powComplex, imaginary part

Percentage Accurate: 40.7% → 74.2%

Time: 17.6s

Alternatives: 20

Speedup: 2.0×

• 36.0% 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: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := t\_1 \cdot y.im\\ t_3 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - t\_0}\\ t_4 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, \frac{t\_1}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot t\_3\\ \mathbf{elif}\;y.re \leq 1250000:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_4 \cdot t\_1, y.im, \sin t\_4\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;\sin t\_2 \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0 + 1}{\sin \left(\left(\frac{t\_2}{y.re} + \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (log (hypot x.im x.re)))
        (t_2 (* t_1 y.im))
        (t_3
         (exp (- (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re) t_0)))
        (t_4 (* (atan2 x.im x.re) y.re)))
   (if (<= y.re -1.45e+43)
     (* (sin (* (fma y.im (/ t_1 y.re) (atan2 x.im x.re)) y.re)) t_3)
     (if (<= y.re 1250000.0)
       (*
        (fma (* (cos t_4) t_1) y.im (sin t_4))
        (exp (* (- y.im) (atan2 x.im x.re))))
       (if (<= y.re 7.8e+174)
         (* (sin t_2) t_3)
         (/
          1.0
          (/
           (+ t_0 1.0)
           (*
            (sin (* (+ (/ t_2 y.re) (atan2 x.im x.re)) y.re))
            (pow (hypot x.im x.re) y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = log(hypot(x_46_im, x_46_re));
	double t_2 = t_1 * y_46_im;
	double t_3 = exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - t_0));
	double t_4 = atan2(x_46_im, x_46_re) * y_46_re;
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = sin((fma(y_46_im, (t_1 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re)) * t_3;
	} else if (y_46_re <= 1250000.0) {
		tmp = fma((cos(t_4) * t_1), y_46_im, sin(t_4)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 7.8e+174) {
		tmp = sin(t_2) * t_3;
	} else {
		tmp = 1.0 / ((t_0 + 1.0) / (sin((((t_2 / y_46_re) + atan2(x_46_im, x_46_re)) * y_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = log(hypot(x_46_im, x_46_re))
	t_2 = Float64(t_1 * y_46_im)
	t_3 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - t_0))
	t_4 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.45e+43)
		tmp = Float64(sin(Float64(fma(y_46_im, Float64(t_1 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)) * t_3);
	elseif (y_46_re <= 1250000.0)
		tmp = Float64(fma(Float64(cos(t_4) * t_1), y_46_im, sin(t_4)) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 7.8e+174)
		tmp = Float64(sin(t_2) * t_3);
	else
		tmp = Float64(1.0 / Float64(Float64(t_0 + 1.0) / Float64(sin(Float64(Float64(Float64(t_2 / y_46_re) + atan(x_46_im, x_46_re)) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+43], N[(N[Sin[N[(N[(y$46$im * N[(t$95$1 / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 1250000.0], N[(N[(N[(N[Cos[t$95$4], $MachinePrecision] * t$95$1), $MachinePrecision] * y$46$im + N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e+174], N[(N[Sin[t$95$2], $MachinePrecision] * t$95$3), $MachinePrecision], N[(1.0 / N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(N[Sin[N[(N[(N[(t$95$2 / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.4500000000000001e43

   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.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if -1.4500000000000001e43 < y.re < 1.25e6

   1. Initial program 39.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in y.re around 0

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

      1. neg-mul-1 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 82.4

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

   8. Applied rewrites 82.4%

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

   if 1.25e6 < y.re < 7.79999999999999962e174

   1. Initial program 34.6%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if 7.79999999999999962e174 < y.re

   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
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-atan2.f64 45.2

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

   7. Applied rewrites 45.2%

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

   8. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 83.9

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

   10. Applied rewrites 83.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1250000:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1}{\sin \left(\left(\frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}{y.re} + \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - t\_0}\\ t_2 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_3 := t\_2 \cdot y.im\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, \frac{t\_2}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot t\_1\\ \mathbf{elif}\;y.re \leq 4.1:\\ \;\;\;\;\frac{-1}{\frac{\frac{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}}{-1}}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;\sin t\_3 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0 + 1}{\sin \left(\left(\frac{t\_3}{y.re} + \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1
         (exp (- (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re) t_0)))
        (t_2 (log (hypot x.im x.re)))
        (t_3 (* t_2 y.im)))
   (if (<= y.re -5e-10)
     (* (sin (* (fma y.im (/ t_2 y.re) (atan2 x.im x.re)) y.re)) t_1)
     (if (<= y.re 4.1)
       (/
        -1.0
        (/
         (/
          (/ (pow (exp y.im) (atan2 x.im x.re)) (pow (hypot x.re x.im) y.re))
          (sin (fma (log (hypot x.re x.im)) y.im (* (atan2 x.im x.re) y.re))))
         -1.0))
       (if (<= y.re 7.8e+174)
         (* (sin t_3) t_1)
         (/
          1.0
          (/
           (+ t_0 1.0)
           (*
            (sin (* (+ (/ t_3 y.re) (atan2 x.im x.re)) y.re))
            (pow (hypot x.im x.re) y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - t_0));
	double t_2 = log(hypot(x_46_im, x_46_re));
	double t_3 = t_2 * y_46_im;
	double tmp;
	if (y_46_re <= -5e-10) {
		tmp = sin((fma(y_46_im, (t_2 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re)) * t_1;
	} else if (y_46_re <= 4.1) {
		tmp = -1.0 / (((pow(exp(y_46_im), atan2(x_46_im, x_46_re)) / pow(hypot(x_46_re, x_46_im), y_46_re)) / sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, (atan2(x_46_im, x_46_re) * y_46_re)))) / -1.0);
	} else if (y_46_re <= 7.8e+174) {
		tmp = sin(t_3) * t_1;
	} else {
		tmp = 1.0 / ((t_0 + 1.0) / (sin((((t_3 / y_46_re) + atan2(x_46_im, x_46_re)) * y_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - t_0))
	t_2 = log(hypot(x_46_im, x_46_re))
	t_3 = Float64(t_2 * y_46_im)
	tmp = 0.0
	if (y_46_re <= -5e-10)
		tmp = Float64(sin(Float64(fma(y_46_im, Float64(t_2 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)) * t_1);
	elseif (y_46_re <= 4.1)
		tmp = Float64(-1.0 / Float64(Float64(Float64((exp(y_46_im) ^ atan(x_46_im, x_46_re)) / (hypot(x_46_re, x_46_im) ^ y_46_re)) / sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re)))) / -1.0));
	elseif (y_46_re <= 7.8e+174)
		tmp = Float64(sin(t_3) * t_1);
	else
		tmp = Float64(1.0 / Float64(Float64(t_0 + 1.0) / Float64(sin(Float64(Float64(Float64(t_3 / y_46_re) + atan(x_46_im, x_46_re)) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -5e-10], N[(N[Sin[N[(N[(y$46$im * N[(t$95$2 / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 4.1], N[(-1.0 / N[(N[(N[(N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e+174], N[(N[Sin[t$95$3], $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 / N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(N[Sin[N[(N[(N[(t$95$3 / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.00000000000000031e-10

