powComplex, imaginary part

Percentage Accurate: 41.1% → 78.1%

Time: 17.1s

Alternatives: 21

Speedup: 2.1×

• 36.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.1% 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: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - t\_0}\\ t_2 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -5:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, \frac{t\_2}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot t\_1\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{e^{t\_0}}{\sin \left(\mathsf{fma}\left(y.im, t\_2, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t\_2 \cdot y.im\right) \cdot t\_1\\ \end{array} \end{array} \]

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))))
   (if (<= y.re -5.0)
     (* (sin (* (fma y.im (/ t_2 y.re) (atan2 x.im x.re)) y.re)) t_1)
     (if (<= y.re 6.4e+59)
       (/
        1.0
        (/
         (exp t_0)
         (*
          (sin (fma y.im t_2 (* (atan2 x.im x.re) y.re)))
          (pow (hypot x.im x.re) y.re))))
       (* (sin (* t_2 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 = 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 tmp;
	if (y_46_re <= -5.0) {
		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 <= 6.4e+59) {
		tmp = 1.0 / (exp(t_0) / (sin(fma(y_46_im, t_2, (atan2(x_46_im, x_46_re) * y_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re)));
	} else {
		tmp = sin((t_2 * y_46_im)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = 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))
	tmp = 0.0
	if (y_46_re <= -5.0)
		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 <= 6.4e+59)
		tmp = Float64(1.0 / Float64(exp(t_0) / Float64(sin(fma(y_46_im, t_2, Float64(atan(x_46_im, x_46_re) * y_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re))));
	else
		tmp = Float64(sin(Float64(t_2 * y_46_im)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[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]}, If[LessEqual[y$46$re, -5.0], 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, 6.4e+59], N[(1.0 / N[(N[Exp[t$95$0], $MachinePrecision] / N[(N[Sin[N[(y$46$im * t$95$2 + 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], N[(N[Sin[N[(t$95$2 * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5

   1. Initial program 38.8%

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

   3. Taylor expanded in y.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 82.2

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

   5. Applied rewrites 82.2%

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

   if -5 < y.re < 6.39999999999999964e59

   1. Initial program 43.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 77.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-exp.f64 77.6

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

   6. Applied rewrites 77.6%

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

   if 6.39999999999999964e59 < y.re

   1. Initial program 44.7%

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

   3. Taylor expanded in y.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 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 14.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{if}\;\sin \left(y.im \cdot t\_1 + t\_0\right) \cdot e^{t\_1 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \leq -0.0002:\\ \;\;\;\;\left(\log x.re + \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re))
        (t_1 (log (sqrt (+ (* x.im x.im) (* x.re x.re))))))
   (if (<=
        (*
         (sin (+ (* y.im t_1) t_0))
         (exp (- (* t_1 y.re) (* y.im (atan2 x.im x.re)))))
        -0.0002)
     (* (+ (log x.re) (/ (* 0.5 (* x.im x.im)) (* x.re 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 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if ((sin(((y_46_im * t_1) + t_0)) * exp(((t_1 * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))))) <= -0.0002) {
		tmp = (log(x_46_re) + ((0.5 * (x_46_im * x_46_im)) / (x_46_re * x_46_re))) * y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46re
    t_1 = log(sqrt(((x_46im * x_46im) + (x_46re * x_46re))))
    if ((sin(((y_46im * t_1) + t_0)) * exp(((t_1 * y_46re) - (y_46im * atan2(x_46im, x_46re))))) <= (-0.0002d0)) then
        tmp = (log(x_46re) + ((0.5d0 * (x_46im * x_46im)) / (x_46re * x_46re))) * y_46im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if ((Math.sin(((y_46_im * t_1) + t_0)) * Math.exp(((t_1 * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))))) <= -0.0002) {
		tmp = (Math.log(x_46_re) + ((0.5 * (x_46_im * x_46_im)) / (x_46_re * 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.atan2(x_46_im, x_46_re) * y_46_re
	t_1 = math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))
	tmp = 0
	if (math.sin(((y_46_im * t_1) + t_0)) * math.exp(((t_1 * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))) <= -0.0002:
		tmp = (math.log(x_46_re) + ((0.5 * (x_46_im * x_46_im)) / (x_46_re * 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(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))
	tmp = 0.0
	if (Float64(sin(Float64(Float64(y_46_im * t_1) + t_0)) * exp(Float64(Float64(t_1 * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))) <= -0.0002)
		tmp = Float64(Float64(log(x_46_re) + Float64(Float64(0.5 * Float64(x_46_im * x_46_im)) / Float64(x_46_re * 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 = atan2(x_46_im, x_46_re) * y_46_re;
	t_1 = log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	tmp = 0.0;
	if ((sin(((y_46_im * t_1) + t_0)) * exp(((t_1 * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))))) <= -0.0002)
		tmp = (log(x_46_re) + ((0.5 * (x_46_im * x_46_im)) / (x_46_re * 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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Sin[N[(N[(y$46$im * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.0002], N[(N[(N[Log[x$46$re], $MachinePrecision] + N[(N[(0.5 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -2.0000000000000001e-4

