46_re))) / math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	else:
		tmp = t_0 / math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -1.75e-25)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 7.8e+68)
		tmp = Float64(Float64(y_46_im * log(hypot(x_46_im, x_46_re))) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = Float64(t_0 / (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_re <= -1.75e-25)
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 7.8e+68)
		tmp = (y_46_im * log(hypot(x_46_im, x_46_re))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	else
		tmp = t_0 / (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.75e-25], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e+68], N[(N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.7500000000000001e-25

   1. Initial program 48.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 84.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 84.1%

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

   if -1.7500000000000001e-25 < y.re < 7.80000000000000037e68

   1. Initial program 39.9%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 79.3%

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

   5. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 66.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 66.3%

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

   8. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \color{blue}{\mathsf{atan2.f64}\left(x.im, x.re\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(\color{blue}{x.im}, x.re\right)\right)\right)\right) \]

      6. hypot-lowering-hypot.f64 72.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(\color{blue}{x.im}, x.re\right)\right)\right)\right) \]

   10. Simplified 72.6%

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

   if 7.80000000000000037e68 < y.re

   1. Initial program 36.1%

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

   4. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 64.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 64.0%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-25}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-58}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.92 \cdot 10^{-50}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -7.5e-58)
     (* t_0 (pow (hypot x.im x.re) y.re))
     (if (<= y.re 1.92e-50)
       (* y.im (log (hypot x.im x.re)))
       (/ t_0 (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re -2.0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -7.5e-58) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 1.92e-50) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 / pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -7.5e-58) {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 1.92e-50) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 / Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_re <= -7.5e-58:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 1.92e-50:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = t_0 / math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -7.5e-58)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 1.92e-50)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(t_0 / (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_re <= -7.5e-58)
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 1.92e-50)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = t_0 / (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e-58], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.92e-50], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.50000000000000002e-58

   1. Initial program 52.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 77.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 77.7%

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

   if -7.50000000000000002e-58 < y.re < 1.92e-50

   1. Initial program 35.3%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 22.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 22.4%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 14.7%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 14.7%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 41.2%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 41.2%

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

   if 1.92e-50 < y.re

   1. Initial program 38.5%

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

   4. Applied egg-rr 33.3%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 60.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 60.4%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.92 \cdot 10^{-50}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{t\_0}}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re -2.0))))
   (if (<= y.re -5.8e-150)
     (/ 1.0 (/ t_1 t_0))
     (if (<= y.re 4.7e-51) (* y.im (log (hypot x.im x.re))) (/ t_0 t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	double tmp;
	if (y_46_re <= -5.8e-150) {
		tmp = 1.0 / (t_1 / t_0);
	} else if (y_46_re <= 4.7e-51) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 / 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.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	double tmp;
	if (y_46_re <= -5.8e-150) {
		tmp = 1.0 / (t_1 / t_0);
	} else if (y_46_re <= 4.7e-51) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 / t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0))
	tmp = 0
	if y_46_re <= -5.8e-150:
		tmp = 1.0 / (t_1 / t_0)
	elif y_46_re <= 4.7e-51:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = t_0 / t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / -2.0)
	tmp = 0.0
	if (y_46_re <= -5.8e-150)
		tmp = Float64(1.0 / Float64(t_1 / t_0));
	elseif (y_46_re <= 4.7e-51)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(t_0 / 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = ((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / -2.0);
	tmp = 0.0;
	if (y_46_re <= -5.8e-150)
		tmp = 1.0 / (t_1 / t_0);
	elseif (y_46_re <= 4.7e-51)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = t_0 / 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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -5.8e-150], N[(1.0 / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.7e-51], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.7999999999999996e-150

   1. Initial program 47.2%

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

   4. Applied egg-rr 32.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 61.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)}\right) \]

      3. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}\right)}{\mathsf{neg}\left(\color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}\right)\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(\frac{y.re}{-2}\right)\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{y.re}{-2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{-2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{-2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(\frac{y.re}{-2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right), \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 66.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 66.1%

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

   if -5.7999999999999996e-150 < y.re < 4.6999999999999997e-51

   1. Initial program 37.2%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 24.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 24.8%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 17.6%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 17.6%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 43.7%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 43.7%