   1. Initial program 43.5%

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

   3. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -5.00000000000000031e-10 < y.re < 4.0999999999999996

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 79.0%

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

      1. /-rgt-identity N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 85.4

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

   6. Applied rewrites 85.4%

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

   if 4.0999999999999996 < y.re < 7.79999999999999962e174

   1. Initial program 33.3%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 7.79999999999999962e174 < y.re

   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
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-atan2.f64 45.2

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

   7. Applied rewrites 45.2%

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

   8. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 83.9

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

   10. Applied rewrites 83.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 4.1:\\ \;\;\;\;\frac{-1}{\frac{\frac{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}}{-1}}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1}{\sin \left(\left(\frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}{y.re} + \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\frac{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}}{-1}}\\ \mathbf{if}\;y.im \leq -0.19:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1}{\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          -1.0
          (/
           (/
            (pow (exp y.im) (atan2 x.im x.re))
            (sin (* (log (hypot x.re x.im)) y.im)))
           -1.0))))
   (if (<= y.im -0.19)
     t_0
     (if (<= y.im 5.2e+98)
       (/
        1.0
        (/
         (+ (* y.im (atan2 x.im x.re)) 1.0)
         (*
          (sin (fma y.im (log (hypot x.im x.re)) (* (atan2 x.im x.re) y.re)))
          (pow (hypot x.im x.re) y.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 = -1.0 / ((pow(exp(y_46_im), atan2(x_46_im, x_46_re)) / sin((log(hypot(x_46_re, x_46_im)) * y_46_im))) / -1.0);
	double tmp;
	if (y_46_im <= -0.19) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e+98) {
		tmp = 1.0 / (((y_46_im * atan2(x_46_im, x_46_re)) + 1.0) / (sin(fma(y_46_im, log(hypot(x_46_im, x_46_re)), (atan2(x_46_im, x_46_re) * y_46_re))) * pow(hypot(x_46_im, x_46_re), y_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(-1.0 / Float64(Float64((exp(y_46_im) ^ atan(x_46_im, x_46_re)) / sin(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))) / -1.0))
	tmp = 0.0
	if (y_46_im <= -0.19)
		tmp = t_0;
	elseif (y_46_im <= 5.2e+98)
		tmp = Float64(1.0 / Float64(Float64(Float64(y_46_im * atan(x_46_im, x_46_re)) + 1.0) / Float64(sin(fma(y_46_im, log(hypot(x_46_im, x_46_re)), Float64(atan(x_46_im, x_46_re) * y_46_re))) * (hypot(x_46_im, x_46_re) ^ y_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[(-1.0 / N[(N[(N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] / N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -0.19], t$95$0, If[LessEqual[y$46$im, 5.2e+98], N[(1.0 / N[(N[(N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -0.19 or 5.1999999999999999e98 < y.im

