powComplex, imaginary part

Percentage Accurate: 40.6% → 65.8%

Time: 11.5s

Alternatives: 21

Speedup: 1.5×

• 35.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := \log \left(\frac{1}{x.re}\right)\\ t_3 := \log \left(-1 \cdot x.re\right)\\ t_4 := e^{-1 \cdot \left(y.re \cdot t\_2\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_1} \cdot \sin \left(t\_3 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \leq 2.05 \cdot 10^{-166}:\\ \;\;\;\;t\_4 \cdot \left(\sin t\_0 + -1 \cdot \left(y.im \cdot \left(\cos t\_0 \cdot t\_2\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 1.08:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_1} \cdot \sin \left(\pi - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_2, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2 (log (/ 1.0 x.re)))
        (t_3 (log (* -1.0 x.re)))
        (t_4 (exp (- (* -1.0 (* y.re t_2)) (* y.im (atan2 x.im x.re))))))
   (if (<= x.re -5e-310)
     (*
      (exp (- (* t_3 y.re) t_1))
      (sin (+ (* t_3 y.im) (* (atan2 x.im x.re) y.re))))
     (if (<= x.re 2.05e-166)
       (* t_4 (+ (sin t_0) (* -1.0 (* y.im (* (cos t_0) t_2)))))
       (if (<= x.re 1.08)
         (*
          (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_1))
          (sin (- PI t_0)))
         (* t_4 (sin (fma -1.0 (* y.im t_2) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = log((1.0 / x_46_re));
	double t_3 = log((-1.0 * x_46_re));
	double t_4 = exp(((-1.0 * (y_46_re * t_2)) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (x_46_re <= -5e-310) {
		tmp = exp(((t_3 * y_46_re) - t_1)) * sin(((t_3 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_re <= 2.05e-166) {
		tmp = t_4 * (sin(t_0) + (-1.0 * (y_46_im * (cos(t_0) * t_2))));
	} else if (x_46_re <= 1.08) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_1)) * sin((((double) M_PI) - t_0));
	} else {
		tmp = t_4 * sin(fma(-1.0, (y_46_im * t_2), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = log(Float64(1.0 / x_46_re))
	t_3 = log(Float64(-1.0 * x_46_re))
	t_4 = exp(Float64(Float64(-1.0 * Float64(y_46_re * t_2)) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (x_46_re <= -5e-310)
		tmp = Float64(exp(Float64(Float64(t_3 * y_46_re) - t_1)) * sin(Float64(Float64(t_3 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_re <= 2.05e-166)
		tmp = Float64(t_4 * Float64(sin(t_0) + Float64(-1.0 * Float64(y_46_im * Float64(cos(t_0) * t_2)))));
	elseif (x_46_re <= 1.08)
		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)) * sin(Float64(pi - t_0)));
	else
		tmp = Float64(t_4 * sin(fma(-1.0, Float64(y_46_im * t_2), 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $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[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(-1.0 * N[(y$46$re * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -5e-310], N[(N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$3 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.05e-166], N[(t$95$4 * N[(N[Sin[t$95$0], $MachinePrecision] + N[(-1.0 * N[(y$46$im * N[(N[Cos[t$95$0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.08], 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] * N[Sin[N[(Pi - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[Sin[N[(-1.0 * N[(y$46$im * t$95$2), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -4.999999999999985e-310

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 18.5

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 33.8

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

   7. Applied rewrites 33.8%

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

   if -4.999999999999985e-310 < x.re < 2.0499999999999999e-166

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-/.f64 32.3

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

   7. Applied rewrites 32.3%

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

   if 2.0499999999999999e-166 < x.re < 1.0800000000000001

   1. Initial program 40.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.6

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

   6. Applied rewrites 49.6%

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

   if 1.0800000000000001 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

Alternative 2: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin \left(\mathsf{fma}\left(\log x.re, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_3 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0}\\ \mathbf{if}\;x.re \leq -1.6 \cdot 10^{-234}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot \sin t\_1\\ \mathbf{elif}\;x.re \leq 4.1 \cdot 10^{-250}:\\ \;\;\;\;t\_3 \cdot \sin \left(-1 \cdot t\_1\right)\\ \mathbf{elif}\;x.re \leq 1.95 \cdot 10^{-166}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x.re \leq 0.04:\\ \;\;\;\;t\_3 \cdot \sin \left(\pi - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2
         (*
          (sin (fma (log x.re) y.im (* (atan2 x.im x.re) y.re)))
          (exp (- (* (log x.re) y.re) (* y.im (atan2 x.im x.re))))))
        (t_3
         (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))))
   (if (<= x.re -1.6e-234)
     (* (exp (- (* (log (* -1.0 x.re)) y.re) t_0)) (sin t_1))
     (if (<= x.re 4.1e-250)
       (* t_3 (sin (* -1.0 t_1)))
       (if (<= x.re 1.95e-166)
         t_2
         (if (<= x.re 0.04) (* t_3 (sin (- PI t_1))) t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = sin(fma(log(x_46_re), y_46_im, (atan2(x_46_im, x_46_re) * y_46_re))) * exp(((log(x_46_re) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_3 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0));
	double tmp;
	if (x_46_re <= -1.6e-234) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(t_1);
	} else if (x_46_re <= 4.1e-250) {
		tmp = t_3 * sin((-1.0 * t_1));
	} else if (x_46_re <= 1.95e-166) {
		tmp = t_2;
	} else if (x_46_re <= 0.04) {
		tmp = t_3 * sin((((double) M_PI) - t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(sin(fma(log(x_46_re), y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re))) * exp(Float64(Float64(log(x_46_re) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))))
	t_3 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_0))
	tmp = 0.0
	if (x_46_re <= -1.6e-234)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(t_1));
	elseif (x_46_re <= 4.1e-250)
		tmp = Float64(t_3 * sin(Float64(-1.0 * t_1)));
	elseif (x_46_re <= 1.95e-166)
		tmp = t_2;
	elseif (x_46_re <= 0.04)
		tmp = Float64(t_3 * sin(Float64(pi - t_1)));
	else
		tmp = t_2;
	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$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(N[Log[x$46$re], $MachinePrecision] * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.6e-234], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.1e-250], N[(t$95$3 * N[Sin[N[(-1.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.95e-166], t$95$2, If[LessEqual[x$46$re, 0.04], N[(t$95$3 * N[Sin[N[(Pi - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -1.5999999999999999e-234