   1. Initial program 53.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 3.9%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 3.9%

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

   12. Taylor expanded in x.im around 0

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

   14. Applied rewrites 24.5%

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

   if -2.0000000000000001e-4 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 40.6%

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

   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 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 17.8%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 13.8%

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

4. Final simplification 15.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \leq -0.0002:\\ \;\;\;\;\left(\log x.re + \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, \frac{t\_1}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot t\_0\\ \mathbf{elif}\;y.re \leq 0.075:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}, t\_1\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t\_1 \cdot y.im\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
           (* y.im (atan2 x.im x.re)))))
        (t_1 (log (hypot x.im x.re))))
   (if (<= y.re -3.8e-8)
     (* (sin (* (fma y.im (/ t_1 y.re) (atan2 x.im x.re)) y.re)) t_0)
     (if (<= y.re 0.075)
       (*
        (sin (* (fma y.re (/ (atan2 x.im x.re) y.im) t_1) y.im))
        (exp (* (- y.im) (atan2 x.im x.re))))
       (* (sin (* t_1 y.im)) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((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 t_1 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -3.8e-8) {
		tmp = sin((fma(y_46_im, (t_1 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re)) * t_0;
	} else if (y_46_re <= 0.075) {
		tmp = sin((fma(y_46_re, (atan2(x_46_im, x_46_re) / y_46_im), t_1) * y_46_im)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = sin((t_1 * y_46_im)) * t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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))))
	t_1 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -3.8e-8)
		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_0);
	elseif (y_46_re <= 0.075)
		tmp = Float64(sin(Float64(fma(y_46_re, Float64(atan(x_46_im, x_46_re) / y_46_im), t_1) * y_46_im)) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(sin(Float64(t_1 * y_46_im)) * t_0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[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]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -3.8e-8], 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$0), $MachinePrecision], If[LessEqual[y$46$re, 0.075], N[(N[Sin[N[(N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(t$95$1 * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.80000000000000028e-8

   1. Initial program 40.5%

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

   3. Taylor expanded in y.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 82.6

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

   5. Applied rewrites 82.6%

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

   if -3.80000000000000028e-8 < y.re < 0.0749999999999999972

   1. Initial program 40.9%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-atan2.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 47.7

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in y.re around 0

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

      1. neg-mul-1 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 74.2

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

   8. Applied rewrites 74.2%

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

   if 0.0749999999999999972 < y.re

   1. Initial program 47.4%

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

   3. Taylor expanded in y.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 79.0

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

   5. Applied rewrites 79.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 0.075:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot t\_1\\ \mathbf{elif}\;y.re \leq 0.075:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}, t\_0\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t\_0 \cdot y.im\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1
         (exp
          (-
           (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
           (* y.im (atan2 x.im x.re))))))
   (if (<= y.re -1.7e-18)
     (* (sin (* (atan2 x.im x.re) y.re)) t_1)
     (if (<= y.re 0.075)
       (*
        (sin (* (fma y.re (/ (atan2 x.im x.re) y.im) t_0) y.im))
        (exp (* (- y.im) (atan2 x.im x.re))))
       (* (sin (* t_0 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 = log(hypot(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) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.7e-18) {
		tmp = sin((atan2(x_46_im, x_46_re) * y_46_re)) * t_1;
	} else if (y_46_re <= 0.075) {
		tmp = sin((fma(y_46_re, (atan2(x_46_im, x_46_re) / y_46_im), t_0) * y_46_im)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = sin((t_0 * y_46_im)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(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) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.7e-18)
		tmp = Float64(sin(Float64(atan(x_46_im, x_46_re) * y_46_re)) * t_1);
	elseif (y_46_re <= 0.075)
		tmp = Float64(sin(Float64(fma(y_46_re, Float64(atan(x_46_im, x_46_re) / y_46_im), t_0) * y_46_im)) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(sin(Float64(t_0 * y_46_im)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $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] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.7e-18], N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 0.075], N[(N[Sin[N[(N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision] + t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.70000000000000001e-18

   1. Initial program 41.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 78.4

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

   5. Applied rewrites 78.4%

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

   if -1.70000000000000001e-18 < y.re < 0.0749999999999999972

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-atan2.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in y.re around 0

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

      1. neg-mul-1 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 75.0

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

   8. Applied rewrites 75.0%

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

   if 0.0749999999999999972 < y.re

   1. Initial program 47.4%

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

   3. Taylor expanded in y.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 79.0

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

   5. Applied rewrites 79.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 0.075:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 0.062:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (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 -1.7e-18)
     t_0
     (if (<= y.re 0.062)
       (*
        (sin
         (*
          (fma y.re (/ (atan2 x.im x.re) y.im) (log (hypot x.im x.re)))
          y.im))
        (exp (* (- y.im) (atan2 x.im x.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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 <= -1.7e-18) {
		tmp = t_0;
	} else if (y_46_re <= 0.062) {
		tmp = sin((fma(y_46_re, (atan2(x_46_im, x_46_re) / y_46_im), log(hypot(x_46_im, x_46_re))) * y_46_im)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(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 <= -1.7e-18)
		tmp = t_0;
	elseif (y_46_re <= 0.062)
		tmp = Float64(sin(Float64(fma(y_46_re, Float64(atan(x_46_im, x_46_re) / y_46_im), log(hypot(x_46_im, x_46_re))) * y_46_im)) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[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, -1.7e-18], t$95$0, If[LessEqual[y$46$re, 0.062], N[(N[Sin[N[(N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision] + N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.70000000000000001e-18 or 0.062 < y.re