   1. Initial program 26.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 61.1

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 65.0%

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

   if -0.19 < y.im < 5.1999999999999999e98

   1. Initial program 47.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 86.6

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

   7. Applied rewrites 86.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.19:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}}{-1}}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1}{\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}}{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{{\left(e^{y.im}\right)}^{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (sin (* (atan2 x.im x.re) y.re))
          (exp
           (-
            (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
            (* y.im (atan2 x.im x.re)))))))
   (if (<= y.re -1.45e+43)
     t_0
     (if (<= y.re 6.2e-129)
       (/
        1.0
        (/
         (/ 1.0 (pow (exp y.im) (- (atan2 x.im x.re))))
         (sin (* (log (hypot x.im x.re)) y.im))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = t_0;
	} else if (y_46_re <= 6.2e-129) {
		tmp = 1.0 / ((1.0 / pow(exp(y_46_im), -atan2(x_46_im, x_46_re))) / sin((log(hypot(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = t_0;
	} else if (y_46_re <= 6.2e-129) {
		tmp = 1.0 / ((1.0 / Math.pow(Math.exp(y_46_im), -Math.atan2(x_46_im, x_46_re))) / Math.sin((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -1.45e+43:
		tmp = t_0
	elif y_46_re <= 6.2e-129:
		tmp = 1.0 / ((1.0 / math.pow(math.exp(y_46_im), -math.atan2(x_46_im, x_46_re))) / math.sin((math.log(math.hypot(x_46_im, x_46_re)) * y_46_im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))))
	tmp = 0.0
	if (y_46_re <= -1.45e+43)
		tmp = t_0;
	elseif (y_46_re <= 6.2e-129)
		tmp = Float64(1.0 / Float64(Float64(1.0 / (exp(y_46_im) ^ Float64(-atan(x_46_im, x_46_re)))) / sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -1.45e+43)
		tmp = t_0;
	elseif (y_46_re <= 6.2e-129)
		tmp = 1.0 / ((1.0 / (exp(y_46_im) ^ -atan2(x_46_im, x_46_re))) / sin((log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+43], t$95$0, If[LessEqual[y$46$re, 6.2e-129], N[(1.0 / N[(N[(1.0 / N[Power[N[Exp[y$46$im], $MachinePrecision], (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.4500000000000001e43 or 6.2000000000000001e-129 < y.re

   1. Initial program 38.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if -1.4500000000000001e43 < y.re < 6.2000000000000001e-129

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 71.2

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

   7. Applied rewrites 71.2%

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

   9. Applied rewrites 71.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{{\left(e^{y.im}\right)}^{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}}{-1}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (sin (* (atan2 x.im x.re) y.re))
          (exp
           (-
            (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
            (* y.im (atan2 x.im x.re)))))))
   (if (<= y.re -1.45e+43)
     t_0
     (if (<= y.re 1.05e-34)
       (/
        -1.0
        (/
         (/
          (pow (exp y.im) (atan2 x.im x.re))
          (sin (* (log (hypot x.re x.im)) y.im)))
         -1.0))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = t_0;
	} else if (y_46_re <= 1.05e-34) {
		tmp = -1.0 / ((pow(exp(y_46_im), atan2(x_46_im, x_46_re)) / sin((log(hypot(x_46_re, x_46_im)) * y_46_im))) / -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = t_0;
	} else if (y_46_re <= 1.05e-34) {
		tmp = -1.0 / ((Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re)) / Math.sin((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im))) / -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -1.45e+43:
		tmp = t_0
	elif y_46_re <= 1.05e-34:
		tmp = -1.0 / ((math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re)) / math.sin((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im))) / -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))))
	tmp = 0.0
	if (y_46_re <= -1.45e+43)
		tmp = t_0;
	elseif (y_46_re <= 1.05e-34)
		tmp = Float64(-1.0 / Float64(Float64((exp(y_46_im) ^ atan(x_46_im, x_46_re)) / sin(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))) / -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -1.45e+43)
		tmp = t_0;
	elseif (y_46_re <= 1.05e-34)
		tmp = -1.0 / (((exp(y_46_im) ^ atan2(x_46_im, x_46_re)) / sin((log(hypot(x_46_re, x_46_im)) * y_46_im))) / -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+43], t$95$0, If[LessEqual[y$46$re, 1.05e-34], N[(-1.0 / N[(N[(N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] / N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.4500000000000001e43 or 1.05e-34 < y.re

   1. Initial program 38.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -1.4500000000000001e43 < y.re < 1.05e-34

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 67.3

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

   7. Applied rewrites 67.3%

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

   9. Applied rewrites 73.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}}{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-129}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (sin (* (atan2 x.im x.re) y.re))
          (exp
           (-
            (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
            (* y.im (atan2 x.im x.re)))))))
   (if (<= y.re -1.45e+43)
     t_0
     (if (<= y.re 6.2e-129)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (sin (* (log (hypot x.im x.re)) y.im)))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = t_0;
	} else if (y_46_re <= 6.2e-129) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * sin((log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = t_0;
	} else if (y_46_re <= 6.2e-129) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -1.45e+43:
		tmp = t_0
	elif y_46_re <= 6.2e-129:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))))
	tmp = 0.0
	if (y_46_re <= -1.45e+43)
		tmp = t_0;
	elseif (y_46_re <= 6.2e-129)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -1.45e+43)
		tmp = t_0;
	elseif (y_46_re <= 6.2e-129)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * sin((log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+43], t$95$0, If[LessEqual[y$46$re, 6.2e-129], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.4500000000000001e43 or 6.2000000000000001e-129 < y.re

   1. Initial program 38.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if -1.4500000000000001e43 < y.re < 6.2000000000000001e-129

   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.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-atan2.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-129}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;{\left(\frac{x.im \cdot x.im}{x.re} \cdot 0.5 + x.re\right)}^{y.re} \cdot t\_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-34}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -1.45e+43)
     (* (pow (+ (* (/ (* x.im x.im) x.re) 0.5) x.re) y.re) t_0)
     (if (<= y.re 1.05e-34)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (sin (* (log (hypot x.im x.re)) y.im)))
       (* t_0 (pow (hypot x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = pow(((((x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re), y_46_re) * t_0;
	} else if (y_46_re <= 1.05e-34) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * sin((log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -1.45e+43) {
		tmp = Math.pow(((((x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re), y_46_re) * t_0;
	} else if (y_46_re <= 1.05e-34) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -1.45e+43:
		tmp = math.pow(((((x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re), y_46_re) * t_0
	elif y_46_re <= 1.05e-34:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.45e+43)
		tmp = Float64((Float64(Float64(Float64(Float64(x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re) ^ y_46_re) * t_0);
	elseif (y_46_re <= 1.05e-34)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -1.45e+43)
		tmp = (((((x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re) ^ y_46_re) * t_0;
	elseif (y_46_re <= 1.05e-34)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * sin((log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+43], N[(N[Power[N[(N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] * 0.5), $MachinePrecision] + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.05e-34], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.4500000000000001e43