   1. Initial program 40.6%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7

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

   4. Applied rewrites 53.7%

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

   6. Applied rewrites 53.7%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 30.8

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

   9. Applied rewrites 30.8%

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

   if -1.5999999999999999e-234 < x.re < 4.10000000000000016e-250

   1. Initial program 40.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 47.0

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

   6. Applied rewrites 47.0%

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

   if 4.10000000000000016e-250 < x.re < 1.95e-166 or 0.0400000000000000008 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

   if 1.95e-166 < x.re < 0.0400000000000000008

   1. Initial program 40.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.6

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

   6. Applied rewrites 49.6%

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

Alternative 3: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\frac{1}{x.re}\right)\\ t_2 := \log \left(-1 \cdot x.re\right)\\ \mathbf{if}\;x.re \leq -2.4 \cdot 10^{-237}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \leq 1:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \pi\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_1\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (log (/ 1.0 x.re)))
        (t_2 (log (* -1.0 x.re))))
   (if (<= x.re -2.4e-237)
     (*
      (exp (- (* t_2 y.re) t_0))
      (sin (+ (* t_2 y.im) (* (atan2 x.im x.re) y.re))))
     (if (<= x.re 1.0)
       (*
        (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
        (sin PI))
       (*
        (exp (- (* -1.0 (* y.re t_1)) (* y.im (atan2 x.im x.re))))
        (sin (fma -1.0 (* y.im t_1) (* y.re (atan2 x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log((1.0 / x_46_re));
	double t_2 = log((-1.0 * x_46_re));
	double tmp;
	if (x_46_re <= -2.4e-237) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_re <= 1.0) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(((double) M_PI));
	} else {
		tmp = exp(((-1.0 * (y_46_re * t_1)) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(-1.0, (y_46_im * t_1), (y_46_re * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(Float64(1.0 / x_46_re))
	t_2 = log(Float64(-1.0 * x_46_re))
	tmp = 0.0
	if (x_46_re <= -2.4e-237)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_re <= 1.0)
		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_0)) * sin(pi));
	else
		tmp = Float64(exp(Float64(Float64(-1.0 * Float64(y_46_re * t_1)) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(-1.0, Float64(y_46_im * t_1), Float64(y_46_re * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -2.4e-237], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.0], 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$0), $MachinePrecision]], $MachinePrecision] * N[Sin[Pi], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(-1.0 * N[(y$46$re * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.4e-237

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 18.5

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 33.8

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

   7. Applied rewrites 33.8%

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

   if -2.4e-237 < x.re < 1

   1. Initial program 40.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.6

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

   6. Applied rewrites 49.6%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-PI.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if 1 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

Alternative 4: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \log \left(-1 \cdot x.re\right)\\ \mathbf{if}\;x.re \leq -2.4 \cdot 10^{-237}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.re \leq 0.55:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \pi\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log x.re, y.im, t\_1\right)\right) \cdot e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* (atan2 x.im x.re) y.re))
        (t_2 (log (* -1.0 x.re))))
   (if (<= x.re -2.4e-237)
     (* (exp (- (* t_2 y.re) t_0)) (sin (+ (* t_2 y.im) t_1)))
     (if (<= x.re 0.55)
       (*
        (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
        (sin PI))
       (*
        (sin (fma (log x.re) y.im t_1))
        (exp (- (* (log x.re) y.re) (* y.im (atan2 x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = log((-1.0 * x_46_re));
	double tmp;
	if (x_46_re <= -2.4e-237) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_1));
	} else if (x_46_re <= 0.55) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(((double) M_PI));
	} else {
		tmp = sin(fma(log(x_46_re), y_46_im, t_1)) * exp(((log(x_46_re) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_2 = log(Float64(-1.0 * x_46_re))
	tmp = 0.0
	if (x_46_re <= -2.4e-237)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + t_1)));
	elseif (x_46_re <= 0.55)
		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_0)) * sin(pi));
	else
		tmp = Float64(sin(fma(log(x_46_re), y_46_im, t_1)) * exp(Float64(Float64(log(x_46_re) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -2.4e-237], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 0.55], 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$0), $MachinePrecision]], $MachinePrecision] * N[Sin[Pi], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[Log[x$46$re], $MachinePrecision] * y$46$im + t$95$1), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.4e-237

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 18.5

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 33.8

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

   7. Applied rewrites 33.8%

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

   if -2.4e-237 < x.re < 0.55000000000000004

   1. Initial program 40.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.6

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

   6. Applied rewrites 49.6%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-PI.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if 0.55000000000000004 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

