powComplex, imaginary part

Percentage Accurate: 39.5% → 76.6%

Time: 12.8s

Alternatives: 19

Speedup: 2.1×

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 39.5% 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: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_2 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-19}:\\ \;\;\;\;t\_1 \cdot \sin \left(\mathsf{fma}\left(y.im, \frac{t\_2}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.4:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \mathsf{fma}\left(\cos t\_0 \cdot t\_2, y.im, \sin t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sin \left(t\_2 \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re))
        (t_1
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im))))
        (t_2 (log (hypot x.im x.re))))
   (if (<= y.re -4.5e-19)
     (* t_1 (sin (* (fma y.im (/ t_2 y.re) (atan2 x.im x.re)) y.re)))
     (if (<= y.re 1.4)
       (*
        (pow (exp (- y.im)) (atan2 x.im x.re))
        (fma (* (cos t_0) t_2) y.im (sin t_0)))
       (* t_1 (sin (* t_2 y.im)))))))

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 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_2 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -4.5e-19) {
		tmp = t_1 * sin((fma(y_46_im, (t_2 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	} else if (y_46_re <= 1.4) {
		tmp = pow(exp(-y_46_im), atan2(x_46_im, x_46_re)) * fma((cos(t_0) * t_2), y_46_im, sin(t_0));
	} else {
		tmp = t_1 * sin((t_2 * y_46_im));
	}
	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 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_2 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -4.5e-19)
		tmp = Float64(t_1 * sin(Float64(fma(y_46_im, Float64(t_2 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)));
	elseif (y_46_re <= 1.4)
		tmp = Float64((exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)) * fma(Float64(cos(t_0) * t_2), y_46_im, sin(t_0)));
	else
		tmp = Float64(t_1 * sin(Float64(t_2 * 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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $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, -4.5e-19], N[(t$95$1 * 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]), $MachinePrecision], If[LessEqual[y$46$re, 1.4], N[(N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Cos[t$95$0], $MachinePrecision] * t$95$2), $MachinePrecision] * y$46$im + N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[N[(t$95$2 * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.50000000000000013e-19

   1. Initial program 51.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   if -4.50000000000000013e-19 < y.re < 1.3999999999999999

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-atan2.f64 86.6

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

   8. Applied rewrites 86.6%

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

   if 1.3999999999999999 < y.re

   1. Initial program 26.9%

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

   3. 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 74.7

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

   5. Applied rewrites 74.7%

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

Alternative 2: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := \sin \left(\mathsf{fma}\left(y.im, \frac{t\_1}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sin \left(t\_1 \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im))))
        (t_1 (log (hypot x.im x.re)))
        (t_2 (sin (* (fma y.im (/ t_1 y.re) (atan2 x.im x.re)) y.re))))
   (if (<= y.re -1e-5)
     (* t_0 t_2)
     (if (<= y.re 2.1e-5)
       (* (pow (exp (- y.im)) (atan2 x.im x.re)) t_2)
       (* t_0 (sin (* t_1 y.im)))))))

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_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_1 = log(hypot(x_46_im, x_46_re));
	double t_2 = sin((fma(y_46_im, (t_1 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	double tmp;
	if (y_46_re <= -1e-5) {
		tmp = t_0 * t_2;
	} else if (y_46_re <= 2.1e-5) {
		tmp = pow(exp(-y_46_im), atan2(x_46_im, x_46_re)) * t_2;
	} else {
		tmp = t_0 * sin((t_1 * y_46_im));
	}
	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_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_1 = log(hypot(x_46_im, x_46_re))
	t_2 = sin(Float64(fma(y_46_im, Float64(t_1 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -1e-5)
		tmp = Float64(t_0 * t_2);
	elseif (y_46_re <= 2.1e-5)
		tmp = Float64((exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)) * t_2);
	else
		tmp = Float64(t_0 * sin(Float64(t_1 * 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[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[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]}, If[LessEqual[y$46$re, -1e-5], N[(t$95$0 * t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-5], N[(N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$0 * N[Sin[N[(t$95$1 * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.00000000000000008e-5

   1. Initial program 52.4%

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

   3. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if -1.00000000000000008e-5 < y.re < 2.09999999999999988e-5

   1. Initial program 37.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 60.1

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

   8. Applied rewrites 60.1%

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

   9. Taylor expanded in y.re around 0

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

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

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

       2. exp-prod N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lower-atan2.f64 74.9

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

   11. Applied rewrites 74.9%

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

   if 2.09999999999999988e-5 < y.re

   1. Initial program 28.6%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 3: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;t\_1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \frac{t\_0}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sin \left(t\_0 \cdot y.im\right)\\ \end{array} \end{array} \]