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 87.2%

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

   if -1.4500000000000001e43 < y.re < 1.05e-34

   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.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-atan2.f64 67.3

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

   5. Applied rewrites 67.3%

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

   if 1.05e-34 < y.re

   1. Initial program 36.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+43}:\\ \;\;\;\;{\left(\frac{x.im \cdot x.im}{x.re} \cdot 0.5 + x.re\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-34}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-169}:\\ \;\;\;\;\frac{1}{\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (* (sin (* (atan2 x.im x.re) y.re)) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -7.2e-156)
     t_0
     (if (<= y.re 3.4e-169)
       (/
        1.0
        (/
         (+ (* y.im (atan2 x.im x.re)) 1.0)
         (sin (* (log (hypot x.im x.re)) y.im))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -7.2e-156) {
		tmp = t_0;
	} else if (y_46_re <= 3.4e-169) {
		tmp = 1.0 / (((y_46_im * atan2(x_46_im, x_46_re)) + 1.0) / sin((log(hypot(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -7.2e-156) {
		tmp = t_0;
	} else if (y_46_re <= 3.4e-169) {
		tmp = 1.0 / (((y_46_im * Math.atan2(x_46_im, x_46_re)) + 1.0) / Math.sin((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -7.2e-156:
		tmp = t_0
	elif y_46_re <= 3.4e-169:
		tmp = 1.0 / (((y_46_im * math.atan2(x_46_im, x_46_re)) + 1.0) / math.sin((math.log(math.hypot(x_46_im, x_46_re)) * y_46_im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(atan(x_46_im, x_46_re) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -7.2e-156)
		tmp = t_0;
	elseif (y_46_re <= 3.4e-169)
		tmp = Float64(1.0 / Float64(Float64(Float64(y_46_im * atan(x_46_im, x_46_re)) + 1.0) / sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -7.2e-156)
		tmp = t_0;
	elseif (y_46_re <= 3.4e-169)
		tmp = 1.0 / (((y_46_im * atan2(x_46_im, x_46_re)) + 1.0) / sin((log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e-156], t$95$0, If[LessEqual[y$46$re, 3.4e-169], N[(1.0 / N[(N[(N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.19999999999999998e-156 or 3.4e-169 < y.re

   1. Initial program 40.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 58.8

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

   5. Applied rewrites 58.8%

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

   if -7.19999999999999998e-156 < y.re < 3.4e-169

   1. Initial program 34.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 73.5

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

   7. Applied rewrites 73.5%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 53.7%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-156}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-169}:\\ \;\;\;\;\frac{1}{\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (* (sin (* (atan2 x.im x.re) y.re)) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -4.6e-156)
     t_0
     (if (<= y.re 2.4e-120)
       (/ 1.0 (/ 1.0 (* (log (hypot x.im x.re)) y.im)))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.6e-156) {
		tmp = t_0;
	} else if (y_46_re <= 2.4e-120) {
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.6e-156) {
		tmp = t_0;
	} else if (y_46_re <= 2.4e-120) {
		tmp = 1.0 / (1.0 / (Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -4.6e-156:
		tmp = t_0
	elif y_46_re <= 2.4e-120:
		tmp = 1.0 / (1.0 / (math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(atan(x_46_im, x_46_re) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -4.6e-156)
		tmp = t_0;
	elseif (y_46_re <= 2.4e-120)
		tmp = Float64(1.0 / Float64(1.0 / Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -4.6e-156)
		tmp = t_0;
	elseif (y_46_re <= 2.4e-120)
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e-156], t$95$0, If[LessEqual[y$46$re, 2.4e-120], N[(1.0 / N[(1.0 / N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.5999999999999999e-156 or 2.3999999999999999e-120 < y.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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if -4.5999999999999999e-156 < y.re < 2.3999999999999999e-120

   1. Initial program 36.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 71.6

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 48.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-156}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{x.im \cdot x.im}{x.re} \cdot 0.5 + x.re\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -6500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (pow (+ (* (/ (* x.im x.im) x.re) 0.5) x.re) y.re)
          (sin (* (atan2 x.im x.re) y.re)))))
   (if (<= y.re -6500.0)
     t_0
     (if (<= y.re 9e-120)
       (/ 1.0 (/ 1.0 (* (log (hypot x.im x.re)) y.im)))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(((((x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re), y_46_re) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -6500.0) {
		tmp = t_0;
	} else if (y_46_re <= 9e-120) {
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(((((x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re), y_46_re) * Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -6500.0) {
		tmp = t_0;
	} else if (y_46_re <= 9e-120) {
		tmp = 1.0 / (1.0 / (Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(((((x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re), y_46_re) * math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -6500.0:
		tmp = t_0
	elif y_46_re <= 9e-120:
		tmp = 1.0 / (1.0 / (math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((Float64(Float64(Float64(Float64(x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re) ^ y_46_re) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -6500.0)
		tmp = t_0;
	elseif (y_46_re <= 9e-120)
		tmp = Float64(1.0 / Float64(1.0 / Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (((((x_46_im * x_46_im) / x_46_re) * 0.5) + x_46_re) ^ y_46_re) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -6500.0)
		tmp = t_0;
	elseif (y_46_re <= 9e-120)
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[N[(N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] * 0.5), $MachinePrecision] + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6500.0], t$95$0, If[LessEqual[y$46$re, 9e-120], N[(1.0 / N[(1.0 / N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6500 or 9e-120 < y.re