Alternative 5: 61.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -9.2 \cdot 10^{-265}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 0.55:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \pi\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log x.re, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -9.2e-265)
     (*
      (exp (- (* (log (* -1.0 x.re)) y.re) t_0))
      (sin (* y.re (atan2 x.im x.re))))
     (if (<= x.re 0.55)
       (*
        (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
        (sin PI))
       (*
        (sin (fma (log x.re) y.im (* (atan2 x.im x.re) y.re)))
        (exp (- (* (log x.re) y.re) (* y.im (atan2 x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -9.2e-265) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (x_46_re <= 0.55) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(((double) M_PI));
	} else {
		tmp = sin(fma(log(x_46_re), y_46_im, (atan2(x_46_im, x_46_re) * y_46_re))) * exp(((log(x_46_re) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -9.2e-265)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (x_46_re <= 0.55)
		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_0)) * sin(pi));
	else
		tmp = Float64(sin(fma(log(x_46_re), y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re))) * exp(Float64(Float64(log(x_46_re) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -9.2e-265], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 0.55], 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$0), $MachinePrecision]], $MachinePrecision] * N[Sin[Pi], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[Log[x$46$re], $MachinePrecision] * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -9.1999999999999996e-265

   1. Initial program 40.6%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7

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

   4. Applied rewrites 53.7%

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

   6. Applied rewrites 53.7%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 30.8

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

   9. Applied rewrites 30.8%

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

   if -9.1999999999999996e-265 < x.re < 0.55000000000000004

   1. Initial program 40.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.6

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

   6. Applied rewrites 49.6%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-PI.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if 0.55000000000000004 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

Alternative 6: 59.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -9.2 \cdot 10^{-265}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot \sin t\_1\\ \mathbf{elif}\;x.re \leq 1.16:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \pi\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_2\right)} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_2, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (/ 1.0 x.re))))
   (if (<= x.re -9.2e-265)
     (* (exp (- (* (log (* -1.0 x.re)) y.re) t_0)) (sin t_1))
     (if (<= x.re 1.16)
       (*
        (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
        (sin PI))
       (* (exp (* -1.0 (* y.re t_2))) (sin (fma -1.0 (* y.im t_2) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log((1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -9.2e-265) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(t_1);
	} else if (x_46_re <= 1.16) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(((double) M_PI));
	} else {
		tmp = exp((-1.0 * (y_46_re * t_2))) * sin(fma(-1.0, (y_46_im * t_2), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(Float64(1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -9.2e-265)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(t_1));
	elseif (x_46_re <= 1.16)
		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_0)) * sin(pi));
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_2))) * sin(fma(-1.0, Float64(y_46_im * t_2), t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -9.2e-265], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.16], 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$0), $MachinePrecision]], $MachinePrecision] * N[Sin[Pi], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(y$46$re * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -9.1999999999999996e-265

   1. Initial program 40.6%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7

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

   4. Applied rewrites 53.7%

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

   6. Applied rewrites 53.7%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 30.8

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

   9. Applied rewrites 30.8%

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

   if -9.1999999999999996e-265 < x.re < 1.15999999999999992

   1. Initial program 40.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.6

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

   6. Applied rewrites 49.6%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-PI.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if 1.15999999999999992 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 25.0

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

   7. Applied rewrites 25.0%

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

Alternative 7: 58.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \pi\\ \mathbf{if}\;y.re \leq -220000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (exp
           (-
            (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          (sin PI))))
   (if (<= y.re -220000.0)
     t_0
     (if (<= y.re 1.2e-11)
       (/ (sin (* (log (fabs x.im)) y.im)) (exp (* y.im (atan2 x.im x.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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(((double) M_PI));
	double tmp;
	if (y_46_re <= -220000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e-11) {
		tmp = sin((log(fabs(x_46_im)) * y_46_im)) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin(Math.PI);
	double tmp;
	if (y_46_re <= -220000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e-11) {
		tmp = Math.sin((Math.log(Math.abs(x_46_im)) * y_46_im)) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.sin(math.pi)
	tmp = 0
	if y_46_re <= -220000.0:
		tmp = t_0
	elif y_46_re <= 1.2e-11:
		tmp = math.sin((math.log(math.fabs(x_46_im)) * y_46_im)) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(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(pi))
	tmp = 0.0
	if (y_46_re <= -220000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.2e-11)
		tmp = Float64(sin(Float64(log(abs(x_46_im)) * y_46_im)) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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(pi);
	tmp = 0.0;
	if (y_46_re <= -220000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.2e-11)
		tmp = sin((log(abs(x_46_im)) * y_46_im)) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[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[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -220000.0], t$95$0, If[LessEqual[y$46$re, 1.2e-11], N[(N[Sin[N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.2e5 or 1.2000000000000001e-11 < y.re

   1. Initial program 40.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.6

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

   6. Applied rewrites 49.6%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-PI.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if -2.2e5 < y.re < 1.2000000000000001e-11