   1. Initial program 43.8%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 68.6

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

   5. Applied rewrites 68.6%

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

   if -1.70000000000000001e-18 < y.re < 0.062

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-atan2.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in y.re around 0

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

      1. neg-mul-1 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 75.0

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

   8. Applied rewrites 75.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 0.062:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - t\_0}\\ \mathbf{if}\;y.re \leq -7 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{\frac{e^{t\_0}}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1
         (*
          (sin (* (atan2 x.im x.re) y.re))
          (exp
           (- (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re) t_0)))))
   (if (<= y.re -7e-73)
     t_1
     (if (<= y.re 3.7e-47)
       (/ 1.0 (/ (exp t_0) (sin (* (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 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = 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) - t_0));
	double tmp;
	if (y_46_re <= -7e-73) {
		tmp = t_1;
	} else if (y_46_re <= 3.7e-47) {
		tmp = 1.0 / (exp(t_0) / sin((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 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = 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) - t_0));
	double tmp;
	if (y_46_re <= -7e-73) {
		tmp = t_1;
	} else if (y_46_re <= 3.7e-47) {
		tmp = 1.0 / (Math.exp(t_0) / Math.sin((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 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = 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) - t_0))
	tmp = 0
	if y_46_re <= -7e-73:
		tmp = t_1
	elif y_46_re <= 3.7e-47:
		tmp = 1.0 / (math.exp(t_0) / math.sin((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 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = 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) - t_0)))
	tmp = 0.0
	if (y_46_re <= -7e-73)
		tmp = t_1;
	elseif (y_46_re <= 3.7e-47)
		tmp = Float64(1.0 / Float64(exp(t_0) / sin(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 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = 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) - t_0));
	tmp = 0.0;
	if (y_46_re <= -7e-73)
		tmp = t_1;
	elseif (y_46_re <= 3.7e-47)
		tmp = 1.0 / (exp(t_0) / sin((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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7e-73], t$95$1, If[LessEqual[y$46$re, 3.7e-47], N[(1.0 / N[(N[Exp[t$95$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$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.9999999999999995e-73 or 3.7e-47 < y.re

   1. Initial program 41.4%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 65.5

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

   5. Applied rewrites 65.5%

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

   if -6.9999999999999995e-73 < y.re < 3.7e-47

   1. Initial program 43.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 78.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-exp.f64 78.8

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

   6. Applied rewrites 78.8%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 64.9

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

   9. Applied rewrites 64.9%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{-73}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -27.0)
     t_0
     (if (<= y.re 4.6e-48)
       (/
        1.0
        (/
         (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)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -27.0) {
		tmp = t_0;
	} else if (y_46_re <= 4.6e-48) {
		tmp = 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;
	}
	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 <= -27.0) {
		tmp = t_0;
	} else if (y_46_re <= 4.6e-48) {
		tmp = 1.0 / (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.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -27.0:
		tmp = t_0
	elif y_46_re <= 4.6e-48:
		tmp = 1.0 / (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)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -27.0)
		tmp = t_0;
	elseif (y_46_re <= 4.6e-48)
		tmp = Float64(1.0 / Float64(exp(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)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -27.0)
		tmp = t_0;
	elseif (y_46_re <= 4.6e-48)
		tmp = 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -27.0], t$95$0, If[LessEqual[y$46$re, 4.6e-48], N[(1.0 / 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]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -27 or 4.6000000000000001e-48 < y.re

   1. Initial program 39.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 64.0

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

   5. Applied rewrites 64.0%

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

   if -27 < y.re < 4.6000000000000001e-48

   1. Initial program 44.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 75.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-exp.f64 75.8

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

   6. Applied rewrites 75.8%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 62.3

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

   9. Applied rewrites 62.3%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -27:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (* (sin (* (atan2 x.im x.re) y.re)) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -27.0)
     t_0
     (if (<= y.re 4.6e-48)
       (*
        (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)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -27.0) {
		tmp = t_0;
	} else if (y_46_re <= 4.6e-48) {
		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.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -27.0) {
		tmp = t_0;
	} else if (y_46_re <= 4.6e-48) {
		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.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -27.0:
		tmp = t_0
	elif y_46_re <= 4.6e-48:
		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)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -27.0)
		tmp = t_0;
	elseif (y_46_re <= 4.6e-48)
		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)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -27.0)
		tmp = t_0;
	elseif (y_46_re <= 4.6e-48)
		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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -27.0], t$95$0, If[LessEqual[y$46$re, 4.6e-48], 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 < -27 or 4.6000000000000001e-48 < y.re