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.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -2.1e-5)
     (* t_1 (sin (* (atan2 x.im x.re) y.re)))
     (if (<= y.re 2.1e-5)
       (*
        (pow (exp (- y.im)) (atan2 x.im x.re))
        (sin (* (fma y.im (/ t_0 y.re) (atan2 x.im x.re)) y.re)))
       (* t_1 (sin (* 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 = log(hypot(x_46_im, x_46_re));
	double t_1 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -2.1e-5) {
		tmp = t_1 * sin((atan2(x_46_im, x_46_re) * y_46_re));
	} else if (y_46_re <= 2.1e-5) {
		tmp = pow(exp(-y_46_im), atan2(x_46_im, x_46_re)) * sin((fma(y_46_im, (t_0 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	} else {
		tmp = t_1 * sin((t_0 * y_46_im));
	}
	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_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.1e-5)
		tmp = Float64(t_1 * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	elseif (y_46_re <= 2.1e-5)
		tmp = Float64((exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)) * sin(Float64(fma(y_46_im, Float64(t_0 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)));
	else
		tmp = Float64(t_1 * sin(Float64(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[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$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e-5], N[(t$95$1 * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-5], N[(N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(y$46$im * N[(t$95$0 / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.09999999999999988e-5

   1. Initial program 52.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

   if -2.09999999999999988e-5 < y.re < 2.09999999999999988e-5

   1. Initial program 37.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 60.1

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

   8. Applied rewrites 60.1%

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

   9. Taylor expanded in y.re around 0

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

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

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

       2. exp-prod N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lower-atan2.f64 74.9

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

   11. Applied rewrites 74.9%

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

   if 2.09999999999999988e-5 < y.re

   1. Initial program 28.6%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 4: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{-89} \lor \neg \left(y.im \leq 4.2 \cdot 10^{-41}\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t\_0 \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t\_0, \frac{y.im}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re))))
   (if (or (<= y.im -8e-89) (not (<= y.im 4.2e-41)))
     (*
      (exp
       (-
        (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
        (* (atan2 x.im x.re) y.im)))
      (sin (* t_0 y.im)))
     (*
      (sin (* (fma t_0 (/ y.im y.re) (atan2 x.im x.re)) y.re))
      (pow (hypot x.im x.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if ((y_46_im <= -8e-89) || !(y_46_im <= 4.2e-41)) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((t_0 * y_46_im));
	} else {
		tmp = sin((fma(t_0, (y_46_im / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_im <= -8e-89) || !(y_46_im <= 4.2e-41))
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(t_0 * y_46_im)));
	else
		tmp = Float64(sin(Float64(fma(t_0, Float64(y_46_im / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y$46$im, -8e-89], N[Not[LessEqual[y$46$im, 4.2e-41]], $MachinePrecision]], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[(t$95$0 * N[(y$46$im / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.00000000000000031e-89 or 4.20000000000000025e-41 < y.im

   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 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 69.3

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

   5. Applied rewrites 69.3%

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

   if -8.00000000000000031e-89 < y.im < 4.20000000000000025e-41

   1. Initial program 35.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 89.7

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

   8. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 89.7

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

   10. Applied rewrites 89.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{-89} \lor \neg \left(y.im \leq 4.2 \cdot 10^{-41}\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \frac{y.im}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \sin \left(t\_0 \cdot y.im\right) \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -2.32 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -102000:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+22}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t\_0, \frac{y.im}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* (sin (* t_0 y.im)) (pow (exp (- y.im)) (atan2 x.im x.re)))))
   (if (<= y.im -2.32e+126)
     t_1
     (if (<= y.im -102000.0)
       (*
        (exp
         (- (* (log (sqrt (* x.im x.im))) y.re) (* (atan2 x.im x.re) y.im)))
        (sin (* (atan2 x.im x.re) y.re)))
       (if (<= y.im 1.02e+22)
         (*
          (sin (* (fma t_0 (/ y.im y.re) (atan2 x.im x.re)) y.re))
          (pow (hypot x.im x.re) 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 = log(hypot(x_46_im, x_46_re));
	double t_1 = sin((t_0 * y_46_im)) * pow(exp(-y_46_im), atan2(x_46_im, x_46_re));
	double tmp;
	if (y_46_im <= -2.32e+126) {
		tmp = t_1;
	} else if (y_46_im <= -102000.0) {
		tmp = exp(((log(sqrt((x_46_im * x_46_im))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	} else if (y_46_im <= 1.02e+22) {
		tmp = sin((fma(t_0, (y_46_im / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_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 = Float64(sin(Float64(t_0 * y_46_im)) * (exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -2.32e+126)
		tmp = t_1;
	elseif (y_46_im <= -102000.0)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(x_46_im * x_46_im))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	elseif (y_46_im <= 1.02e+22)
		tmp = Float64(sin(Float64(fma(t_0, Float64(y_46_im / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = 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[(N[Sin[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.32e+126], t$95$1, If[LessEqual[y$46$im, -102000.0], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.02e+22], N[(N[Sin[N[(N[(t$95$0 * N[(y$46$im / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.31999999999999995e126 or 1.02e22 < y.im