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

   if 4.6999999999999997e-51 < y.re

   1. Initial program 38.5%

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

   4. Applied egg-rr 33.3%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 60.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 60.4%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -7 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\sin t\_0}}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-80}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -7e-145)
     (/ 1.0 (/ (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re -2.0)) (sin t_0)))
     (if (<= y.re 1.15e-80)
       (* y.im (log (hypot x.im x.re)))
       (* t_0 (pow (hypot x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -7e-145) {
		tmp = 1.0 / (pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0)) / sin(t_0));
	} else if (y_46_re <= 1.15e-80) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -7e-145) {
		tmp = 1.0 / (Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0)) / Math.sin(t_0));
	} else if (y_46_re <= 1.15e-80) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_re <= -7e-145:
		tmp = 1.0 / (math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0)) / math.sin(t_0))
	elif y_46_re <= 1.15e-80:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -7e-145)
		tmp = Float64(1.0 / Float64((Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / -2.0)) / sin(t_0)));
	elseif (y_46_re <= 1.15e-80)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_re <= -7e-145)
		tmp = 1.0 / ((((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / -2.0)) / sin(t_0));
	elseif (y_46_re <= 1.15e-80)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7e-145], N[(1.0 / N[(N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.15e-80], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.99999999999999994e-145

   1. Initial program 47.2%

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

   4. Applied egg-rr 32.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 61.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)}\right) \]

      3. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}\right)}{\mathsf{neg}\left(\color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}\right)\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(\frac{y.re}{-2}\right)\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{y.re}{-2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{-2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{-2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(\frac{y.re}{-2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right), \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 66.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 66.1%

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

   if -6.99999999999999994e-145 < y.re < 1.1499999999999999e-80

   1. Initial program 38.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 26.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 26.1%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 18.3%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 18.3%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 46.4%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 46.4%

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

   if 1.1499999999999999e-80 < y.re

   1. Initial program 37.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 54.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 54.5%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 55.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-80}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-147}:\\ \;\;\;\;\sin t\_0 \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-81}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -3.4e-147)
     (* (sin t_0) (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0)))
     (if (<= y.re 1.9e-81)
       (* y.im (log (hypot x.im x.re)))
       (* t_0 (pow (hypot x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -3.4e-147) {
		tmp = sin(t_0) * pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	} else if (y_46_re <= 1.9e-81) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -3.4e-147) {
		tmp = Math.sin(t_0) * Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	} else if (y_46_re <= 1.9e-81) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_re <= -3.4e-147:
		tmp = math.sin(t_0) * math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	elif y_46_re <= 1.9e-81:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -3.4e-147)
		tmp = Float64(sin(t_0) * (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0)));
	elseif (y_46_re <= 1.9e-81)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.4e-147)
		tmp = sin(t_0) * (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0));
	elseif (y_46_re <= 1.9e-81)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.4e-147], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.9e-81], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.39999999999999996e-147

   1. Initial program 47.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 65.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 65.1%

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

      4. sqrt-pow2 N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      6. sqrt-pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(\frac{y.re}{2}\right)\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{y.re}{2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      15. atan2-lowering-atan2.f64 66.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Applied egg-rr 66.1%

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

   if -3.39999999999999996e-147 < y.re < 1.8999999999999999e-81

   1. Initial program 38.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 26.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 26.1%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 18.3%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 18.3%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 46.4%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 46.4%

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

   if 1.8999999999999999e-81 < y.re

   1. Initial program 37.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 54.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 54.5%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 55.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-147}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-81}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -7 \cdot 10^{-145}:\\ \;\;\;\;\frac{t\_0}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-81}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -7e-145)
     (/ t_0 (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re -2.0)))
     (if (<= y.re 2.7e-81)
       (* y.im (log (hypot x.im x.re)))
       (* t_0 (pow (hypot x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -7e-145) {
		tmp = t_0 / pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	} else if (y_46_re <= 2.7e-81) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -7e-145) {
		tmp = t_0 / Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	} else if (y_46_re <= 2.7e-81) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_re <= -7e-145:
		tmp = t_0 / math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0))
	elif y_46_re <= 2.7e-81:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -7e-145)
		tmp = Float64(t_0 / (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / -2.0)));
	elseif (y_46_re <= 2.7e-81)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_re <= -7e-145)
		tmp = t_0 / (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / -2.0));
	elseif (y_46_re <= 2.7e-81)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7e-145], N[(t$95$0 / N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.7e-81], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.99999999999999994e-145