   1. Initial program 39.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 64.0

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

   5. Applied rewrites 64.0%

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

   if -27 < y.re < 4.6000000000000001e-48

   1. Initial program 44.8%

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

   3. Taylor expanded in y.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 62.3

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

   5. Applied rewrites 62.3%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -27:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-294}:\\ \;\;\;\;\left(-e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \sin \left(\log \left(\frac{-1}{x.im}\right) \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5.5e-41)
     t_0
     (if (<= y.re -4.8e-294)
       (*
        (- (exp (* (- y.im) (atan2 x.im x.re))))
        (sin (* (log (/ -1.0 x.im)) y.im)))
       (if (<= y.re 5.8e-186) (* (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 <= -5.5e-41) {
		tmp = t_0;
	} else if (y_46_re <= -4.8e-294) {
		tmp = -exp((-y_46_im * atan2(x_46_im, x_46_re))) * sin((log((-1.0 / x_46_im)) * y_46_im));
	} else if (y_46_re <= 5.8e-186) {
		tmp = 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 <= -5.5e-41) {
		tmp = t_0;
	} else if (y_46_re <= -4.8e-294) {
		tmp = -Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((Math.log((-1.0 / x_46_im)) * y_46_im));
	} else if (y_46_re <= 5.8e-186) {
		tmp = 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 <= -5.5e-41:
		tmp = t_0
	elif y_46_re <= -4.8e-294:
		tmp = -math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((math.log((-1.0 / x_46_im)) * y_46_im))
	elif y_46_re <= 5.8e-186:
		tmp = 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 <= -5.5e-41)
		tmp = t_0;
	elseif (y_46_re <= -4.8e-294)
		tmp = Float64(Float64(-exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))) * sin(Float64(log(Float64(-1.0 / x_46_im)) * y_46_im)));
	elseif (y_46_re <= 5.8e-186)
		tmp = 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 <= -5.5e-41)
		tmp = t_0;
	elseif (y_46_re <= -4.8e-294)
		tmp = -exp((-y_46_im * atan2(x_46_im, x_46_re))) * sin((log((-1.0 / x_46_im)) * y_46_im));
	elseif (y_46_re <= 5.8e-186)
		tmp = 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, -5.5e-41], t$95$0, If[LessEqual[y$46$re, -4.8e-294], N[((-N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[Sin[N[(N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e-186], N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.50000000000000022e-41 or 5.80000000000000038e-186 < y.re

   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 56.2

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

   5. Applied rewrites 56.2%

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

   if -5.50000000000000022e-41 < y.re < -4.79999999999999994e-294

   1. Initial program 35.4%

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

   3. Taylor expanded in x.im around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. sub-neg N/A

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 34.1%

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

   if -4.79999999999999994e-294 < y.re < 5.80000000000000038e-186

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 52.3%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 49.2%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-41}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-294}:\\ \;\;\;\;\left(-e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \sin \left(\log \left(\frac{-1}{x.im}\right) \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1.75e-72)
     t_0
     (if (<= y.re 5.8e-186) (* (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 <= -1.75e-72) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-186) {
		tmp = 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 <= -1.75e-72) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-186) {
		tmp = 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 <= -1.75e-72:
		tmp = t_0
	elif y_46_re <= 5.8e-186:
		tmp = 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 <= -1.75e-72)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-186)
		tmp = 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 <= -1.75e-72)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-186)
		tmp = 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, -1.75e-72], t$95$0, If[LessEqual[y$46$re, 5.8e-186], N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.75e-72 or 5.80000000000000038e-186 < y.re

   1. Initial program 44.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 54.4

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

   5. Applied rewrites 54.4%

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

   if -1.75e-72 < y.re < 5.80000000000000038e-186

   1. Initial program 38.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 34.1%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-72}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \sin t\_0\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{-72}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re} \cdot t\_1\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re)) (t_1 (sin t_0)))
   (if (<= y.re -9e-72)
     (* (pow (fma 0.5 (/ (* x.re x.re) x.im) x.im) y.re) t_1)
     (if (<= y.re 5.8e-186)
       (* (log (hypot x.im x.re)) y.im)
       (if (<= y.re 1e-9) t_0 (* (pow x.im y.re) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = sin(t_0);
	double tmp;
	if (y_46_re <= -9e-72) {
		tmp = pow(fma(0.5, ((x_46_re * x_46_re) / x_46_im), x_46_im), y_46_re) * t_1;
	} else if (y_46_re <= 5.8e-186) {
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	} else if (y_46_re <= 1e-9) {
		tmp = t_0;
	} else {
		tmp = pow(x_46_im, y_46_re) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = sin(t_0)
	tmp = 0.0
	if (y_46_re <= -9e-72)
		tmp = Float64((fma(0.5, Float64(Float64(x_46_re * x_46_re) / x_46_im), x_46_im) ^ y_46_re) * t_1);
	elseif (y_46_re <= 5.8e-186)
		tmp = Float64(log(hypot(x_46_im, x_46_re)) * y_46_im);
	elseif (y_46_re <= 1e-9)
		tmp = t_0;
	else
		tmp = Float64((x_46_im ^ y_46_re) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$46$re, -9e-72], N[(N[Power[N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] + x$46$im), $MachinePrecision], y$46$re], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 5.8e-186], N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1e-9], t$95$0, N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9e-72