   1. Initial program 33.9%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y.re around 0

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

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

      11. exp-prod N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if -2.31999999999999995e126 < y.im < -102000

   1. Initial program 32.1%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in x.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.1

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

   8. Applied rewrites 68.1%

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

   if -102000 < y.im < 1.02e22

   1. Initial program 42.8%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 42.5

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 87.9

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

   8. Applied rewrites 87.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.9

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

   10. Applied rewrites 87.9%

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

Alternative 6: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \sin \left(t\_0 \cdot y.im\right) \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -2.32 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -102000:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+22}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im, \frac{t\_0}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* (sin (* t_0 y.im)) (pow (exp (- y.im)) (atan2 x.im x.re)))))
   (if (<= y.im -2.32e+126)
     t_1
     (if (<= y.im -102000.0)
       (*
        (exp
         (- (* (log (sqrt (* x.im x.im))) y.re) (* (atan2 x.im x.re) y.im)))
        (sin (* (atan2 x.im x.re) y.re)))
       (if (<= y.im 1.02e+22)
         (*
          (pow (hypot x.im x.re) y.re)
          (sin (* (fma y.im (/ t_0 y.re) (atan2 x.im x.re)) 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 = log(hypot(x_46_im, x_46_re));
	double t_1 = sin((t_0 * y_46_im)) * pow(exp(-y_46_im), atan2(x_46_im, x_46_re));
	double tmp;
	if (y_46_im <= -2.32e+126) {
		tmp = t_1;
	} else if (y_46_im <= -102000.0) {
		tmp = exp(((log(sqrt((x_46_im * x_46_im))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	} else if (y_46_im <= 1.02e+22) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) * sin((fma(y_46_im, (t_0 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(sin(Float64(t_0 * y_46_im)) * (exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -2.32e+126)
		tmp = t_1;
	elseif (y_46_im <= -102000.0)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(x_46_im * x_46_im))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	elseif (y_46_im <= 1.02e+22)
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) * sin(Float64(fma(y_46_im, Float64(t_0 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)));
	else
		tmp = 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[(N[Sin[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.32e+126], t$95$1, If[LessEqual[y$46$im, -102000.0], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.02e+22], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(N[(y$46$im * N[(t$95$0 / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.31999999999999995e126 or 1.02e22 < y.im

   1. Initial program 33.9%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y.re around 0

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

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

      11. exp-prod N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if -2.31999999999999995e126 < y.im < -102000

   1. Initial program 32.1%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in x.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.1

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

   8. Applied rewrites 68.1%

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

   if -102000 < y.im < 1.02e22

   1. Initial program 42.8%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 42.5

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 87.9

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

   8. Applied rewrites 87.9%

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

Alternative 7: 65.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \sin \left(t\_0 \cdot y.im\right)\\ \mathbf{if}\;y.re \leq -4.3 \cdot 10^{-11}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq -1.82 \cdot 10^{-218}:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(y.im, \frac{t\_0}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+14}:\\ \;\;\;\;t\_1 \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re))) (t_1 (sin (* t_0 y.im))))
   (if (<= y.re -4.3e-11)
     (*
      (exp
       (-
        (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
        (* (atan2 x.im x.re) y.im)))
      (sin (* (atan2 x.im x.re) y.re)))
     (if (<= y.re -1.82e-218)
       (* 1.0 (sin (* (fma y.im (/ t_0 y.re) (atan2 x.im x.re)) y.re)))
       (if (<= y.re 1.9e+14)
         (* t_1 (pow (exp (- y.im)) (atan2 x.im x.re)))
         (* (pow (hypot x.im x.re) 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 = log(hypot(x_46_im, x_46_re));
	double t_1 = sin((t_0 * y_46_im));
	double tmp;
	if (y_46_re <= -4.3e-11) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	} else if (y_46_re <= -1.82e-218) {
		tmp = 1.0 * sin((fma(y_46_im, (t_0 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	} else if (y_46_re <= 1.9e+14) {
		tmp = t_1 * pow(exp(-y_46_im), atan2(x_46_im, x_46_re));
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) * 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 = sin(Float64(t_0 * y_46_im))
	tmp = 0.0
	if (y_46_re <= -4.3e-11)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	elseif (y_46_re <= -1.82e-218)
		tmp = Float64(1.0 * sin(Float64(fma(y_46_im, Float64(t_0 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)));
	elseif (y_46_re <= 1.9e+14)
		tmp = Float64(t_1 * (exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)));
	else
		tmp = Float64((hypot(x_46_im, x_46_re) ^ 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[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -4.3e-11], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.82e-218], N[(1.0 * N[Sin[N[(N[(y$46$im * N[(t$95$0 / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.9e+14], N[(t$95$1 * N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.30000000000000001e-11

   1. Initial program 53.1%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   if -4.30000000000000001e-11 < y.re < -1.81999999999999998e-218