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

Alternative 8: 42.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \log \left(\frac{1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)}{t\_0}\\ \mathbf{elif}\;x.re \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{t\_0}\\ \mathbf{elif}\;x.re \leq 4.6 \cdot 10^{+89}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_1\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(-1 \cdot \left(y.im \cdot t\_1\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.im (atan2 x.im x.re)))) (t_1 (log (/ 1.0 x.re))))
   (if (<= x.re -4.4e-46)
     (/ (sin (* -1.0 (* y.im (log (/ -1.0 x.re))))) t_0)
     (if (<= x.re 1.7e-13)
       (/ (sin (* (log (fabs x.im)) y.im)) t_0)
       (if (<= x.re 4.6e+89)
         (* (exp (* -1.0 (* y.re t_1))) (sin (* y.re (atan2 x.im x.re))))
         (/ (sin (* -1.0 (* y.im t_1))) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double t_1 = log((1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -4.4e-46) {
		tmp = sin((-1.0 * (y_46_im * log((-1.0 / x_46_re))))) / t_0;
	} else if (x_46_re <= 1.7e-13) {
		tmp = sin((log(fabs(x_46_im)) * y_46_im)) / t_0;
	} else if (x_46_re <= 4.6e+89) {
		tmp = exp((-1.0 * (y_46_re * t_1))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = sin((-1.0 * (y_46_im * t_1))) / t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp((y_46im * atan2(x_46im, x_46re)))
    t_1 = log((1.0d0 / x_46re))
    if (x_46re <= (-4.4d-46)) then
        tmp = sin(((-1.0d0) * (y_46im * log(((-1.0d0) / x_46re))))) / t_0
    else if (x_46re <= 1.7d-13) then
        tmp = sin((log(abs(x_46im)) * y_46im)) / t_0
    else if (x_46re <= 4.6d+89) then
        tmp = exp(((-1.0d0) * (y_46re * t_1))) * sin((y_46re * atan2(x_46im, x_46re)))
    else
        tmp = sin(((-1.0d0) * (y_46im * t_1))) / 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.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.log((1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -4.4e-46) {
		tmp = Math.sin((-1.0 * (y_46_im * Math.log((-1.0 / x_46_re))))) / t_0;
	} else if (x_46_re <= 1.7e-13) {
		tmp = Math.sin((Math.log(Math.abs(x_46_im)) * y_46_im)) / t_0;
	} else if (x_46_re <= 4.6e+89) {
		tmp = Math.exp((-1.0 * (y_46_re * t_1))) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = Math.sin((-1.0 * (y_46_im * t_1))) / t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	t_1 = math.log((1.0 / x_46_re))
	tmp = 0
	if x_46_re <= -4.4e-46:
		tmp = math.sin((-1.0 * (y_46_im * math.log((-1.0 / x_46_re))))) / t_0
	elif x_46_re <= 1.7e-13:
		tmp = math.sin((math.log(math.fabs(x_46_im)) * y_46_im)) / t_0
	elif x_46_re <= 4.6e+89:
		tmp = math.exp((-1.0 * (y_46_re * t_1))) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = math.sin((-1.0 * (y_46_im * t_1))) / t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_1 = log(Float64(1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -4.4e-46)
		tmp = Float64(sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_re))))) / t_0);
	elseif (x_46_re <= 1.7e-13)
		tmp = Float64(sin(Float64(log(abs(x_46_im)) * y_46_im)) / t_0);
	elseif (x_46_re <= 4.6e+89)
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_1))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(sin(Float64(-1.0 * Float64(y_46_im * t_1))) / 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 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	t_1 = log((1.0 / x_46_re));
	tmp = 0.0;
	if (x_46_re <= -4.4e-46)
		tmp = sin((-1.0 * (y_46_im * log((-1.0 / x_46_re))))) / t_0;
	elseif (x_46_re <= 1.7e-13)
		tmp = sin((log(abs(x_46_im)) * y_46_im)) / t_0;
	elseif (x_46_re <= 4.6e+89)
		tmp = exp((-1.0 * (y_46_re * t_1))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	else
		tmp = sin((-1.0 * (y_46_im * t_1))) / 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[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -4.4e-46], N[(N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x$46$re, 1.7e-13], N[(N[Sin[N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x$46$re, 4.6e+89], N[(N[Exp[N[(-1.0 * N[(y$46$re * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -4.4000000000000002e-46

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in x.re around -inf

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-log.f64 N/A

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

       5. lower-/.f64 20.1

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

   12. Applied rewrites 20.1%

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

   if -4.4000000000000002e-46 < x.re < 1.70000000000000008e-13

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   if 1.70000000000000008e-13 < x.re < 4.5999999999999998e89

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-atan2.f64 18.9

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

   7. Applied rewrites 18.9%

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

   if 4.5999999999999998e89 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in x.re around inf

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-log.f64 N/A

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

       5. lower-/.f64 17.7

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

   12. Applied rewrites 17.7%

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

Alternative 9: 41.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.re \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)}{t\_0}\\ \mathbf{elif}\;x.re \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.im (atan2 x.im x.re)))))
   (if (<= x.re -4.4e-46)
     (/ (sin (* -1.0 (* y.im (log (/ -1.0 x.re))))) t_0)
     (if (<= x.re 1.7e-13)
       (/ (sin (* (log (fabs x.im)) y.im)) t_0)
       (*
        (exp (* -1.0 (* y.re (log (/ 1.0 x.re)))))
        (sin (* y.re (atan2 x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -4.4e-46) {
		tmp = sin((-1.0 * (y_46_im * log((-1.0 / x_46_re))))) / t_0;
	} else if (x_46_re <= 1.7e-13) {
		tmp = sin((log(fabs(x_46_im)) * y_46_im)) / t_0;
	} else {
		tmp = exp((-1.0 * (y_46_re * log((1.0 / x_46_re))))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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 = exp((y_46im * atan2(x_46im, x_46re)))
    if (x_46re <= (-4.4d-46)) then
        tmp = sin(((-1.0d0) * (y_46im * log(((-1.0d0) / x_46re))))) / t_0
    else if (x_46re <= 1.7d-13) then
        tmp = sin((log(abs(x_46im)) * y_46im)) / t_0
    else
        tmp = exp(((-1.0d0) * (y_46re * log((1.0d0 / x_46re))))) * sin((y_46re * atan2(x_46im, x_46re)))
    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.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -4.4e-46) {
		tmp = Math.sin((-1.0 * (y_46_im * Math.log((-1.0 / x_46_re))))) / t_0;
	} else if (x_46_re <= 1.7e-13) {
		tmp = Math.sin((Math.log(Math.abs(x_46_im)) * y_46_im)) / t_0;
	} else {
		tmp = Math.exp((-1.0 * (y_46_re * Math.log((1.0 / x_46_re))))) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_re <= -4.4e-46:
		tmp = math.sin((-1.0 * (y_46_im * math.log((-1.0 / x_46_re))))) / t_0
	elif x_46_re <= 1.7e-13:
		tmp = math.sin((math.log(math.fabs(x_46_im)) * y_46_im)) / t_0
	else:
		tmp = math.exp((-1.0 * (y_46_re * math.log((1.0 / x_46_re))))) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_re <= -4.4e-46)
		tmp = Float64(sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_re))))) / t_0);
	elseif (x_46_re <= 1.7e-13)
		tmp = Float64(sin(Float64(log(abs(x_46_im)) * y_46_im)) / t_0);
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * log(Float64(1.0 / x_46_re))))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_re <= -4.4e-46)
		tmp = sin((-1.0 * (y_46_im * log((-1.0 / x_46_re))))) / t_0;
	elseif (x_46_re <= 1.7e-13)
		tmp = sin((log(abs(x_46_im)) * y_46_im)) / t_0;
	else
		tmp = exp((-1.0 * (y_46_re * log((1.0 / x_46_re))))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -4.4e-46], N[(N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x$46$re, 1.7e-13], N[(N[Sin[N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(y$46$re * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -4.4000000000000002e-46