   1. Initial program 41.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-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 66.5

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

   5. Applied rewrites 66.5%

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

   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 62.9%

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

   if -9e-72 < y.re < 5.80000000000000038e-186

   1. Initial program 38.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 34.1%

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

   if 5.80000000000000038e-186 < y.re < 1.00000000000000006e-9

   1. Initial program 50.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 30.5

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 12.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 30.5%

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

   if 1.00000000000000006e-9 < y.re

   1. Initial program 44.3%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x.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 45.9%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{-72}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \sin t\_0\\ \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot t\_1\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re)) (t_1 (sin t_0)))
   (if (<= y.re -1.5e-51)
     (* (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re) t_1)
     (if (<= y.re 5.8e-186)
       (* (log (hypot x.im x.re)) y.im)
       (if (<= y.re 1e-9) t_0 (* (pow x.im y.re) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = sin(t_0);
	double tmp;
	if (y_46_re <= -1.5e-51) {
		tmp = pow(fma(0.5, ((x_46_im * x_46_im) / x_46_re), x_46_re), y_46_re) * t_1;
	} else if (y_46_re <= 5.8e-186) {
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	} else if (y_46_re <= 1e-9) {
		tmp = t_0;
	} else {
		tmp = pow(x_46_im, y_46_re) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = sin(t_0)
	tmp = 0.0
	if (y_46_re <= -1.5e-51)
		tmp = Float64((fma(0.5, Float64(Float64(x_46_im * x_46_im) / x_46_re), x_46_re) ^ y_46_re) * t_1);
	elseif (y_46_re <= 5.8e-186)
		tmp = Float64(log(hypot(x_46_im, x_46_re)) * y_46_im);
	elseif (y_46_re <= 1e-9)
		tmp = t_0;
	else
		tmp = Float64((x_46_im ^ y_46_re) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$46$re, -1.5e-51], N[(N[Power[N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 5.8e-186], N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1e-9], t$95$0, N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.50000000000000001e-51

   1. Initial program 40.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   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 63.5%

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

   if -1.50000000000000001e-51 < y.re < 5.80000000000000038e-186

   1. Initial program 38.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 42.9%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 33.7%

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

   if 5.80000000000000038e-186 < y.re < 1.00000000000000006e-9

   1. Initial program 50.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 30.5

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 12.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 30.5%

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

   if 1.00000000000000006e-9 < y.re

   1. Initial program 44.3%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x.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 45.9%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \sin t\_0\\ \mathbf{if}\;y.re \leq -1.55 \cdot 10^{-40}:\\ \;\;\;\;{x.re}^{y.re} \cdot t\_1\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re)) (t_1 (sin t_0)))
   (if (<= y.re -1.55e-40)
     (* (pow x.re y.re) t_1)
     (if (<= y.re 5.8e-186)
       (* (log (hypot x.im x.re)) y.im)
       (if (<= y.re 1e-9) t_0 (* (pow x.im y.re) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = sin(t_0);
	double tmp;
	if (y_46_re <= -1.55e-40) {
		tmp = pow(x_46_re, y_46_re) * t_1;
	} else if (y_46_re <= 5.8e-186) {
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	} else if (y_46_re <= 1e-9) {
		tmp = t_0;
	} else {
		tmp = pow(x_46_im, y_46_re) * t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y_46_re <= -1.55e-40) {
		tmp = Math.pow(x_46_re, y_46_re) * t_1;
	} else if (y_46_re <= 5.8e-186) {
		tmp = Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im;
	} else if (y_46_re <= 1e-9) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_re
	t_1 = math.sin(t_0)
	tmp = 0
	if y_46_re <= -1.55e-40:
		tmp = math.pow(x_46_re, y_46_re) * t_1
	elif y_46_re <= 5.8e-186:
		tmp = math.log(math.hypot(x_46_im, x_46_re)) * y_46_im
	elif y_46_re <= 1e-9:
		tmp = t_0
	else:
		tmp = math.pow(x_46_im, y_46_re) * t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = sin(t_0)
	tmp = 0.0
	if (y_46_re <= -1.55e-40)
		tmp = Float64((x_46_re ^ y_46_re) * t_1);
	elseif (y_46_re <= 5.8e-186)
		tmp = Float64(log(hypot(x_46_im, x_46_re)) * y_46_im);
	elseif (y_46_re <= 1e-9)
		tmp = t_0;
	else
		tmp = Float64((x_46_im ^ y_46_re) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y_46_re <= -1.55e-40)
		tmp = (x_46_re ^ y_46_re) * t_1;
	elseif (y_46_re <= 5.8e-186)
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	elseif (y_46_re <= 1e-9)
		tmp = t_0;
	else
		tmp = (x_46_im ^ y_46_re) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$46$re, -1.55e-40], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 5.8e-186], N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1e-9], t$95$0, N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.55000000000000005e-40