   1. Initial program 27.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 25.0

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

   5. Applied rewrites 25.0%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 63.0

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 63.0%

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

   if -1.81999999999999998e-218 < y.re < 1.9e14

   1. Initial program 41.7%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in y.re around 0

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

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

      11. exp-prod N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 75.9

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

   8. Applied rewrites 75.9%

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

   if 1.9e14 < y.re

   1. Initial program 26.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 20.5

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

   5. Applied rewrites 20.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 67.4

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

   8. Applied rewrites 67.4%

      \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -4.3 \cdot 10^{-11}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_1} \cdot t\_2\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-63}:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(y.im, \frac{t\_0}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+38}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im}\right) \cdot y.re - t\_1} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(t\_0 \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -4.3e-11)
     (*
      (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_1))
      t_2)
     (if (<= y.re 8.5e-63)
       (* 1.0 (sin (* (fma y.im (/ t_0 y.re) (atan2 x.im x.re)) y.re)))
       (if (<= y.re 4e+38)
         (* (exp (- (* (log (sqrt (* x.im x.im))) y.re) t_1)) t_2)
         (* (pow (hypot x.im x.re) y.re) (sin (* 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 = log(hypot(x_46_im, x_46_re));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -4.3e-11) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_1)) * t_2;
	} else if (y_46_re <= 8.5e-63) {
		tmp = 1.0 * sin((fma(y_46_im, (t_0 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	} else if (y_46_re <= 4e+38) {
		tmp = exp(((log(sqrt((x_46_im * x_46_im))) * y_46_re) - t_1)) * t_2;
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) * sin((t_0 * y_46_im));
	}
	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 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -4.3e-11)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_1)) * t_2);
	elseif (y_46_re <= 8.5e-63)
		tmp = Float64(1.0 * sin(Float64(fma(y_46_im, Float64(t_0 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)));
	elseif (y_46_re <= 4e+38)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(x_46_im * x_46_im))) * y_46_re) - t_1)) * t_2);
	else
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) * sin(Float64(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[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -4.3e-11], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 8.5e-63], N[(1.0 * N[Sin[N[(N[(y$46$im * N[(t$95$0 / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4e+38], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.30000000000000001e-11

   1. Initial program 53.1%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   if -4.30000000000000001e-11 < y.re < 8.49999999999999969e-63

   1. Initial program 35.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 27.6

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

   5. Applied rewrites 27.6%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 62.4

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 62.4%

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

   if 8.49999999999999969e-63 < y.re < 3.99999999999999991e38

   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

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

   if 3.99999999999999991e38 < y.re

   1. Initial program 26.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 70.4

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

   8. Applied rewrites 70.4%

      \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -3.55 \cdot 10^{-12}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-63}:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(y.im, \frac{t\_1}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+38}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sin \left(t\_1 \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (log (hypot x.im x.re)))
        (t_2 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -3.55e-12)
     (* t_0 t_2)
     (if (<= y.re 8.5e-63)
       (* 1.0 (sin (* (fma y.im (/ t_1 y.re) (atan2 x.im x.re)) y.re)))
       (if (<= y.re 4e+38)
         (*
          (exp
           (- (* (log (sqrt (* x.im x.im))) y.re) (* (atan2 x.im x.re) y.im)))
          t_2)
         (* t_0 (sin (* t_1 y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = log(hypot(x_46_im, x_46_re));
	double t_2 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -3.55e-12) {
		tmp = t_0 * t_2;
	} else if (y_46_re <= 8.5e-63) {
		tmp = 1.0 * sin((fma(y_46_im, (t_1 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	} else if (y_46_re <= 4e+38) {
		tmp = exp(((log(sqrt((x_46_im * x_46_im))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_2;
	} else {
		tmp = t_0 * sin((t_1 * y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = log(hypot(x_46_im, x_46_re))
	t_2 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.55e-12)
		tmp = Float64(t_0 * t_2);
	elseif (y_46_re <= 8.5e-63)
		tmp = Float64(1.0 * sin(Float64(fma(y_46_im, Float64(t_1 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)));
	elseif (y_46_re <= 4e+38)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(x_46_im * x_46_im))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * t_2);
	else
		tmp = Float64(t_0 * sin(Float64(t_1 * 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -3.55e-12], N[(t$95$0 * t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 8.5e-63], N[(1.0 * 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]), $MachinePrecision], If[LessEqual[y$46$re, 4e+38], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$0 * N[Sin[N[(t$95$1 * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.55e-12