   1. Initial program 39.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 62.4%

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

   if -6500 < y.re < 9e-120

   1. Initial program 37.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 40.9%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6500:\\ \;\;\;\;{\left(\frac{x.im \cdot x.im}{x.re} \cdot 0.5 + x.re\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x.im \cdot x.im}{x.re} \cdot 0.5 + x.re\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{-154}:\\ \;\;\;\;{\left(\frac{x.re \cdot x.re}{x.im} \cdot 0.5 + x.im\right)}^{y.re} \cdot t\_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -6e-154)
     (* (pow (+ (* (/ (* x.re x.re) x.im) 0.5) x.im) y.re) t_0)
     (if (<= y.re 1.05e-119)
       (/ 1.0 (/ 1.0 (* (log (hypot x.im x.re)) y.im)))
       (* (pow x.im y.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 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -6e-154) {
		tmp = pow(((((x_46_re * x_46_re) / x_46_im) * 0.5) + x_46_im), y_46_re) * t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = pow(x_46_im, y_46_re) * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -6e-154) {
		tmp = Math.pow(((((x_46_re * x_46_re) / x_46_im) * 0.5) + x_46_im), y_46_re) * t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = 1.0 / (1.0 / (Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -6e-154:
		tmp = math.pow(((((x_46_re * x_46_re) / x_46_im) * 0.5) + x_46_im), y_46_re) * t_0
	elif y_46_re <= 1.05e-119:
		tmp = 1.0 / (1.0 / (math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = math.pow(x_46_im, y_46_re) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -6e-154)
		tmp = Float64((Float64(Float64(Float64(Float64(x_46_re * x_46_re) / x_46_im) * 0.5) + x_46_im) ^ y_46_re) * t_0);
	elseif (y_46_re <= 1.05e-119)
		tmp = Float64(1.0 / Float64(1.0 / Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -6e-154)
		tmp = (((((x_46_re * x_46_re) / x_46_im) * 0.5) + x_46_im) ^ y_46_re) * t_0;
	elseif (y_46_re <= 1.05e-119)
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = (x_46_im ^ y_46_re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -6e-154], N[(N[Power[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] * 0.5), $MachinePrecision] + x$46$im), $MachinePrecision], y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.05e-119], N[(1.0 / N[(1.0 / N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.0000000000000005e-154

   1. Initial program 44.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 67.2%

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

   if -6.0000000000000005e-154 < y.re < 1.05e-119

   1. Initial program 36.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 48.0%

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

   if 1.05e-119 < y.re

   1. Initial program 36.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 43.9%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-154}:\\ \;\;\;\;{\left(\frac{x.re \cdot x.re}{x.im} \cdot 0.5 + x.im\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_1 := {x.im}^{y.re} \cdot t\_0\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot t\_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))) (t_1 (* (pow x.im y.re) t_0)))
   (if (<= y.re -3.5e+271)
     t_1
     (if (<= y.re -3.8e-8)
       (* (pow (- x.re) y.re) t_0)
       (if (<= y.re 1.05e-119)
         (/ 1.0 (/ 1.0 (* (log (hypot x.im x.re)) y.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 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = pow(x_46_im, y_46_re) * t_0;
	double tmp;
	if (y_46_re <= -3.5e+271) {
		tmp = t_1;
	} else if (y_46_re <= -3.8e-8) {
		tmp = pow(-x_46_re, y_46_re) * t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = Math.pow(x_46_im, y_46_re) * t_0;
	double tmp;
	if (y_46_re <= -3.5e+271) {
		tmp = t_1;
	} else if (y_46_re <= -3.8e-8) {
		tmp = Math.pow(-x_46_re, y_46_re) * t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = 1.0 / (1.0 / (Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	t_1 = math.pow(x_46_im, y_46_re) * t_0
	tmp = 0
	if y_46_re <= -3.5e+271:
		tmp = t_1
	elif y_46_re <= -3.8e-8:
		tmp = math.pow(-x_46_re, y_46_re) * t_0
	elif y_46_re <= 1.05e-119:
		tmp = 1.0 / (1.0 / (math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	t_1 = Float64((x_46_im ^ y_46_re) * t_0)
	tmp = 0.0
	if (y_46_re <= -3.5e+271)
		tmp = t_1;
	elseif (y_46_re <= -3.8e-8)
		tmp = Float64((Float64(-x_46_re) ^ y_46_re) * t_0);
	elseif (y_46_re <= 1.05e-119)
		tmp = Float64(1.0 / Float64(1.0 / Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	t_1 = (x_46_im ^ y_46_re) * t_0;
	tmp = 0.0;
	if (y_46_re <= -3.5e+271)
		tmp = t_1;
	elseif (y_46_re <= -3.8e-8)
		tmp = (-x_46_re ^ y_46_re) * t_0;
	elseif (y_46_re <= 1.05e-119)
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+271], t$95$1, If[LessEqual[y$46$re, -3.8e-8], N[(N[Power[(-x$46$re), y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.05e-119], N[(1.0 / N[(1.0 / N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.4999999999999999e271 or 1.05e-119 < y.re

