powComplex, imaginary part

Percentage Accurate: 40.7% → 66.9%

Time: 18.3s

Alternatives: 14

Speedup: 2.8×

• 35.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 40.7% accurate, 1.0× speedup

Math

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

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: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\frac{-1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -2 \cdot 10^{-308}:\\ \;\;\;\;e^{-\mathsf{fma}\left(y.re, t\_1, t\_0\right)} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t\_1 \cdot \left(-y.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re))) (t_1 (log (/ -1.0 x.re))))
   (if (<= x.re -2e-308)
     (*
      (exp (- (fma y.re t_1 t_0)))
      (sin (fma y.re (atan2 x.im x.re) (* t_1 (- y.im)))))
     (*
      (exp (- (* y.re (log x.re)) t_0))
      (sin (fma y.im (log x.re) (* y.re (atan2 x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = log((-1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -2e-308) {
		tmp = exp(-fma(y_46_re, t_1, t_0)) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (t_1 * -y_46_im)));
	} else {
		tmp = exp(((y_46_re * log(x_46_re)) - t_0)) * sin(fma(y_46_im, log(x_46_re), (y_46_re * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -2e-308)
		tmp = Float64(exp(Float64(-fma(y_46_re, t_1, t_0))) * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(t_1 * Float64(-y_46_im)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0)) * sin(fma(y_46_im, log(x_46_re), Float64(y_46_re * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x.re < -1.9999999999999998e-308

   1. Initial program 42.9%

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

   3. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 N/A

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

      12. lower-sin.f64 N/A

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

   5. Applied rewrites 69.5%

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

   if -1.9999999999999998e-308 < x.re

   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. Add Preprocessing

   3. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 68.3

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

   5. Applied rewrites 68.3%

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

Alternative 2: 58.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_1 := e^{y.re \cdot t\_0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot t\_0\right)\\ t_3 := t\_2 \cdot t\_1\\ t_4 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_5 := t\_1 \cdot \sin \left(y.im \cdot \log t\_4\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-236}:\\ \;\;\;\;{t\_4}^{y.re} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_1 (exp (- (* y.re t_0) (* y.im (atan2 x.im x.re)))))
        (t_2 (sin (+ (* y.re (atan2 x.im x.re)) (* y.im t_0))))
        (t_3 (* t_2 t_1))
        (t_4 (sqrt (fma x.im x.im (* x.re x.re))))
        (t_5 (* t_1 (sin (* y.im (log t_4))))))
   (if (<= t_3 (- INFINITY))
     t_5
     (if (<= t_3 -1e-236)
       (* (pow t_4 y.re) t_2)
       (if (<= t_3 INFINITY)
         t_5
         (*
          y.re
          (* (atan2 x.im x.re) (pow (exp (- y.im)) (atan2 x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_1 = exp(((y_46_re * t_0) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_2 = sin(((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * t_0)));
	double t_3 = t_2 * t_1;
	double t_4 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_5 = t_1 * sin((y_46_im * log(t_4)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_3 <= -1e-236) {
		tmp = pow(t_4, y_46_re) * t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * pow(exp(-y_46_im), atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_1 = exp(Float64(Float64(y_46_re * t_0) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	t_2 = sin(Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) + Float64(y_46_im * t_0)))
	t_3 = Float64(t_2 * t_1)
	t_4 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_5 = Float64(t_1 * sin(Float64(y_46_im * log(t_4))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_3 <= -1e-236)
		tmp = Float64((t_4 ^ y_46_re) * t_2);
	elseif (t_3 <= Inf)
		tmp = t_5;
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * (exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(y$46$re * t$95$0), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * N[Sin[N[(y$46$im * N[Log[t$95$4], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$5, If[LessEqual[t$95$3, -1e-236], N[(N[Power[t$95$4, y$46$re], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$5, N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 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 or -1e-236 < (*.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 81.1%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   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)))) < -1e-236