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in x.re around -inf

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

       1. lower-sin.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-log.f64 N/A

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

       5. lower-/.f64 20.1

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

   12. Applied rewrites 20.1%

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

   if -4.4000000000000002e-46 < x.re < 1.70000000000000008e-13

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   if 1.70000000000000008e-13 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-atan2.f64 18.9

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

   7. Applied rewrites 18.9%

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

Alternative 10: 41.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 1.7e-13)
   (/ (sin (* (log (fabs x.im)) y.im)) (exp (* y.im (atan2 x.im x.re))))
   (*
    (exp (* -1.0 (* y.re (log (/ 1.0 x.re)))))
    (sin (* y.re (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1.7e-13) {
		tmp = sin((log(fabs(x_46_im)) * y_46_im)) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = exp((-1.0 * (y_46_re * log((1.0 / x_46_re))))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 1.7d-13) then
        tmp = sin((log(abs(x_46im)) * y_46im)) / exp((y_46im * atan2(x_46im, x_46re)))
    else
        tmp = exp(((-1.0d0) * (y_46re * log((1.0d0 / x_46re))))) * sin((y_46re * atan2(x_46im, x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1.7e-13) {
		tmp = Math.sin((Math.log(Math.abs(x_46_im)) * y_46_im)) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = Math.exp((-1.0 * (y_46_re * Math.log((1.0 / x_46_re))))) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 1.7e-13:
		tmp = math.sin((math.log(math.fabs(x_46_im)) * y_46_im)) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = math.exp((-1.0 * (y_46_re * math.log((1.0 / x_46_re))))) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 1.7e-13)
		tmp = Float64(sin(Float64(log(abs(x_46_im)) * y_46_im)) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * log(Float64(1.0 / x_46_re))))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.7e-13)
		tmp = sin((log(abs(x_46_im)) * y_46_im)) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = exp((-1.0 * (y_46_re * log((1.0 / x_46_re))))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 1.7e-13], N[(N[Sin[N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(y$46$re * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.70000000000000008e-13

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   if 1.70000000000000008e-13 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-atan2.f64 18.9

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

   7. Applied rewrites 18.9%

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

Alternative 11: 37.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 2.25e+77)
   (/ (sin (* (log (fabs x.im)) y.im)) (exp (* y.im (atan2 x.im x.re))))
   (*
    1.0
    (sin (fma -1.0 (* y.im (log (/ 1.0 x.re))) (* y.re (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 2.25e+77) {
		tmp = sin((log(fabs(x_46_im)) * y_46_im)) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * sin(fma(-1.0, (y_46_im * log((1.0 / x_46_re))), (y_46_re * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 2.25e+77)
		tmp = Float64(sin(Float64(log(abs(x_46_im)) * y_46_im)) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * sin(fma(-1.0, Float64(y_46_im * log(Float64(1.0 / x_46_re))), Float64(y_46_re * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 2.25e+77], N[(N[Sin[N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.25000000000000012e77

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   if 2.25000000000000012e77 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.6

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in x.re around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 10.5

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

   10. Applied rewrites 10.5%

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

Alternative 12: 26.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.1 \cdot 10^{-305}:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{-1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 3.5 \cdot 10^{-179}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -1.1e-305)
   (*
    1.0
    (sin (fma -1.0 (* y.im (log (/ -1.0 x.im))) (* y.re (atan2 x.im x.re)))))
   (if (<= x.im 3.5e-179)
     (* 1.0 (sin (* -1.0 (* y.im (log (/ 1.0 x.re))))))
     (* (exp (- (* y.im (atan2 x.im x.re)))) (sin (* y.im (log x.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1.1e-305) {
		tmp = 1.0 * sin(fma(-1.0, (y_46_im * log((-1.0 / x_46_im))), (y_46_re * atan2(x_46_im, x_46_re))));
	} else if (x_46_im <= 3.5e-179) {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_re)))));
	} else {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(x_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -1.1e-305)
		tmp = Float64(1.0 * sin(fma(-1.0, Float64(y_46_im * log(Float64(-1.0 / x_46_im))), Float64(y_46_re * atan(x_46_im, x_46_re)))));
	elseif (x_46_im <= 3.5e-179)
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_re))))));
	else
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_im * log(x_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -1.1e-305], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 3.5e-179], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.09999999999999998e-305