   1. Initial program 40.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   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 54.5%

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

   if -1.55000000000000005e-40 < y.re < 5.80000000000000038e-186

   1. Initial program 38.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 40.4%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 31.7%

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

   if 5.80000000000000038e-186 < y.re < 1.00000000000000006e-9

   1. Initial program 50.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 30.5

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 12.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 30.5%

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

   if 1.00000000000000006e-9 < y.re

   1. Initial program 44.3%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x.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 45.9%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{-40}:\\ \;\;\;\;{x.re}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := {x.im}^{y.re} \cdot \sin t\_0\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re)) (t_1 (* (pow x.im y.re) (sin t_0))))
   (if (<= y.re -2.5e-68)
     t_1
     (if (<= y.re 5.8e-186)
       (* (log (hypot x.im x.re)) y.im)
       (if (<= y.re 1e-9) t_0 t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = pow(x_46_im, y_46_re) * sin(t_0);
	double tmp;
	if (y_46_re <= -2.5e-68) {
		tmp = t_1;
	} else if (y_46_re <= 5.8e-186) {
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	} else if (y_46_re <= 1e-9) {
		tmp = t_0;
	} 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.atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = Math.pow(x_46_im, y_46_re) * Math.sin(t_0);
	double tmp;
	if (y_46_re <= -2.5e-68) {
		tmp = t_1;
	} else if (y_46_re <= 5.8e-186) {
		tmp = Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im;
	} else if (y_46_re <= 1e-9) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_re
	t_1 = math.pow(x_46_im, y_46_re) * math.sin(t_0)
	tmp = 0
	if y_46_re <= -2.5e-68:
		tmp = t_1
	elif y_46_re <= 5.8e-186:
		tmp = math.log(math.hypot(x_46_im, x_46_re)) * y_46_im
	elif y_46_re <= 1e-9:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = Float64((x_46_im ^ y_46_re) * sin(t_0))
	tmp = 0.0
	if (y_46_re <= -2.5e-68)
		tmp = t_1;
	elseif (y_46_re <= 5.8e-186)
		tmp = Float64(log(hypot(x_46_im, x_46_re)) * y_46_im);
	elseif (y_46_re <= 1e-9)
		tmp = t_0;
	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 = atan2(x_46_im, x_46_re) * y_46_re;
	t_1 = (x_46_im ^ y_46_re) * sin(t_0);
	tmp = 0.0;
	if (y_46_re <= -2.5e-68)
		tmp = t_1;
	elseif (y_46_re <= 5.8e-186)
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	elseif (y_46_re <= 1e-9)
		tmp = t_0;
	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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.5e-68], t$95$1, If[LessEqual[y$46$re, 5.8e-186], N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1e-9], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.49999999999999986e-68 or 1.00000000000000006e-9 < y.re

   1. Initial program 42.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 60.4

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

   5. Applied rewrites 60.4%

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

   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 47.3%

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

   if -2.49999999999999986e-68 < y.re < 5.80000000000000038e-186

   1. Initial program 38.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 34.1%

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

   if 5.80000000000000038e-186 < y.re < 1.00000000000000006e-9

   1. Initial program 50.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 30.5

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 12.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 30.5%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-68}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 24.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2}\right) \cdot 0.5\right) \cdot y.im\\ \end{array} \end{array} \]

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 -1.75e-72)
     t_0
     (if (<= y.re 5.8e-186)
       (* (log (hypot x.im x.re)) y.im)
       (if (<= y.re 1.3e+54)
         t_0
         (* (* (log (pow (hypot x.im x.re) 2.0)) 0.5) 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 <= -1.75e-72) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-186) {
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	} else if (y_46_re <= 1.3e+54) {
		tmp = t_0;
	} else {
		tmp = (log(pow(hypot(x_46_im, x_46_re), 2.0)) * 0.5) * y_46_im;
	}
	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 = 1.0 * Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -1.75e-72) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-186) {
		tmp = Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im;
	} else if (y_46_re <= 1.3e+54) {
		tmp = t_0;
	} else {
		tmp = (Math.log(Math.pow(Math.hypot(x_46_im, x_46_re), 2.0)) * 0.5) * y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -1.75e-72:
		tmp = t_0
	elif y_46_re <= 5.8e-186:
		tmp = math.log(math.hypot(x_46_im, x_46_re)) * y_46_im
	elif y_46_re <= 1.3e+54:
		tmp = t_0
	else:
		tmp = (math.log(math.pow(math.hypot(x_46_im, x_46_re), 2.0)) * 0.5) * 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 <= -1.75e-72)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-186)
		tmp = Float64(log(hypot(x_46_im, x_46_re)) * y_46_im);
	elseif (y_46_re <= 1.3e+54)
		tmp = t_0;
	else
		tmp = Float64(Float64(log((hypot(x_46_im, x_46_re) ^ 2.0)) * 0.5) * y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -1.75e-72)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-186)
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	elseif (y_46_re <= 1.3e+54)
		tmp = t_0;
	else
		tmp = (log((hypot(x_46_im, x_46_re) ^ 2.0)) * 0.5) * y_46_im;
	end
	tmp_2 = 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, -1.75e-72], t$95$0, If[LessEqual[y$46$re, 5.8e-186], N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.3e+54], t$95$0, N[(N[(N[Log[N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.75e-72 or 5.80000000000000038e-186 < y.re < 1.30000000000000003e54