   1. Initial program 77.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-atan2.f64 43.8

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

   8. Applied rewrites 43.8%

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

      1. lift-neg.f64 N/A

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

      2. rem-log-exp N/A

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

      3. lift-atan2.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 44.6

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

   10. Applied rewrites 44.6%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \leq -\infty:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)\\ \mathbf{elif}\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \leq -1 \cdot 10^{-236}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{elif}\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \leq \infty:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{-1}{x.im}\right)\\ \mathbf{if}\;x.im \leq -1.35 \cdot 10^{-267}:\\ \;\;\;\;e^{-\mathsf{fma}\left(y.re, t\_2, t\_0\right)} \cdot \sin \left(\mathsf{fma}\left(t\_2, -y.im, t\_1\right)\right)\\ \mathbf{elif}\;x.im \leq 1.2 \cdot 10^{-297}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(t\_1 + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t\_0} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (/ -1.0 x.im))))
   (if (<= x.im -1.35e-267)
     (* (exp (- (fma y.re t_2 t_0))) (sin (fma t_2 (- y.im) t_1)))
     (if (<= x.im 1.2e-297)
       (*
        (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re)
        (sin (+ t_1 (* y.im (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))))
       (*
        (exp (- (* y.re (log x.im)) t_0))
        (sin (fma y.im (log x.im) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log((-1.0 / x_46_im));
	double tmp;
	if (x_46_im <= -1.35e-267) {
		tmp = exp(-fma(y_46_re, t_2, t_0)) * sin(fma(t_2, -y_46_im, t_1));
	} else if (x_46_im <= 1.2e-297) {
		tmp = pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re) * sin((t_1 + (y_46_im * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0)) * sin(fma(y_46_im, log(x_46_im), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(Float64(-1.0 / x_46_im))
	tmp = 0.0
	if (x_46_im <= -1.35e-267)
		tmp = Float64(exp(Float64(-fma(y_46_re, t_2, t_0))) * sin(fma(t_2, Float64(-y_46_im), t_1)));
	elseif (x_46_im <= 1.2e-297)
		tmp = Float64((sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re) * sin(Float64(t_1 + Float64(y_46_im * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0)) * sin(fma(y_46_im, log(x_46_im), t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $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[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -1.35e-267], N[(N[Exp[(-N[(y$46$re * t$95$2 + t$95$0), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(t$95$2 * (-y$46$im) + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.2e-297], N[(N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.34999999999999994e-267

   1. Initial program 41.2%

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

   3. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 N/A

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

      12. lower-sin.f64 N/A

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

   5. Applied rewrites 68.2%

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

   if -1.34999999999999994e-267 < x.im < 1.2e-297

   1. Initial program 62.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 1.2e-297 < x.im

   1. Initial program 38.6%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 68.1

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

   5. Applied rewrites 68.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.35 \cdot 10^{-267}:\\ \;\;\;\;e^{-\mathsf{fma}\left(y.re, \log \left(\frac{-1}{x.im}\right), y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\frac{-1}{x.im}\right), -y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 1.2 \cdot 10^{-297}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq 6.8 \cdot 10^{-298}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0} \cdot \sin t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re))) (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= x.re 6.8e-298)
     (*
      (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
      (sin t_1))
     (* (exp (- (* y.re (log x.re)) t_0)) (sin (fma y.im (log x.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_im * atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= 6.8e-298) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin(t_1);
	} else {
		tmp = exp(((y_46_re * log(x_46_re)) - t_0)) * sin(fma(y_46_im, log(x_46_re), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= 6.8e-298)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * sin(t_1));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0)) * sin(fma(y_46_im, log(x_46_re), t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, 6.8e-298], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.8e-298