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.6

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in x.im around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 9.1

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

   10. Applied rewrites 9.1%

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

   if -1.09999999999999998e-305 < x.im < 3.50000000000000024e-179

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.6

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 8.2

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

   10. Applied rewrites 8.2%

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

   11. Taylor expanded in x.re around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 N/A

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

       4. lower-/.f64 6.9

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

   13. Applied rewrites 6.9%

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

   if 3.50000000000000024e-179 < x.im

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

   8. Taylor expanded in x.im around 0

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

      1. lower-log.f64 18.7

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

   10. Applied rewrites 18.7%

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

Alternative 13: 20.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -8.2 \cdot 10^{-238}:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{-1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.re \leq 7 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{1 + y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -8.2e-238)
   (*
    1.0
    (sin (fma -1.0 (* y.im (log (/ -1.0 x.re))) (* y.re (atan2 x.im x.re)))))
   (if (<= x.re 7e+77)
     (/ (sin (* (log (fabs x.im)) y.im)) (+ 1.0 (* y.im (atan2 x.im x.re))))
     (* 1.0 (sin (* -1.0 (* y.im (log (/ 1.0 x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -8.2e-238) {
		tmp = 1.0 * sin(fma(-1.0, (y_46_im * log((-1.0 / x_46_re))), (y_46_re * atan2(x_46_im, x_46_re))));
	} else if (x_46_re <= 7e+77) {
		tmp = sin((log(fabs(x_46_im)) * y_46_im)) / (1.0 + (y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_re)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -8.2e-238)
		tmp = Float64(1.0 * sin(fma(-1.0, Float64(y_46_im * log(Float64(-1.0 / x_46_re))), Float64(y_46_re * atan(x_46_im, x_46_re)))));
	elseif (x_46_re <= 7e+77)
		tmp = Float64(sin(Float64(log(abs(x_46_im)) * y_46_im)) / Float64(1.0 + Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_re))))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -8.2e-238], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7e+77], N[(N[Sin[N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -8.2000000000000002e-238

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.6

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 9.7

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

   10. Applied rewrites 9.7%

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

   if -8.2000000000000002e-238 < x.re < 7.0000000000000003e77

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 18.7

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

   7. Applied rewrites 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in y.im around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-atan2.f64 10.0

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

   12. Applied rewrites 10.0%

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

   if 7.0000000000000003e77 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.6

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 8.2

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

   10. Applied rewrites 8.2%

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

   11. Taylor expanded in x.re around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 N/A

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

       4. lower-/.f64 6.9

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

   13. Applied rewrites 6.9%

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

Alternative 14: 17.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= x.re -1e-308)
     (* 1.0 (sin (fma -1.0 (* y.im (log (/ -1.0 x.re))) t_0)))
     (* 1.0 (sin (fma -1.0 (* y.im (log (/ 1.0 x.re))) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -1e-308) {
		tmp = 1.0 * sin(fma(-1.0, (y_46_im * log((-1.0 / x_46_re))), t_0));
	} else {
		tmp = 1.0 * sin(fma(-1.0, (y_46_im * log((1.0 / x_46_re))), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= -1e-308)
		tmp = Float64(1.0 * sin(fma(-1.0, Float64(y_46_im * log(Float64(-1.0 / x_46_re))), t_0)));
	else
		tmp = Float64(1.0 * sin(fma(-1.0, Float64(y_46_im * log(Float64(1.0 / x_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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1e-308], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -9.9999999999999991e-309

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.6

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 9.7

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

   10. Applied rewrites 9.7%

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

   if -9.9999999999999991e-309 < x.re

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.6

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in x.re around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 10.5

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

   10. Applied rewrites 10.5%

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

Alternative 15: 13.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.9 \cdot 10^{-46}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 7 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{1 + y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -2.9e-46)
   (* 1.0 (sin (* -1.0 (* y.im (log (/ -1.0 x.re))))))
   (if (<= x.re 7e+77)
     (/ (sin (* (log (fabs x.im)) y.im)) (+ 1.0 (* y.im (atan2 x.im x.re))))
     (* 1.0 (sin (* -1.0 (* y.im (log (/ 1.0 x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -2.9e-46) {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_re)))));
	} else if (x_46_re <= 7e+77) {
		tmp = sin((log(fabs(x_46_im)) * y_46_im)) / (1.0 + (y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_re)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-2.9d-46)) then
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log(((-1.0d0) / x_46re)))))
    else if (x_46re <= 7d+77) then
        tmp = sin((log(abs(x_46im)) * y_46im)) / (1.0d0 + (y_46im * atan2(x_46im, x_46re)))
    else
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log((1.0d0 / x_46re)))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -2.9e-46) {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((-1.0 / x_46_re)))));
	} else if (x_46_re <= 7e+77) {
		tmp = Math.sin((Math.log(Math.abs(x_46_im)) * y_46_im)) / (1.0 + (y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((1.0 / x_46_re)))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -2.9e-46:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((-1.0 / x_46_re)))))
	elif x_46_re <= 7e+77:
		tmp = math.sin((math.log(math.fabs(x_46_im)) * y_46_im)) / (1.0 + (y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((1.0 / x_46_re)))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -2.9e-46)
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_re))))));
	elseif (x_46_re <= 7e+77)
		tmp = Float64(sin(Float64(log(abs(x_46_im)) * y_46_im)) / Float64(1.0 + Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_re))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -2.9e-46)
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_re)))));
	elseif (x_46_re <= 7e+77)
		tmp = sin((log(abs(x_46_im)) * y_46_im)) / (1.0 + (y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_re)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -2.9e-46], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7e+77], N[(N[Sin[N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.90000000000000005e-46