   1. Initial program 37.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 47.5%

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

   if -3.4999999999999999e271 < y.re < -3.80000000000000028e-8

   1. Initial program 44.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in x.re around -inf

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

   8. Applied rewrites 69.6%

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

   if -3.80000000000000028e-8 < y.re < 1.05e-119

   1. Initial program 38.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 68.5

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

   7. Applied rewrites 68.5%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 41.4%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+271}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_1 := {x.im}^{y.re} \cdot t\_0\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -18000000000:\\ \;\;\;\;{x.re}^{y.re} \cdot t\_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))) (t_1 (* (pow x.im y.re) t_0)))
   (if (<= y.re -3.5e+271)
     t_1
     (if (<= y.re -18000000000.0)
       (* (pow x.re y.re) t_0)
       (if (<= y.re 1.05e-119)
         (/ 1.0 (/ 1.0 (* (log (hypot x.im x.re)) y.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 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = pow(x_46_im, y_46_re) * t_0;
	double tmp;
	if (y_46_re <= -3.5e+271) {
		tmp = t_1;
	} else if (y_46_re <= -18000000000.0) {
		tmp = pow(x_46_re, y_46_re) * t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = Math.pow(x_46_im, y_46_re) * t_0;
	double tmp;
	if (y_46_re <= -3.5e+271) {
		tmp = t_1;
	} else if (y_46_re <= -18000000000.0) {
		tmp = Math.pow(x_46_re, y_46_re) * t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = 1.0 / (1.0 / (Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	t_1 = math.pow(x_46_im, y_46_re) * t_0
	tmp = 0
	if y_46_re <= -3.5e+271:
		tmp = t_1
	elif y_46_re <= -18000000000.0:
		tmp = math.pow(x_46_re, y_46_re) * t_0
	elif y_46_re <= 1.05e-119:
		tmp = 1.0 / (1.0 / (math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	t_1 = Float64((x_46_im ^ y_46_re) * t_0)
	tmp = 0.0
	if (y_46_re <= -3.5e+271)
		tmp = t_1;
	elseif (y_46_re <= -18000000000.0)
		tmp = Float64((x_46_re ^ y_46_re) * t_0);
	elseif (y_46_re <= 1.05e-119)
		tmp = Float64(1.0 / Float64(1.0 / Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	t_1 = (x_46_im ^ y_46_re) * t_0;
	tmp = 0.0;
	if (y_46_re <= -3.5e+271)
		tmp = t_1;
	elseif (y_46_re <= -18000000000.0)
		tmp = (x_46_re ^ y_46_re) * t_0;
	elseif (y_46_re <= 1.05e-119)
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+271], t$95$1, If[LessEqual[y$46$re, -18000000000.0], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.05e-119], N[(1.0 / N[(1.0 / N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.4999999999999999e271 or 1.05e-119 < y.re

   1. Initial program 37.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 47.5%

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

   if -3.4999999999999999e271 < y.re < -1.8e10

   1. Initial program 44.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 70.4%

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

   if -1.8e10 < y.re < 1.05e-119

   1. Initial program 38.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 69.2

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

   7. Applied rewrites 69.2%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 40.5%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+271}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq -18000000000:\\ \;\;\;\;{x.re}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow x.im y.re) (sin (* (atan2 x.im x.re) y.re)))))
   (if (<= y.re -6e-154)
     t_0
     (if (<= y.re 1.05e-119)
       (/ 1.0 (/ 1.0 (* (log (hypot x.im x.re)) y.im)))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(x_46_im, y_46_re) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -6e-154) {
		tmp = t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(x_46_im, y_46_re) * Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -6e-154) {
		tmp = t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = 1.0 / (1.0 / (Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(x_46_im, y_46_re) * math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -6e-154:
		tmp = t_0
	elif y_46_re <= 1.05e-119:
		tmp = 1.0 / (1.0 / (math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((x_46_im ^ y_46_re) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -6e-154)
		tmp = t_0;
	elseif (y_46_re <= 1.05e-119)
		tmp = Float64(1.0 / Float64(1.0 / Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im ^ y_46_re) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -6e-154)
		tmp = t_0;
	elseif (y_46_re <= 1.05e-119)
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6e-154], t$95$0, If[LessEqual[y$46$re, 1.05e-119], N[(1.0 / N[(1.0 / N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.0000000000000005e-154 or 1.05e-119 < y.re

   1. Initial program 40.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 45.0%

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

   if -6.0000000000000005e-154 < y.re < 1.05e-119

   1. Initial program 36.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 48.0%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-154}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 22.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 52000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{y.im \cdot \left(x.re \cdot x.re\right)}{x.im \cdot x.im}, \log x.im \cdot y.im\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (sin (* (atan2 x.im x.re) y.re)))))
   (if (<= y.re -4.6e-156)
     t_0
     (if (<= y.re 2.4e-120)
       (/ 1.0 (/ 1.0 (* (log (hypot x.im x.re)) y.im)))
       (if (<= y.re 52000000000.0)
         t_0
         (/
          1.0
          (/
           1.0
           (fma
            0.5
            (/ (* y.im (* x.re x.re)) (* x.im x.im))
            (* (log x.im) y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -4.6e-156) {
		tmp = t_0;
	} else if (y_46_re <= 2.4e-120) {
		tmp = 1.0 / (1.0 / (log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else if (y_46_re <= 52000000000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (1.0 / fma(0.5, ((y_46_im * (x_46_re * x_46_re)) / (x_46_im * x_46_im)), (log(x_46_im) * y_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -4.6e-156)
		tmp = t_0;
	elseif (y_46_re <= 2.4e-120)
		tmp = Float64(1.0 / Float64(1.0 / Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	elseif (y_46_re <= 52000000000.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(0.5, Float64(Float64(y_46_im * Float64(x_46_re * x_46_re)) / Float64(x_46_im * x_46_im)), Float64(log(x_46_im) * y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e-156], t$95$0, If[LessEqual[y$46$re, 2.4e-120], N[(1.0 / N[(1.0 / N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 52000000000.0], t$95$0, N[(1.0 / N[(1.0 / N[(0.5 * N[(N[(y$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.5999999999999999e-156 or 2.3999999999999999e-120 < y.re < 5.2e10