   1. Initial program 42.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 55.2

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

   5. Applied rewrites 55.2%

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

   if 6.8e-298 < x.re

   1. Initial program 40.2%

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

   3. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 68.9

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

   5. Applied rewrites 68.9%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 6.8 \cdot 10^{-298}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -2.65 \cdot 10^{+138}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-147}:\\ \;\;\;\;e^{y.re \cdot t\_0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin t\_1\\ \mathbf{elif}\;y.im \leq 2400000000000:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(t\_1 + y.im \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= y.im -2.65e+138)
     (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))
     (if (<= y.im 4.3e-147)
       (* (exp (- (* y.re t_0) (* y.im (atan2 x.im x.re)))) (sin t_1))
       (if (<= y.im 2400000000000.0)
         (*
          (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re)
          (sin (+ t_1 (* y.im t_0))))
         (*
          y.re
          (* (atan2 x.im x.re) (pow (exp (- y.im)) (atan2 x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= -2.65e+138) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	} else if (y_46_im <= 4.3e-147) {
		tmp = exp(((y_46_re * t_0) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(t_1);
	} else if (y_46_im <= 2400000000000.0) {
		tmp = pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re) * sin((t_1 + (y_46_im * t_0)));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * pow(exp(-y_46_im), atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= -2.65e+138)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	elseif (y_46_im <= 4.3e-147)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_0) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(t_1));
	elseif (y_46_im <= 2400000000000.0)
		tmp = Float64((sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re) * sin(Float64(t_1 + Float64(y_46_im * t_0))));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * (exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.65e+138], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.3e-147], N[(N[Exp[N[(N[(y$46$re * t$95$0), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2400000000000.0], N[(N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(t$95$1 + N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.64999999999999992e138

   1. Initial program 31.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 54.7

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-atan2.f64 68.7

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

   8. Applied rewrites 68.7%

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

   if -2.64999999999999992e138 < y.im < 4.3000000000000001e-147

   1. Initial program 39.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 54.5

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

   5. Applied rewrites 54.5%

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

   if 4.3000000000000001e-147 < y.im < 2.4e12

   1. Initial program 72.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if 2.4e12 < y.im

   1. Initial program 34.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-atan2.f64 59.2

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

   8. Applied rewrites 59.2%

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

      1. lift-neg.f64 N/A

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

      2. rem-log-exp N/A

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

      3. lift-atan2.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 61.9

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

   10. Applied rewrites 61.9%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.65 \cdot 10^{+138}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-147}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 2400000000000:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+126}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-147}:\\ \;\;\;\;t\_0 \cdot \sin t\_1\\ \mathbf{elif}\;y.im \leq 2400000000000:\\ \;\;\;\;t\_0 \cdot \sin \left(t\_1 + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
        (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= y.im -5e+126)
     (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))
     (if (<= y.im 4.3e-147)
       (* t_0 (sin t_1))
       (if (<= y.im 2400000000000.0)
         (*
          t_0
          (sin (+ t_1 (* y.im (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))))
         (*
          y.re
          (* (atan2 x.im x.re) (pow (exp (- y.im)) (atan2 x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= -5e+126) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	} else if (y_46_im <= 4.3e-147) {
		tmp = t_0 * sin(t_1);
	} else if (y_46_im <= 2400000000000.0) {
		tmp = t_0 * sin((t_1 + (y_46_im * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))))));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * pow(exp(-y_46_im), atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= -5e+126)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	elseif (y_46_im <= 4.3e-147)
		tmp = Float64(t_0 * sin(t_1));
	elseif (y_46_im <= 2400000000000.0)
		tmp = Float64(t_0 * sin(Float64(t_1 + Float64(y_46_im * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))))));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * (exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5e+126], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.3e-147], N[(t$95$0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2400000000000.0], N[(t$95$0 * N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.99999999999999977e126

   1. Initial program 32.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-atan2.f64 67.7

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

   8. Applied rewrites 67.7%

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

   if -4.99999999999999977e126 < y.im < 4.3000000000000001e-147

   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. 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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 53.0

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

   5. Applied rewrites 53.0%

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

   if 4.3000000000000001e-147 < y.im < 2.4e12

   1. Initial program 72.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if 2.4e12 < y.im

   1. Initial program 34.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-atan2.f64 59.2

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

   8. Applied rewrites 59.2%

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

      1. lift-neg.f64 N/A

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

      2. rem-log-exp N/A

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

      3. lift-atan2.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 61.9

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

   10. Applied rewrites 61.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+126}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-147}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 2400000000000:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;t\_1 \cdot \sin t\_0\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re)))
   (if (<= y.re -6.5e+31)
     (* t_1 (sin t_0))
     (if (<= y.re 6.2e-7)
       (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))
       (* 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -6.5e+31) {
		tmp = t_1 * sin(t_0);
	} else if (y_46_re <= 6.2e-7) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	} 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 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -6.5e+31)
		tmp = Float64(t_1 * sin(t_0));
	elseif (y_46_re <= 6.2e-7)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	else
		tmp = Float64(t_0 * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -6.5e+31], N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-7], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.5000000000000004e31