   1. Initial program 44.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-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 57.9

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

   5. Applied rewrites 57.9%

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

   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 15.0%

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

   if -1.75e-72 < y.re < 5.80000000000000038e-186

   1. Initial program 38.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 34.1%

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

   if 1.30000000000000003e54 < y.re

   1. Initial program 42.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 7.9%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 2.5%

       \[\leadsto y.im \cdot \color{blue}{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 28.1%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-72}:\\ \;\;\;\;1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+54}:\\ \;\;\;\;1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2}\right) \cdot 0.5\right) \cdot y.im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 17.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(-1, \log \left(\frac{-1}{x.re}\right), t\_0\right) \cdot y.im\\ \mathbf{elif}\;x.re \leq 6.5 \cdot 10^{-136}:\\ \;\;\;\;\left(\log x.re + t\_0\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* x.im x.im)) (* x.re x.re))))
   (if (<= x.re -5e-309)
     (* (fma -1.0 (log (/ -1.0 x.re)) t_0) y.im)
     (if (<= x.re 6.5e-136)
       (* (+ (log x.re) t_0) y.im)
       (* (log (hypot x.im x.re)) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (0.5 * (x_46_im * x_46_im)) / (x_46_re * x_46_re);
	double tmp;
	if (x_46_re <= -5e-309) {
		tmp = fma(-1.0, log((-1.0 / x_46_re)), t_0) * y_46_im;
	} else if (x_46_re <= 6.5e-136) {
		tmp = (log(x_46_re) + t_0) * y_46_im;
	} else {
		tmp = log(hypot(x_46_im, x_46_re)) * y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(0.5 * Float64(x_46_im * x_46_im)) / Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (x_46_re <= -5e-309)
		tmp = Float64(fma(-1.0, log(Float64(-1.0 / x_46_re)), t_0) * y_46_im);
	elseif (x_46_re <= 6.5e-136)
		tmp = Float64(Float64(log(x_46_re) + t_0) * y_46_im);
	else
		tmp = Float64(log(hypot(x_46_im, 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[(N[(0.5 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -5e-309], N[(N[(-1.0 * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[x$46$re, 6.5e-136], N[(N[(N[Log[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision], N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -4.9999999999999995e-309

   1. Initial program 41.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 23.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 12.9%

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

   12. Taylor expanded in x.re around -inf

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

   14. Applied rewrites 16.8%

       \[\leadsto y.im \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{x.re}\right), \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\right) \]

   if -4.9999999999999995e-309 < x.re < 6.50000000000000011e-136

   1. Initial program 52.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 15.0%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 8.5%

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

   12. Taylor expanded in x.im around 0

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

   14. Applied rewrites 34.9%

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

   if 6.50000000000000011e-136 < x.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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 19.3%

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

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(-1, \log \left(\frac{-1}{x.re}\right), \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\right) \cdot y.im\\ \mathbf{elif}\;x.re \leq 6.5 \cdot 10^{-136}:\\ \;\;\;\;\left(\log x.re + \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 17.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(-1, \log \left(\frac{-1}{x.re}\right), t\_0\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\left(\log x.re + t\_0\right) \cdot y.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* x.im x.im)) (* x.re x.re))))
   (if (<= x.re -5e-309)
     (* (fma -1.0 (log (/ -1.0 x.re)) t_0) y.im)
     (* (+ (log x.re) t_0) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (0.5 * (x_46_im * x_46_im)) / (x_46_re * x_46_re);
	double tmp;
	if (x_46_re <= -5e-309) {
		tmp = fma(-1.0, log((-1.0 / x_46_re)), t_0) * y_46_im;
	} else {
		tmp = (log(x_46_re) + t_0) * y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(0.5 * Float64(x_46_im * x_46_im)) / Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (x_46_re <= -5e-309)
		tmp = Float64(fma(-1.0, log(Float64(-1.0 / x_46_re)), t_0) * y_46_im);
	else
		tmp = Float64(Float64(log(x_46_re) + t_0) * 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[(N[(0.5 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -5e-309], N[(N[(-1.0 * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision], N[(N[(N[Log[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.9999999999999995e-309

   1. Initial program 41.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 44.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 23.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 12.9%

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

   12. Taylor expanded in x.re around -inf

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

   14. Applied rewrites 16.8%

       \[\leadsto y.im \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{x.re}\right), \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\right) \]

   if -4.9999999999999995e-309 < x.re

   1. Initial program 43.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 23.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 16.2%