   1. Initial program 53.1%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 89.2

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

   8. Applied rewrites 89.2%

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

   if -3.55e-12 < y.re < 8.49999999999999969e-63

   1. Initial program 35.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 27.6

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

   5. Applied rewrites 27.6%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 62.4

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 62.4%

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

   if 8.49999999999999969e-63 < y.re < 3.99999999999999991e38

   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

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

   if 3.99999999999999991e38 < y.re

   1. Initial program 26.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 70.4

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

   8. Applied rewrites 70.4%

      \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -3.55 \cdot 10^{-12}:\\ \;\;\;\;t\_0 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 0.046:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(y.im, \frac{t\_1}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sin \left(t\_1 \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re)) (t_1 (log (hypot x.im x.re))))
   (if (<= y.re -3.55e-12)
     (* t_0 (sin (* (atan2 x.im x.re) y.re)))
     (if (<= y.re 0.046)
       (* 1.0 (sin (* (fma y.im (/ t_1 y.re) (atan2 x.im x.re)) y.re)))
       (* t_0 (sin (* t_1 y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -3.55e-12) {
		tmp = t_0 * sin((atan2(x_46_im, x_46_re) * y_46_re));
	} else if (y_46_re <= 0.046) {
		tmp = 1.0 * sin((fma(y_46_im, (t_1 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	} else {
		tmp = t_0 * sin((t_1 * y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -3.55e-12)
		tmp = Float64(t_0 * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	elseif (y_46_re <= 0.046)
		tmp = Float64(1.0 * sin(Float64(fma(y_46_im, Float64(t_1 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)));
	else
		tmp = Float64(t_0 * sin(Float64(t_1 * 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -3.55e-12], N[(t$95$0 * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.046], N[(1.0 * 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]), $MachinePrecision], N[(t$95$0 * N[Sin[N[(t$95$1 * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.55e-12

   1. Initial program 53.1%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 89.2

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

   8. Applied rewrites 89.2%

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

   if -3.55e-12 < y.re < 0.045999999999999999

   1. Initial program 37.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 26.2

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

   5. Applied rewrites 26.2%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 59.1

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

   8. Applied rewrites 59.1%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 58.8%

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

   if 0.045999999999999999 < y.re

   1. Initial program 26.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 19.6

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

   5. Applied rewrites 19.6%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 67.4

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

   8. Applied rewrites 67.4%

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

Alternative 11: 56.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -3.55 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(y.im, \frac{t\_1}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 7.6 \cdot 10^{+223}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \tan^{-1}_* \frac{x.im}{x.re}, y.re, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow (hypot x.im x.re) y.re) (sin (* (atan2 x.im x.re) y.re))))
        (t_1 (log (hypot x.im x.re))))
   (if (<= y.re -3.55e-12)
     t_0
     (if (<= y.re 3.1e-15)
       (* 1.0 (sin (* (fma y.im (/ t_1 y.re) (atan2 x.im x.re)) y.re)))
       (if (<= y.re 7.6e+223)
         t_0
         (* (fma (* t_1 (atan2 x.im x.re)) y.re (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 = pow(hypot(x_46_im, x_46_re), y_46_re) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -3.55e-12) {
		tmp = t_0;
	} else if (y_46_re <= 3.1e-15) {
		tmp = 1.0 * sin((fma(y_46_im, (t_1 / y_46_re), atan2(x_46_im, x_46_re)) * y_46_re));
	} else if (y_46_re <= 7.6e+223) {
		tmp = t_0;
	} else {
		tmp = fma((t_1 * atan2(x_46_im, x_46_re)), y_46_re, atan2(x_46_im, x_46_re)) * y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)))
	t_1 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -3.55e-12)
		tmp = t_0;
	elseif (y_46_re <= 3.1e-15)
		tmp = Float64(1.0 * sin(Float64(fma(y_46_im, Float64(t_1 / y_46_re), atan(x_46_im, x_46_re)) * y_46_re)));
	elseif (y_46_re <= 7.6e+223)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(t_1 * atan(x_46_im, x_46_re)), y_46_re, atan(x_46_im, x_46_re)) * y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $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, -3.55e-12], t$95$0, If[LessEqual[y$46$re, 3.1e-15], N[(1.0 * 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]), $MachinePrecision], If[LessEqual[y$46$re, 7.6e+223], t$95$0, N[(N[(N[(t$95$1 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.55e-12 or 3.0999999999999999e-15 < y.re < 7.6000000000000001e223

   1. Initial program 44.2%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 69.4

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

   8. Applied rewrites 69.4%

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

   if -3.55e-12 < y.re < 3.0999999999999999e-15

   1. Initial program 36.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 26.2

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

   5. Applied rewrites 26.2%

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

   6. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 60.2

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

   8. Applied rewrites 60.2%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 60.1%

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

   if 7.6000000000000001e223 < y.re

   1. Initial program 13.3%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 26.7

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 53.8%

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

Alternative 12: 46.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.im \leq -0.084:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= y.im -0.084)
     (* (pow (* (fma (/ 0.5 x.re) (/ (* x.im x.im) x.re) 1.0) x.re) y.re) t_0)
     (* (pow (hypot x.im x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_im <= -0.084) {
		tmp = pow((fma((0.5 / x_46_re), ((x_46_im * x_46_im) / x_46_re), 1.0) * x_46_re), y_46_re) * t_0;
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) * t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_im <= -0.084)
		tmp = Float64((Float64(fma(Float64(0.5 / x_46_re), Float64(Float64(x_46_im * x_46_im) / x_46_re), 1.0) * x_46_re) ^ y_46_re) * t_0);
	else
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) * t_0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -0.084], N[(N[Power[N[(N[(N[(0.5 / x$46$re), $MachinePrecision] * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + 1.0), $MachinePrecision] * x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -0.0840000000000000052