   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. 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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if -6.5000000000000004e31 < y.re < 6.1999999999999999e-7

   1. Initial program 41.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 37.6

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-atan2.f64 51.6

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

   8. Applied rewrites 51.6%

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

   if 6.1999999999999999e-7 < y.re

   1. Initial program 43.2%

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

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 53.0

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

   8. Applied rewrites 53.0%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+52}:\\ \;\;\;\;\sin t\_0 \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \end{array} \]

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 -1.3e+52)
     (* (sin t_0) (pow x.re y.re))
     (if (<= y.re 6.2e-7)
       (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))
       (* t_0 (pow (sqrt (fma x.im x.im (* x.re 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 <= -1.3e+52) {
		tmp = sin(t_0) * pow(x_46_re, y_46_re);
	} else if (y_46_re <= 6.2e-7) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	} else {
		tmp = t_0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * 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 <= -1.3e+52)
		tmp = Float64(sin(t_0) * (x_46_re ^ y_46_re));
	elseif (y_46_re <= 6.2e-7)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	else
		tmp = Float64(t_0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.3e+52], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-7], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.3e52

   1. Initial program 35.6%

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

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 73.6

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

   8. Applied rewrites 73.6%

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

   if -1.3e52 < y.re < 6.1999999999999999e-7

   1. Initial program 42.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-atan2.f64 51.6

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

   8. Applied rewrites 51.6%

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

   if 6.1999999999999999e-7 < y.re

   1. Initial program 43.2%

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

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 53.0

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

   8. Applied rewrites 53.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+52}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq -1.15 \cdot 10^{+106}:\\ \;\;\;\;\sin t\_0 \cdot {\left(-x.re\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= x.re -1.15e+106)
     (* (sin t_0) (pow (- x.re) y.re))
     (* t_0 (pow (sqrt (fma x.im x.im (* x.re 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 (x_46_re <= -1.15e+106) {
		tmp = sin(t_0) * pow(-x_46_re, y_46_re);
	} else {
		tmp = t_0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * 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 (x_46_re <= -1.15e+106)
		tmp = Float64(sin(t_0) * (Float64(-x_46_re) ^ y_46_re));
	else
		tmp = Float64(t_0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1.15e+106], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[(-x$46$re), y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.1500000000000001e106

   1. Initial program 17.2%

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

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 41.4

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

   5. Applied rewrites 41.4%

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

   6. Taylor expanded in x.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 45.4

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

   8. Applied rewrites 45.4%

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

   if -1.1500000000000001e106 < x.re

   1. Initial program 47.7%

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

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 45.8

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

   8. Applied rewrites 45.8%

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

Alternative 10: 45.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{-179}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-91}:\\ \;\;\;\;\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (* y.re (atan2 x.im x.re))
          (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -4.4e-179)
     t_0
     (if (<= y.re 6e-91)
       (* (* y.im 0.5) (log (fma x.re x.re (* x.im x.im))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * atan2(x_46_im, x_46_re)) * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -4.4e-179) {
		tmp = t_0;
	} else if (y_46_re <= 6e-91) {
		tmp = (y_46_im * 0.5) * log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -4.4e-179)
		tmp = t_0;
	elseif (y_46_re <= 6e-91)
		tmp = Float64(Float64(y_46_im * 0.5) * log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.4e-179], t$95$0, If[LessEqual[y$46$re, 6e-91], N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.40000000000000009e-179 or 6.0000000000000004e-91 < y.re

   1. Initial program 42.6%

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

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 54.6

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

   8. Applied rewrites 54.6%

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

   if -4.40000000000000009e-179 < y.re < 6.0000000000000004e-91

   1. Initial program 38.7%

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

   4. Applied rewrites 38.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 35.0

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

   7. Applied rewrites 35.0%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 28.3

         \[\leadsto \left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot x.im}\right)\right) \]