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

   12. Taylor expanded in x.im around 0

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

   14. Applied rewrites 19.4%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(-1, \log \left(\frac{-1}{x.re}\right), \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\left(\log x.re + \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re \cdot x.re}\right) \cdot y.im\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 14.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 6.2 \cdot 10^{-233}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5 \cdot \left(x.re \cdot x.re\right)}{x.im \cdot x.im} + \log x.im\right) \cdot y.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im 6.2e-233)
   (* (atan2 x.im x.re) y.re)
   (* (+ (/ (* 0.5 (* 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 tmp;
	if (x_46_im <= 6.2e-233) {
		tmp = atan2(x_46_im, x_46_re) * y_46_re;
	} else {
		tmp = (((0.5 * (x_46_re * x_46_re)) / (x_46_im * x_46_im)) + log(x_46_im)) * 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_46im <= 6.2d-233) then
        tmp = atan2(x_46im, x_46re) * y_46re
    else
        tmp = (((0.5d0 * (x_46re * x_46re)) / (x_46im * x_46im)) + log(x_46im)) * 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_im <= 6.2e-233) {
		tmp = Math.atan2(x_46_im, x_46_re) * y_46_re;
	} else {
		tmp = (((0.5 * (x_46_re * x_46_re)) / (x_46_im * x_46_im)) + Math.log(x_46_im)) * 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_im <= 6.2e-233:
		tmp = math.atan2(x_46_im, x_46_re) * y_46_re
	else:
		tmp = (((0.5 * (x_46_re * x_46_re)) / (x_46_im * x_46_im)) + math.log(x_46_im)) * 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_im <= 6.2e-233)
		tmp = Float64(atan(x_46_im, x_46_re) * y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(x_46_re * x_46_re)) / Float64(x_46_im * x_46_im)) + log(x_46_im)) * 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_im <= 6.2e-233)
		tmp = atan2(x_46_im, x_46_re) * y_46_re;
	else
		tmp = (((0.5 * (x_46_re * x_46_re)) / (x_46_im * x_46_im)) + log(x_46_im)) * 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$46$im, 6.2e-233], N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] + N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 6.2000000000000003e-233

   1. Initial program 41.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 18.6%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 16.0%

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

   if 6.2000000000000003e-233 < x.im

   1. Initial program 43.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 13.1%

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

   12. Taylor expanded in x.re around 0

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

   14. Applied rewrites 12.0%

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

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6.2 \cdot 10^{-233}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5 \cdot \left(x.re \cdot x.re\right)}{x.im \cdot x.im} + \log x.im\right) \cdot y.im\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 9.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 8.8 \cdot 10^{-198}:\\ \;\;\;\;\log x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\log x.re \cdot y.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 8.8e-198) (* (log x.im) y.im) (* (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 <= 8.8e-198) {
		tmp = log(x_46_im) * y_46_im;
	} else {
		tmp = 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 <= 8.8d-198) then
        tmp = log(x_46im) * y_46im
    else
        tmp = 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 <= 8.8e-198) {
		tmp = Math.log(x_46_im) * y_46_im;
	} else {
		tmp = 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 <= 8.8e-198:
		tmp = math.log(x_46_im) * y_46_im
	else:
		tmp = 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 <= 8.8e-198)
		tmp = Float64(log(x_46_im) * y_46_im);
	else
		tmp = 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 <= 8.8e-198)
		tmp = log(x_46_im) * y_46_im;
	else
		tmp = 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, 8.8e-198], N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision], N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 8.8000000000000001e-198

   1. Initial program 43.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 22.8%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 13.0%

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

   12. Taylor expanded in x.re around 0

       \[\leadsto y.im \cdot \log x.im \]
   13. Step-by-step derivation

   14. Applied rewrites 4.4%

       \[\leadsto y.im \cdot \log x.im \]

   if 8.8000000000000001e-198 < x.re

   1. Initial program 40.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 24.9%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 16.7%

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

   12. Taylor expanded in x.im around 0

       \[\leadsto y.im \cdot \log x.re \]
   13. Step-by-step derivation

   14. Applied rewrites 13.8%

       \[\leadsto y.im \cdot \log x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 8.8 \cdot 10^{-198}:\\ \;\;\;\;\log x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\log x.re \cdot y.im\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 13.6% accurate, 6.4× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* (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 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 = 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 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 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(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 = 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]

Derivation

1. Initial program 42.2%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
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 42.5

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

5. Applied rewrites 42.5%

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

6. Taylor expanded in x.im around 0

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

8. Applied rewrites 16.1%

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

9. Taylor expanded in y.re around 0

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

11. Applied rewrites 12.4%

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

12. Final simplification 12.4%

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

Alternative 21: 4.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \log x.im \cdot y.im \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* (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 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 = 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 Math.log(x_46_im) * y_46_im;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 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(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 = log(x_46_im) * y_46_im;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision]

Derivation

1. Initial program 42.2%

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

3. Taylor expanded in y.im around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 46.4%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 23.7%

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

9. Taylor expanded in y.re around 0

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

11. Applied rewrites 14.6%

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

12. Taylor expanded in x.re around 0

    \[\leadsto y.im \cdot \log x.im \]
13. Step-by-step derivation

14. Applied rewrites 3.7%

    \[\leadsto y.im \cdot \log x.im \]

15. Final simplification 3.7%

    \[\leadsto \log x.im \cdot y.im \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(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)))))