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.6

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 8.2

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

   10. Applied rewrites 8.2%

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

   11. Taylor expanded in x.re around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 N/A

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

       4. lower-/.f64 4.5

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

   13. Applied rewrites 4.5%

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

   if -2.90000000000000005e-46 < x.re < 7.0000000000000003e77

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 21.7

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 21.7%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   5. Taylor expanded in x.re around 0

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \]
   6. Step-by-step derivation

      1. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \]

      3. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \]

      5. lower-pow.f64 18.7

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \cdot \frac{1}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      7. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \cdot \frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      8. lift-*.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2}}\right)\right) \cdot \frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   9. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   10. Taylor expanded in y.im around 0

       \[\leadsto \frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{1 + \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{1 + y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{1 + y.im \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}} \]

       3. lower-atan2.f64 10.0

          \[\leadsto \frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{1 + y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   12. Applied rewrites 10.0%

       \[\leadsto \frac{\sin \left(\log \left(\left|x.im\right|\right) \cdot y.im\right)}{1 + \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if 7.0000000000000003e77 < x.re

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.re around inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.re}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

       4. lower-/.f64 6.9

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

   13. Applied rewrites 6.9%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 12.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 4.9 \cdot 10^{+85}:\\ \;\;\;\;1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log \left(-x.im\right) \cdot y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 4.9e+85)
   (* 1.0 (sin (fma (atan2 x.im x.re) y.re (* (log (- x.im)) y.im))))
   (* 1.0 (sin (* -1.0 (* y.im (log (/ 1.0 x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 4.9e+85) {
		tmp = 1.0 * sin(fma(atan2(x_46_im, x_46_re), y_46_re, (log(-x_46_im) * y_46_im)));
	} else {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_re)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 4.9e+85)
		tmp = Float64(1.0 * sin(fma(atan(x_46_im, x_46_re), y_46_re, Float64(log(Float64(-x_46_im)) * y_46_im))));
	else
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_re))))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 4.9e+85], N[(1.0 * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.8999999999999997e85

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in x.im around -inf

      \[\leadsto 1 \cdot \sin \color{blue}{\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \log \left(\frac{-1}{x.im}\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{-1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{-1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{-1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. lower-atan2.f64 9.1

         \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{-1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   10. Applied rewrites 9.1%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{-1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   11. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right) \]

       2. +-commutative N/A

          \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right) \]

       3. add-flip N/A

          \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right)}\right) \]

       4. sub-flip N/A

          \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right)\right)\right)}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right)}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto 1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right)}\right)\right)\right) \]

       7. distribute-lft-neg-out N/A

          \[\leadsto 1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto 1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\mathsf{neg}\left(1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right)\right) \]

       9. distribute-lft-neg-out N/A

          \[\leadsto 1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right) \]

       10. metadata-eval N/A

           \[\leadsto 1 \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + -1 \cdot \left(\color{blue}{y.im} \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{y.re}, -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right) \]

       12. mul-1-neg N/A

           \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \mathsf{neg}\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right) \]

       13. lift-*.f64 N/A

           \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \mathsf{neg}\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \mathsf{neg}\left(\log \left(\frac{-1}{x.im}\right) \cdot y.im\right)\right)\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \left(\mathsf{neg}\left(\log \left(\frac{-1}{x.im}\right)\right)\right) \cdot y.im\right)\right) \]

       16. lower-*.f64 N/A

           \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \left(\mathsf{neg}\left(\log \left(\frac{-1}{x.im}\right)\right)\right) \cdot y.im\right)\right) \]

   12. Applied rewrites 9.1%

       \[\leadsto 1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{y.re}, \log \left(-x.im\right) \cdot y.im\right)\right) \]

   if 4.8999999999999997e85 < x.re

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.re around inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.re}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

       4. lower-/.f64 6.9

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

   13. Applied rewrites 6.9%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 11.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -8.2 \cdot 10^{-54}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -8.2e-54)
   (* 1.0 (sin (* -1.0 (* y.im (log (/ -1.0 x.re))))))
   (if (<= x.re 1.4e-155)
     (* 1.0 (sin (* -1.0 (* y.im (log (/ 1.0 x.im))))))
     (* 1.0 (sin (* -1.0 (* y.im (log (/ 1.0 x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -8.2e-54) {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_re)))));
	} else if (x_46_re <= 1.4e-155) {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_im)))));
	} else {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_re)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-8.2d-54)) then
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log(((-1.0d0) / x_46re)))))
    else if (x_46re <= 1.4d-155) then
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log((1.0d0 / x_46im)))))
    else
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log((1.0d0 / x_46re)))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -8.2e-54) {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((-1.0 / x_46_re)))));
	} else if (x_46_re <= 1.4e-155) {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((1.0 / x_46_im)))));
	} else {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((1.0 / x_46_re)))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -8.2e-54:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((-1.0 / x_46_re)))))
	elif x_46_re <= 1.4e-155:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((1.0 / x_46_im)))))
	else:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((1.0 / x_46_re)))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -8.2e-54)
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_re))))));
	elseif (x_46_re <= 1.4e-155)
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_im))))));
	else
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_re))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -8.2e-54)
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_re)))));
	elseif (x_46_re <= 1.4e-155)
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_im)))));
	else
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_re)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -8.2e-54], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.4e-155], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -8.2000000000000001e-54

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.re around -inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.re}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right) \]