   1. Initial program 47.2%

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

   4. Applied egg-rr 32.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 61.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]

   8. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. atan2-lowering-atan2.f64 64.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   10. Simplified 64.5%

       \[\leadsto \frac{-\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]

   if -6.99999999999999994e-145 < y.re < 2.6999999999999999e-81

   1. Initial program 38.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 26.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 26.1%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 18.3%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 18.3%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 46.4%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 46.4%

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

   if 2.6999999999999999e-81 < y.re

   1. Initial program 37.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 54.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 54.5%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 55.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{-145}:\\ \;\;\;\;\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-81}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 25.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{-46}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + y.re \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.5e-58)
   (sin (* y.re (atan2 x.im x.re)))
   (if (<= y.re 2.25e-46)
     (* y.im (log (hypot x.im x.re)))
     (*
      y.re
      (*
       (atan2 x.im x.re)
       (+ 1.0 (* y.re (* (log (+ (* x.re x.re) (* x.im x.im))) 0.5))))))))

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 <= -3.5e-58) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 2.25e-46) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + (y_46_re * (log(((x_46_re * x_46_re) + (x_46_im * x_46_im))) * 0.5))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.5e-58) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 2.25e-46) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * (1.0 + (y_46_re * (Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im))) * 0.5))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3.5e-58:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 2.25e-46:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * (1.0 + (y_46_re * (math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im))) * 0.5))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.5e-58)
		tmp = sin(Float64(y_46_re * atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 2.25e-46)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(1.0 + Float64(y_46_re * Float64(log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3.5e-58)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 2.25e-46)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + (y_46_re * (log(((x_46_re * x_46_re) + (x_46_im * x_46_im))) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.5e-58], N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 2.25e-46], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(1.0 + N[(y$46$re * N[(N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.4999999999999999e-58

   1. Initial program 52.8%

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

   4. Applied egg-rr 35.7%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 69.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]

   8. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{1}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 15.8%

       \[\leadsto \frac{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - \color{blue}{1}} \]

   if -3.4999999999999999e-58 < y.re < 2.25e-46

   1. Initial program 35.3%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 22.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 22.4%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 14.7%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 14.7%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 41.2%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 41.2%

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

   if 2.25e-46 < y.re

   1. Initial program 38.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 57.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 57.8%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. atan2-lowering-atan2.f64 25.3%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 25.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{y.re}\right) \]

   10. Applied egg-rr 25.3%

       \[\leadsto \color{blue}{\left(\left(y.re \cdot \left(\log \left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot 0.5\right) + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{-46}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + y.re \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (* y.re (atan2 x.im x.re))
          (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re -2.0)))))
   (if (<= y.re -1.5e-146)
     t_0
     (if (<= y.re 4.7e-51) (* y.im (log (hypot x.im 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 = (y_46_re * atan2(x_46_im, x_46_re)) / pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	double tmp;
	if (y_46_re <= -1.5e-146) {
		tmp = t_0;
	} else if (y_46_re <= 4.7e-51) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * Math.atan2(x_46_im, x_46_re)) / Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0));
	double tmp;
	if (y_46_re <= -1.5e-146) {
		tmp = t_0;
	} else if (y_46_re <= 4.7e-51) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * math.atan2(x_46_im, x_46_re)) / math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / -2.0))
	tmp = 0
	if y_46_re <= -1.5e-146:
		tmp = t_0
	elif y_46_re <= 4.7e-51:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	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(y_46_re * atan(x_46_im, x_46_re)) / (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / -2.0)))
	tmp = 0.0
	if (y_46_re <= -1.5e-146)
		tmp = t_0;
	elseif (y_46_re <= 4.7e-51)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	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 = (y_46_re * atan2(x_46_im, x_46_re)) / (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / -2.0));
	tmp = 0.0;
	if (y_46_re <= -1.5e-146)
		tmp = t_0;
	elseif (y_46_re <= 4.7e-51)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	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[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.5e-146], t$95$0, If[LessEqual[y$46$re, 4.7e-51], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.50000000000000009e-146 or 4.6999999999999997e-51 < y.re