   1. Initial program 32.0%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 33.2

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 40.3%

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

   if -0.0840000000000000052 < y.im

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 47.5

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

   8. Applied rewrites 47.5%

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

Alternative 13: 40.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;x.re \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= x.re -3.5e-45)
     (* (pow (- x.re) y.re) t_0)
     (*
      (pow (* (fma (/ 0.5 x.re) (/ (* x.im x.im) x.re) 1.0) x.re) y.re)
      t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (x_46_re <= -3.5e-45) {
		tmp = pow(-x_46_re, y_46_re) * t_0;
	} else {
		tmp = pow((fma((0.5 / x_46_re), ((x_46_im * x_46_im) / x_46_re), 1.0) * x_46_re), y_46_re) * t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (x_46_re <= -3.5e-45)
		tmp = Float64((Float64(-x_46_re) ^ y_46_re) * t_0);
	else
		tmp = Float64((Float64(fma(Float64(0.5 / x_46_re), Float64(Float64(x_46_im * x_46_im) / x_46_re), 1.0) * x_46_re) ^ y_46_re) * t_0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -3.5e-45], N[(N[Power[(-x$46$re), y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Power[N[(N[(N[(0.5 / x$46$re), $MachinePrecision] * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + 1.0), $MachinePrecision] * x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.5e-45

   1. Initial program 32.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 50.5

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in x.re around -inf

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

   11. Applied rewrites 50.5%

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

   if -3.5e-45 < x.re

   1. Initial program 41.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 39.0%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{0.5}{x.re}, \frac{x.im \cdot x.im}{x.re}, 1\right) \cdot x.re\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 40.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;x.re \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= x.re -3.5e-45)
     (* (pow (- x.re) y.re) t_0)
     (* (pow (fma (/ (* x.im x.im) x.re) 0.5 x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (x_46_re <= -3.5e-45) {
		tmp = pow(-x_46_re, y_46_re) * t_0;
	} else {
		tmp = pow(fma(((x_46_im * x_46_im) / x_46_re), 0.5, x_46_re), y_46_re) * t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (x_46_re <= -3.5e-45)
		tmp = Float64((Float64(-x_46_re) ^ y_46_re) * t_0);
	else
		tmp = Float64((fma(Float64(Float64(x_46_im * x_46_im) / x_46_re), 0.5, x_46_re) ^ y_46_re) * t_0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -3.5e-45], N[(N[Power[(-x$46$re), y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Power[N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] * 0.5 + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.5e-45

   1. Initial program 32.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 50.5

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in x.re around -inf

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

   11. Applied rewrites 50.5%

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

   if -3.5e-45 < x.re

   1. Initial program 41.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 38.3%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 42.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;x.im \leq -1.05 \cdot 10^{-32}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot t\_0\\ \mathbf{elif}\;x.im \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= x.im -1.05e-32)
     (* (pow (- x.im) y.re) t_0)
     (if (<= x.im 4.8e-9)
       (* (pow (- x.re) y.re) t_0)
       (* (pow x.im y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (x_46_im <= -1.05e-32) {
		tmp = pow(-x_46_im, y_46_re) * t_0;
	} else if (x_46_im <= 4.8e-9) {
		tmp = pow(-x_46_re, y_46_re) * t_0;
	} else {
		tmp = pow(x_46_im, y_46_re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((atan2(x_46im, x_46re) * y_46re))
    if (x_46im <= (-1.05d-32)) then
        tmp = (-x_46im ** y_46re) * t_0
    else if (x_46im <= 4.8d-9) then
        tmp = (-x_46re ** y_46re) * t_0
    else
        tmp = (x_46im ** y_46re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (x_46_im <= -1.05e-32) {
		tmp = Math.pow(-x_46_im, y_46_re) * t_0;
	} else if (x_46_im <= 4.8e-9) {
		tmp = Math.pow(-x_46_re, y_46_re) * t_0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if x_46_im <= -1.05e-32:
		tmp = math.pow(-x_46_im, y_46_re) * t_0
	elif x_46_im <= 4.8e-9:
		tmp = math.pow(-x_46_re, y_46_re) * t_0
	else:
		tmp = math.pow(x_46_im, y_46_re) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (x_46_im <= -1.05e-32)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * t_0);
	elseif (x_46_im <= 4.8e-9)
		tmp = Float64((Float64(-x_46_re) ^ y_46_re) * t_0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (x_46_im <= -1.05e-32)
		tmp = (-x_46_im ^ y_46_re) * t_0;
	elseif (x_46_im <= 4.8e-9)
		tmp = (-x_46_re ^ y_46_re) * t_0;
	else
		tmp = (x_46_im ^ y_46_re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -1.05e-32], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x$46$im, 4.8e-9], N[(N[Power[(-x$46$re), y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.05e-32