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 15.8%

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

   if -4.5999999999999999e-156 < y.re < 2.3999999999999999e-120

   1. Initial program 36.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 71.6

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 48.7%

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

   if 5.2e10 < y.re

   1. Initial program 36.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 11.8%

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{\left(x.re \cdot x.re\right) \cdot y.im}{\color{blue}{x.im \cdot x.im}}, y.im \cdot \log x.im\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-156}:\\ \;\;\;\;1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 52000000000:\\ \;\;\;\;1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{y.im \cdot \left(x.re \cdot x.re\right)}{x.im \cdot x.im}, \log x.im \cdot y.im\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 16.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;x.re \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(-y.im, \log \left(\frac{-1}{x.re}\right), \frac{t\_0 \cdot 0.5}{x.re \cdot x.re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{t\_0}{x.re \cdot x.re}, \log x.re \cdot y.im\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (* x.im x.im))))
   (if (<= x.re -1e-308)
     (/
      1.0
      (/ 1.0 (fma (- y.im) (log (/ -1.0 x.re)) (/ (* t_0 0.5) (* x.re x.re)))))
     (/ 1.0 (/ 1.0 (fma 0.5 (/ t_0 (* x.re x.re)) (* (log x.re) y.im)))))))

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 * (x_46_im * x_46_im);
	double tmp;
	if (x_46_re <= -1e-308) {
		tmp = 1.0 / (1.0 / fma(-y_46_im, log((-1.0 / x_46_re)), ((t_0 * 0.5) / (x_46_re * x_46_re))));
	} else {
		tmp = 1.0 / (1.0 / fma(0.5, (t_0 / (x_46_re * x_46_re)), (log(x_46_re) * y_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * Float64(x_46_im * x_46_im))
	tmp = 0.0
	if (x_46_re <= -1e-308)
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(-y_46_im), log(Float64(-1.0 / x_46_re)), Float64(Float64(t_0 * 0.5) / Float64(x_46_re * x_46_re)))));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(0.5, Float64(t_0 / Float64(x_46_re * x_46_re)), Float64(log(x_46_re) * y_46_im))));
	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[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1e-308], N[(1.0 / N[(1.0 / N[((-y$46$im) * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision] + N[(N[(t$95$0 * 0.5), $MachinePrecision] / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(0.5 * N[(t$95$0 / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -9.9999999999999991e-309

   1. Initial program 39.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 46.1

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

   7. Applied rewrites 46.1%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 12.2%

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

   11. Taylor expanded in x.re around -inf

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

   13. Applied rewrites 16.0%

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(-1 \cdot y.im, \log \left(\frac{-1}{x.re}\right), \frac{0.5 \cdot \left(\left(x.im \cdot x.im\right) \cdot y.im\right)}{x.re \cdot x.re}\right)}} \]

   if -9.9999999999999991e-309 < x.re

   1. Initial program 38.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 43.3

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 20.9%

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

   11. Taylor expanded in x.im around 0

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

   13. Applied rewrites 23.2%

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{\left(x.im \cdot x.im\right) \cdot y.im}{\color{blue}{x.re \cdot x.re}}, y.im \cdot \log x.re\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(-y.im, \log \left(\frac{-1}{x.re}\right), \frac{\left(y.im \cdot \left(x.im \cdot x.im\right)\right) \cdot 0.5}{x.re \cdot x.re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{y.im \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}, \log x.re \cdot y.im\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 15.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 8.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{1}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{y.im \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}, \log x.re \cdot y.im\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 8.6e-263)
   (/ 1.0 (/ 1.0 (* (atan2 x.im x.re) y.re)))
   (/
    1.0
    (/
     1.0
     (fma 0.5 (/ (* y.im (* x.im x.im)) (* x.re x.re)) (* (log x.re) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 8.6e-263) {
		tmp = 1.0 / (1.0 / (atan2(x_46_im, x_46_re) * y_46_re));
	} else {
		tmp = 1.0 / (1.0 / fma(0.5, ((y_46_im * (x_46_im * x_46_im)) / (x_46_re * x_46_re)), (log(x_46_re) * y_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 8.6e-263)
		tmp = Float64(1.0 / Float64(1.0 / Float64(atan(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(0.5, Float64(Float64(y_46_im * Float64(x_46_im * x_46_im)) / Float64(x_46_re * x_46_re)), Float64(log(x_46_re) * y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 8.6e-263], N[(1.0 / N[(1.0 / N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(0.5 * N[(N[(y$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 8.5999999999999994e-263

   1. Initial program 37.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 63.1%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 50.4