   1. Initial program 43.1%

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

   4. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 61.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 61.0%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]

   8. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. atan2-lowering-atan2.f64 62.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   10. Simplified 62.0%

       \[\leadsto \frac{-\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]

   if -1.50000000000000009e-146 < y.re < 4.6999999999999997e-51

   1. Initial program 37.2%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 24.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 24.8%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 17.6%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 17.6%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 43.7%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 43.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 18.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{-58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -9.5e-58)
   (sin (* y.re (atan2 x.im x.re)))
   (* y.im (log (hypot x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -9.5e-58) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -9.5e-58) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -9.5e-58:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -9.5e-58)
		tmp = sin(Float64(y_46_re * atan(x_46_im, x_46_re)));
	else
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -9.5e-58)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	else
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -9.5e-58], N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -9.4999999999999994e-58

   1. Initial program 52.8%

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

   4. Applied egg-rr 35.7%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\frac{y.im}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}}} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

      5. atan2-lowering-atan2.f64 69.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, -2\right)\right)\right)\right) \]

   7. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{-2}\right)}} \]

   8. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{1}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 15.8%

       \[\leadsto \frac{-\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{0 - \color{blue}{1}} \]

   if -9.4999999999999994e-58 < y.re

   1. Initial program 36.6%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 28.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 28.1%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 9.3%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 9.3%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 25.2%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 25.2%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{-58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 18.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{-58}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.3e-58)
   (* y.re (atan2 x.im x.re))
   (* y.im (log (hypot x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.3e-58) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.3e-58) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3.3e-58:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	else:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.3e-58)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	else
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3.3e-58)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	else
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.3e-58], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.30000000000000026e-58

   1. Initial program 52.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 77.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 77.7%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. atan2-lowering-atan2.f64 15.5%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 15.5%

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

   if -3.30000000000000026e-58 < y.re

   1. Initial program 36.6%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 28.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 28.1%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 9.3%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 9.3%

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

   9. Taylor expanded in y.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 25.2%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   11. Simplified 25.2%

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

Alternative 15: 13.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+26}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im 5e+26) (* y.re (atan2 x.im x.re)) (sin (* y.im (log x.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_im <= 5e+26) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else {
		tmp = sin((y_46_im * log(x_46_im)));
	}
	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) :: tmp
    if (x_46im <= 5d+26) then
        tmp = y_46re * atan2(x_46im, x_46re)
    else
        tmp = sin((y_46im * log(x_46im)))
    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 tmp;
	if (x_46_im <= 5e+26) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else {
		tmp = Math.sin((y_46_im * Math.log(x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= 5e+26:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	else:
		tmp = math.sin((y_46_im * math.log(x_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_im <= 5e+26)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	else
		tmp = sin(Float64(y_46_im * log(x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= 5e+26)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	else
		tmp = sin((y_46_im * log(x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, 5e+26], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000001e26

   1. Initial program 47.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 47.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 47.6%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. atan2-lowering-atan2.f64 13.6%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 13.6%

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

   if 5.0000000000000001e26 < x.im

   1. Initial program 25.3%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)}\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      4. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 17.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.im\right)}, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right)\right)\right)\right) \]

   5. Simplified 17.1%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 5.1%

         \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 5.1%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{\sin \left(y.im \cdot \log x.im\right)} \]
   10. Step-by-step derivation

       1. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log x.im\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log x.im\right)\right) \]

       3. log-lowering-log.f64 16.5%

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.im\right)\right)\right) \]

   11. Simplified 16.5%

       \[\leadsto \color{blue}{\sin \left(y.im \cdot \log x.im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 13.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

3. Taylor expanded in y.im around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

   2. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

   4. atan2-lowering-atan2.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

   8. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

   9. hypot-lowering-hypot.f64 45.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

5. Simplified 45.1%

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

6. Taylor expanded in y.re around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   2. atan2-lowering-atan2.f64 11.6%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

8. Simplified 11.6%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))