   1. Initial program 26.1%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 39.3

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

   8. Applied rewrites 39.3%

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

   9. Taylor expanded in x.im around -inf

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

   11. Applied rewrites 39.3%

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

   if -1.05e-32 < x.im < 4.8e-9

   1. Initial program 49.8%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 44.8

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

   8. Applied rewrites 44.8%

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

   9. Taylor expanded in x.re around -inf

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

   11. Applied rewrites 42.4%

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

   if 4.8e-9 < x.im

   1. Initial program 27.8%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 47.1

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

   8. Applied rewrites 47.1%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 47.0%

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

Alternative 16: 35.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -2.65 \cdot 10^{+26} \lor \neg \left(y.re \leq 5800000000\right):\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (or (<= y.re -2.65e+26) (not (<= y.re 5800000000.0)))
     (* (pow x.im y.re) t_0)
     (* 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if ((y_46_re <= -2.65e+26) || !(y_46_re <= 5800000000.0)) {
		tmp = pow(x_46_im, y_46_re) * t_0;
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((atan2(x_46im, x_46re) * y_46re))
    if ((y_46re <= (-2.65d+26)) .or. (.not. (y_46re <= 5800000000.0d0))) then
        tmp = (x_46im ** y_46re) * t_0
    else
        tmp = 1.0d0 * 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.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if ((y_46_re <= -2.65e+26) || !(y_46_re <= 5800000000.0)) {
		tmp = Math.pow(x_46_im, y_46_re) * t_0;
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if (y_46_re <= -2.65e+26) or not (y_46_re <= 5800000000.0):
		tmp = math.pow(x_46_im, y_46_re) * t_0
	else:
		tmp = 1.0 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if ((y_46_re <= -2.65e+26) || !(y_46_re <= 5800000000.0))
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	else
		tmp = Float64(1.0 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if ((y_46_re <= -2.65e+26) || ~((y_46_re <= 5800000000.0)))
		tmp = (x_46_im ^ y_46_re) * t_0;
	else
		tmp = 1.0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y$46$re, -2.65e+26], N[Not[LessEqual[y$46$re, 5800000000.0]], $MachinePrecision]], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.64999999999999984e26 or 5.8e9 < y.re

   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.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 64.6

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

   8. Applied rewrites 64.6%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 52.5%

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

   if -2.64999999999999984e26 < y.re < 5.8e9

   1. Initial program 38.3%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 25.4

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

   8. Applied rewrites 25.4%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 21.1%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.65 \cdot 10^{+26} \lor \neg \left(y.re \leq 5800000000\right):\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 40.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;x.im \leq -8.3 \cdot 10^{-33}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot t\_0\\ \mathbf{elif}\;x.im \leq 10^{-162}:\\ \;\;\;\;{x.re}^{y.re} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= x.im -8.3e-33)
     (* (pow (- x.im) y.re) t_0)
     (if (<= x.im 1e-162) (* (pow x.re y.re) t_0) (* (pow x.im y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (x_46_im <= -8.3e-33) {
		tmp = pow(-x_46_im, y_46_re) * t_0;
	} else if (x_46_im <= 1e-162) {
		tmp = pow(x_46_re, y_46_re) * t_0;
	} else {
		tmp = pow(x_46_im, y_46_re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((atan2(x_46im, x_46re) * y_46re))
    if (x_46im <= (-8.3d-33)) then
        tmp = (-x_46im ** y_46re) * t_0
    else if (x_46im <= 1d-162) then
        tmp = (x_46re ** y_46re) * t_0
    else
        tmp = (x_46im ** y_46re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (x_46_im <= -8.3e-33) {
		tmp = Math.pow(-x_46_im, y_46_re) * t_0;
	} else if (x_46_im <= 1e-162) {
		tmp = Math.pow(x_46_re, y_46_re) * t_0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if x_46_im <= -8.3e-33:
		tmp = math.pow(-x_46_im, y_46_re) * t_0
	elif x_46_im <= 1e-162:
		tmp = math.pow(x_46_re, y_46_re) * t_0
	else:
		tmp = math.pow(x_46_im, y_46_re) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (x_46_im <= -8.3e-33)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * t_0);
	elseif (x_46_im <= 1e-162)
		tmp = Float64((x_46_re ^ y_46_re) * t_0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (x_46_im <= -8.3e-33)
		tmp = (-x_46_im ^ y_46_re) * t_0;
	elseif (x_46_im <= 1e-162)
		tmp = (x_46_re ^ y_46_re) * t_0;
	else
		tmp = (x_46_im ^ y_46_re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -8.3e-33], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x$46$im, 1e-162], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -8.29999999999999985e-33