   10. Applied rewrites 28.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{-179}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-91}:\\ \;\;\;\;\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 21.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-163}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.6e-163)
   (* y.re (* (atan2 x.im x.re) (- 1.0 (* y.im (atan2 x.im x.re)))))
   (* (* y.im 0.5) (log (fma x.re x.re (* x.im x.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.6e-163) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 - (y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = (y_46_im * 0.5) * log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.6e-163)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(1.0 - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	else
		tmp = Float64(Float64(y_46_im * 0.5) * log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.6e-163], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(1.0 - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.5999999999999999e-163

   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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-atan2.f64 46.7

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

   8. Applied rewrites 46.7%

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

   9. Taylor expanded in y.im around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-atan2.f64 16.2

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

   11. Applied rewrites 16.2%

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

   if -4.5999999999999999e-163 < y.re

   1. Initial program 40.6%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 24.7

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

   7. Applied rewrites 24.7%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 26.2

         \[\leadsto \left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot x.im}\right)\right) \]

   10. Applied rewrites 26.2%

       \[\leadsto \color{blue}{\left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-163}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 21.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{if}\;y.im \leq -2.7 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (* y.im 0.5) (log (fma x.re x.re (* x.im x.im))))))
   (if (<= y.im -2.7e-113)
     t_0
     (if (<= y.im 1.35e-145) (sin (* y.re (atan2 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_im * 0.5) * log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	double tmp;
	if (y_46_im <= -2.7e-113) {
		tmp = t_0;
	} else if (y_46_im <= 1.35e-145) {
		tmp = sin((y_46_re * atan2(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_im * 0.5) * log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))))
	tmp = 0.0
	if (y_46_im <= -2.7e-113)
		tmp = t_0;
	elseif (y_46_im <= 1.35e-145)
		tmp = sin(Float64(y_46_re * atan(x_46_im, x_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.7e-113], t$95$0, If[LessEqual[y$46$im, 1.35e-145], N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.69999999999999996e-113 or 1.35e-145 < y.im

   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

   4. Applied rewrites 34.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 28.1

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 20.9

         \[\leadsto \left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot x.im}\right)\right) \]

   10. Applied rewrites 20.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)} \]

   if -2.69999999999999996e-113 < y.im < 1.35e-145

   1. Initial program 42.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 35.1

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

   8. Applied rewrites 35.1%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 31.0%

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

       1. lift-atan2.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. *-rgt-identity 31.0

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 31.0

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

   13. Applied rewrites 31.0%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{-113}:\\ \;\;\;\;\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 21.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{if}\;y.im \leq -2.7 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (* y.im 0.5) (log (fma x.re x.re (* x.im x.im))))))
   (if (<= y.im -2.7e-113)
     t_0
     (if (<= y.im 1.35e-145) (* y.re (atan2 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_im * 0.5) * log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	double tmp;
	if (y_46_im <= -2.7e-113) {
		tmp = t_0;
	} else if (y_46_im <= 1.35e-145) {
		tmp = y_46_re * atan2(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_im * 0.5) * log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))))
	tmp = 0.0
	if (y_46_im <= -2.7e-113)
		tmp = t_0;
	elseif (y_46_im <= 1.35e-145)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.7e-113], t$95$0, If[LessEqual[y$46$im, 1.35e-145], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.69999999999999996e-113 or 1.35e-145 < y.im

   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

   4. Applied rewrites 34.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 28.1

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 20.9

         \[\leadsto \left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot x.im}\right)\right) \]

   10. Applied rewrites 20.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)} \]

   if -2.69999999999999996e-113 < y.im < 1.35e-145

   1. Initial program 42.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 55.4

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

   5. Applied rewrites 55.4%

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

   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. lower-*.f64 N/A

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

      2. lower-atan2.f64 30.9

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

   8. Applied rewrites 30.9%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{-113}:\\ \;\;\;\;\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 13.9% accurate, 6.4× 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 41.5%

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

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 45.4

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

5. Applied rewrites 45.4%

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

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. lower-*.f64 N/A

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

   2. lower-atan2.f64 11.2

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

8. Applied rewrites 11.2%

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

Reproduce

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