       4. lower-/.f64 4.5

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)}\right) \]

   if -8.2000000000000001e-54 < x.re < 1.4e-155

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.im around inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.im}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \]

       4. lower-/.f64 4.6

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \]

   13. Applied rewrites 4.6%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)}\right) \]

   if 1.4e-155 < x.re

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.re around inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.re}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

       4. lower-/.f64 6.9

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

   13. Applied rewrites 6.9%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 11.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ \mathbf{if}\;e^{t\_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t\_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \leq \infty:\\ \;\;\;\;\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)\\ \end{array} \end{array} \]

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))))))
   (if (<=
        (*
         (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
         (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))
        INFINITY)
     (* (sin (* (log (sqrt (fma x.im x.im (* x.re x.re)))) y.im)) 1.0)
     (* 1.0 (sin (* -1.0 (* y.im (log (/ 1.0 x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double tmp;
	if ((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)))) <= ((double) INFINITY)) {
		tmp = sin((log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))) * y_46_im)) * 1.0;
	} else {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_re)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	tmp = 0.0
	if (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)))) <= Inf)
		tmp = Float64(sin(Float64(log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))) * y_46_im)) * 1.0);
	else
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_re))))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[Sin[N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   12. Applied rewrites 8.2%

       \[\leadsto \color{blue}{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right) \cdot 1} \]

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.re around inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.re}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

       4. lower-/.f64 6.9

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \]

   13. Applied rewrites 6.9%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 9.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{-308}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -1e-308)
   (* 1.0 (sin (* -1.0 (* y.im (log (/ -1.0 x.im))))))
   (* 1.0 (sin (* -1.0 (* y.im (log (/ 1.0 x.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1e-308) {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_im)))));
	} else {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_im)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-1d-308)) then
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log(((-1.0d0) / x_46im)))))
    else
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log((1.0d0 / x_46im)))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1e-308) {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((-1.0 / x_46_im)))));
	} else {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((1.0 / x_46_im)))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -1e-308:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((-1.0 / x_46_im)))))
	else:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((1.0 / x_46_im)))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -1e-308)
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_im))))));
	else
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -1e-308)
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_im)))));
	else
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((1.0 / x_46_im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -1e-308], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -9.9999999999999991e-309

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.im around -inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

       4. lower-/.f64 4.8

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

   13. Applied rewrites 4.8%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right) \]

   if -9.9999999999999991e-309 < x.im

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.im around inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.im}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \]

       4. lower-/.f64 4.6

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \]

   13. Applied rewrites 4.6%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 7.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5.6 \cdot 10^{-93}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -5.6e-93)
   (* 1.0 (sin (* -1.0 (* y.im (log (/ -1.0 x.re))))))
   (* 1.0 (sin (* -1.0 (* y.im (log (/ -1.0 x.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -5.6e-93) {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_re)))));
	} else {
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_im)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-5.6d-93)) then
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log(((-1.0d0) / x_46re)))))
    else
        tmp = 1.0d0 * sin(((-1.0d0) * (y_46im * log(((-1.0d0) / x_46im)))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -5.6e-93) {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((-1.0 / x_46_re)))));
	} else {
		tmp = 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((-1.0 / x_46_im)))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -5.6e-93:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((-1.0 / x_46_re)))))
	else:
		tmp = 1.0 * math.sin((-1.0 * (y_46_im * math.log((-1.0 / x_46_im)))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -5.6e-93)
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_re))))));
	else
		tmp = Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -5.6e-93)
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_re)))));
	else
		tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -5.6e-93], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -5.59999999999999997e-93

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.re around -inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.re}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right) \]

       4. lower-/.f64 4.5

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)}\right) \]

   if -5.59999999999999997e-93 < x.re

   1. Initial program 40.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   3. Step-by-step derivation

      1. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lower-atan2.f64 26.6

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Applied rewrites 26.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in y.im around 0

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-pow.f64 8.2

         \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. Applied rewrites 8.2%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   11. Taylor expanded in x.im around -inf

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

       4. lower-/.f64 4.8

          \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

   13. Applied rewrites 4.8%

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 4.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* 1.0 (sin (* -1.0 (* y.im (log (/ -1.0 x.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_im)))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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(((-1.0d0) * (y_46im * log(((-1.0d0) / x_46im)))))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 * Math.sin((-1.0 * (y_46_im * Math.log((-1.0 / x_46_im)))));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 * math.sin((-1.0 * (y_46_im * math.log((-1.0 / x_46_im)))))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 * sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_im))))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 * sin((-1.0 * (y_46_im * log((-1.0 / x_46_im)))));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.6%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

2. Taylor expanded in y.re around 0

   \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
3. Step-by-step derivation

   1. lower-exp.f64 N/A

      \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. lower-neg.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. lower-*.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. lower-atan2.f64 26.6

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

4. Applied rewrites 26.6%

   \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

5. Taylor expanded in y.im around 0

   \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
6. Step-by-step derivation

7. Applied rewrites 13.0%

   \[\leadsto 1 \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

8. Taylor expanded in y.re around 0

   \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 1 \cdot \sin \left(y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

   2. lower-log.f64 N/A

      \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   6. lower-pow.f64 8.2

      \[\leadsto 1 \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

10. Applied rewrites 8.2%

    \[\leadsto 1 \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

11. Taylor expanded in x.im around -inf

    \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right)\right) \]

    2. lower-*.f64 N/A

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

    3. lower-log.f64 N/A

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

    4. lower-/.f64 4.8

       \[\leadsto 1 \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right) \]

13. Applied rewrites 4.8%

    \[\leadsto 1 \cdot \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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)))))