   1. Initial program 26.1%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 N/A

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

      13. lower-sin.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

   5. Applied rewrites 51.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 39.3

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   8. Applied rewrites 39.3%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   9. Taylor expanded in x.im around -inf

      \[\leadsto {\left(-1 \cdot x.im\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 39.3%

       \[\leadsto {\left(-x.im\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   if -8.29999999999999985e-33 < x.im < 9.99999999999999954e-163

   1. Initial program 49.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot y.im} + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-atan2.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. lower-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-hypot.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 46.5

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   8. Applied rewrites 46.5%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   9. Taylor expanded in x.im around 0

      \[\leadsto {x.re}^{y.re} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.9%

       \[\leadsto {x.re}^{y.re} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   if 9.99999999999999954e-163 < x.im

   1. Initial program 35.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot y.im} + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-atan2.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. lower-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-hypot.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

   5. Applied rewrites 56.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 44.4

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   8. Applied rewrites 44.4%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   9. Taylor expanded in x.re around 0

      \[\leadsto {x.im}^{y.re} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.4%

       \[\leadsto {x.im}^{y.re} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 37.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -4500:\\ \;\;\;\;{x.re}^{y.re} \cdot t\_0\\ \mathbf{elif}\;y.re \leq 5800000000:\\ \;\;\;\;1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -4500.0)
     (* (pow x.re y.re) t_0)
     (if (<= y.re 5800000000.0) (* 1.0 t_0) (* (pow x.im y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -4500.0) {
		tmp = pow(x_46_re, y_46_re) * t_0;
	} else if (y_46_re <= 5800000000.0) {
		tmp = 1.0 * t_0;
	} else {
		tmp = pow(x_46_im, y_46_re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((atan2(x_46im, x_46re) * y_46re))
    if (y_46re <= (-4500.0d0)) then
        tmp = (x_46re ** y_46re) * t_0
    else if (y_46re <= 5800000000.0d0) then
        tmp = 1.0d0 * t_0
    else
        tmp = (x_46im ** y_46re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -4500.0) {
		tmp = Math.pow(x_46_re, y_46_re) * t_0;
	} else if (y_46_re <= 5800000000.0) {
		tmp = 1.0 * t_0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -4500.0:
		tmp = math.pow(x_46_re, y_46_re) * t_0
	elif y_46_re <= 5800000000.0:
		tmp = 1.0 * t_0
	else:
		tmp = math.pow(x_46_im, y_46_re) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -4500.0)
		tmp = Float64((x_46_re ^ y_46_re) * t_0);
	elseif (y_46_re <= 5800000000.0)
		tmp = Float64(1.0 * t_0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -4500.0)
		tmp = (x_46_re ^ y_46_re) * t_0;
	elseif (y_46_re <= 5800000000.0)
		tmp = 1.0 * t_0;
	else
		tmp = (x_46_im ^ y_46_re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -4500.0], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 5800000000.0], N[(1.0 * t$95$0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4500

   1. Initial program 51.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot y.im} + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-atan2.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. lower-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-hypot.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

   5. Applied rewrites 90.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 90.1

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   8. Applied rewrites 90.1%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   9. Taylor expanded in x.im around 0

      \[\leadsto {x.re}^{y.re} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.0%

       \[\leadsto {x.re}^{y.re} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   if -4500 < y.re < 5.8e9

   1. Initial program 38.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot y.im} + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-atan2.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. lower-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-hypot.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

   5. Applied rewrites 48.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 22.6

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   8. Applied rewrites 22.6%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   9. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 20.3%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   if 5.8e9 < y.re

   1. Initial program 27.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot y.im} + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-atan2.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. lower-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-hypot.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

   5. Applied rewrites 60.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   7. 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 44.2

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   8. Applied rewrites 44.2%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   9. Taylor expanded in x.re around 0

      \[\leadsto {x.im}^{y.re} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 42.9%

       \[\leadsto {x.im}^{y.re} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 13.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* 1.0 (sin (* (atan2 x.im x.re) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 * sin((atan2(x_46_im, x_46_re) * y_46_re));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0 * sin((atan2(x_46im, x_46re) * y_46re))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 * Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 * math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 * sin((atan2(x_46_im, x_46_re) * y_46_re));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Add Preprocessing

3. Taylor expanded in y.im around 0

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot y.im} + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   5. lower-cos.f64 N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. lower-atan2.f64 N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. lower-log.f64 N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right), y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   12. lower-hypot.f64 N/A

       \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}, y.im, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   13. lower-sin.f64 N/A

       \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \]

5. Applied rewrites 61.2%

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.im, \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

6. Taylor expanded in y.im around 0

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
7. 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 44.0

       \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

8. Applied rewrites 44.0%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

9. Taylor expanded in y.re around 0

   \[\leadsto 1 \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
10. Step-by-step derivation

11. Applied rewrites 13.1%

    \[\leadsto 1 \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(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)))))