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

   7. Applied rewrites 50.4%

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

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 9.4%

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

   if 8.5999999999999994e-263 < x.re

   1. Initial program 40.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 43.5

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

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 20.1%

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

   11. Taylor expanded in x.im around 0

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

   13. Applied rewrites 24.3%

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{\left(x.im \cdot x.im\right) \cdot y.im}{\color{blue}{x.re \cdot x.re}}, y.im \cdot \log x.re\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 8.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{1}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \frac{y.im \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}, \log x.re \cdot y.im\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 14.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 6.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{1}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\log x.re \cdot y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 6.8e-103)
   (/ 1.0 (/ 1.0 (* (atan2 x.im x.re) y.re)))
   (/ 1.0 (/ 1.0 (* (log x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 6.8e-103) {
		tmp = 1.0 / (1.0 / (atan2(x_46_im, x_46_re) * y_46_re));
	} else {
		tmp = 1.0 / (1.0 / (log(x_46_re) * y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 6.8d-103) then
        tmp = 1.0d0 / (1.0d0 / (atan2(x_46im, x_46re) * y_46re))
    else
        tmp = 1.0d0 / (1.0d0 / (log(x_46re) * y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 6.8e-103) {
		tmp = 1.0 / (1.0 / (Math.atan2(x_46_im, x_46_re) * y_46_re));
	} else {
		tmp = 1.0 / (1.0 / (Math.log(x_46_re) * y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 6.8e-103:
		tmp = 1.0 / (1.0 / (math.atan2(x_46_im, x_46_re) * y_46_re))
	else:
		tmp = 1.0 / (1.0 / (math.log(x_46_re) * y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 6.8e-103)
		tmp = Float64(1.0 / Float64(1.0 / Float64(atan(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(log(x_46_re) * y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 6.8e-103)
		tmp = 1.0 / (1.0 / (atan2(x_46_im, x_46_re) * y_46_re));
	else
		tmp = 1.0 / (1.0 / (log(x_46_re) * y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 6.8e-103], N[(1.0 / N[(1.0 / N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.80000000000000006e-103

   1. Initial program 38.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 49.7

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

   7. Applied rewrites 49.7%

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

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 9.2%

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

   if 6.80000000000000006e-103 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-atan2.f64 N/A

         \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}}} \]

      8. lower-log.f64 N/A

         \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right)}} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right)}} \]

      11. lower-hypot.f64 43.4

          \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}} \]

   7. Applied rewrites 43.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.8%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}} \]

   11. Taylor expanded in x.re around inf

       \[\leadsto \frac{1}{\frac{1}{-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.re}\right)}\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 20.5%

       \[\leadsto \frac{1}{\frac{1}{-y.im \cdot \left(-\log x.re\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 6.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{1}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\log x.re \cdot y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 13.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ 1.0 (/ 1.0 (* (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) {
	return 1.0 / (1.0 / (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
    code = 1.0d0 / (1.0d0 / (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) {
	return 1.0 / (1.0 / (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):
	return 1.0 / (1.0 / (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)
	return Float64(1.0 / Float64(1.0 / 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)
	tmp = 1.0 / (1.0 / (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_] := N[(1.0 / N[(1.0 / N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   4. lift--.f64 N/A

      \[\leadsto \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   5. exp-diff N/A

      \[\leadsto \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

4. Applied rewrites 68.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}}} \]

5. Taylor expanded in y.im around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}} \]

   5. lower-atan2.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{1}{\frac{1}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{\frac{1}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re}}} \]

   9. lower-hypot.f64 47.8

      \[\leadsto \frac{1}{\frac{1}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}}} \]

7. Applied rewrites 47.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}} \]

8. Taylor expanded in y.re around 0

   \[\leadsto \frac{1}{\frac{1}{\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
9. Step-by-step derivation

10. Applied rewrites 10.5%

    \[\leadsto \frac{1}{\frac{1}{\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

11. Final simplification 10.5%

    \[\leadsto \frac{1}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}} \]
12. Add Preprocessing

Alternative 20: 4.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\log x.im \cdot y.im}} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ 1.0 (/ 1.0 (* (log x.im) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 / (1.0 / (log(x_46_im) * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0 / (1.0d0 / (log(x_46im) * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 / (1.0 / (Math.log(x_46_im) * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 / (1.0 / (math.log(x_46_im) * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 / Float64(1.0 / Float64(log(x_46_im) * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 / (1.0 / (log(x_46_im) * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 / N[(1.0 / N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   4. lift--.f64 N/A

      \[\leadsto \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   5. exp-diff N/A

      \[\leadsto \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

4. Applied rewrites 68.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}}} \]

5. Taylor expanded in y.re around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}}} \]

   2. exp-prod N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}} \]

   4. lower-exp.f64 N/A

      \[\leadsto \frac{1}{\frac{{\color{blue}{\left(e^{y.im}\right)}}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}} \]

   5. lower-atan2.f64 N/A

      \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}}} \]

   8. lower-log.f64 N/A

      \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)}} \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right)}} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right)}} \]

   11. lower-hypot.f64 44.7

       \[\leadsto \frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}} \]

7. Applied rewrites 44.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}} \]

8. Taylor expanded in y.im around 0

   \[\leadsto \frac{1}{\frac{1}{\color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}} \]
9. Step-by-step derivation

10. Applied rewrites 16.6%

    \[\leadsto \frac{1}{\frac{1}{\color{blue}{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}} \]

11. Taylor expanded in x.re around 0

    \[\leadsto \frac{1}{\frac{1}{y.im \cdot \log x.im}} \]
12. Step-by-step derivation

13. Applied rewrites 4.5%

    \[\leadsto \frac{1}{\frac{1}{y.im \cdot \log x.im}} \]

14. Final simplification 4.5%

    \[\leadsto \frac{1}{\frac{1}{\log x.im \cdot y.im}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024284 
(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)))))