powComplex, real part

Percentage Accurate: 41.4% → 82.7%

Time: 11.9s

Alternatives: 6

Speedup: N/A×

• 35.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)))
    (cos (+ (* 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))) * cos(((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))) * cos(((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.cos(((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.cos(((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))) * cos(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))) * cos(((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[Cos[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 41.4% accurate, 1.0× speedup

Math

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

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)))
    (cos (+ (* 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))) * cos(((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))) * cos(((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.cos(((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.cos(((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))) * cos(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))) * cos(((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[Cos[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: 82.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.im \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;t\_0 \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(-1 \cdot x.im, x.re\right)\right), \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - y.im \cdot \left(\log \left(\mathsf{hypot}\left(-1 \cdot x.im, -1 \cdot x.re\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (- (* (log (hypot x.re x.im)) y.re) (* (atan2 x.im x.re) y.im)))))
   (if (<= y.im 6.8e+30)
     (* t_0 (sin (fma y.im (log (hypot (* -1.0 x.im) x.re)) (/ PI 2.0))))
     (*
      t_0
      (-
       (sin (fma y.re (atan2 x.im x.re) (/ PI 2.0)))
       (*
        y.im
        (*
         (log (hypot (* -1.0 x.im) (* -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(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_im <= 6.8e+30) {
		tmp = t_0 * sin(fma(y_46_im, log(hypot((-1.0 * x_46_im), x_46_re)), (((double) M_PI) / 2.0)));
	} else {
		tmp = t_0 * (sin(fma(y_46_re, atan2(x_46_im, x_46_re), (((double) M_PI) / 2.0))) - (y_46_im * (log(hypot((-1.0 * x_46_im), (-1.0 * x_46_re))) * sin((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 = exp(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_im <= 6.8e+30)
		tmp = Float64(t_0 * sin(fma(y_46_im, log(hypot(Float64(-1.0 * x_46_im), x_46_re)), Float64(pi / 2.0))));
	else
		tmp = Float64(t_0 * Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(pi / 2.0))) - Float64(y_46_im * Float64(log(hypot(Float64(-1.0 * x_46_im), Float64(-1.0 * x_46_re))) * sin(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[Exp[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, 6.8e+30], N[(t$95$0 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[N[(-1.0 * x$46$im), $MachinePrecision] ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(y$46$im * N[(N[Log[N[Sqrt[N[(-1.0 * x$46$im), $MachinePrecision] ^ 2 + N[(-1.0 * x$46$re), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < 6.8000000000000005e30

   1. Initial program 45.8%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. lower--.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-atan2.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 52.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-hypot.f64 52.5

         \[\leadsto e^{\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

   7. Applied rewrites 52.5%

      \[\leadsto e^{\color{blue}{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow1/2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-neg.f64 86.1

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in y.re around 0

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

       1. sin-+PI/2-rev N/A

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

       2. lower-sin.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lower-fma.f64 N/A

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

   12. Applied rewrites 92.4%

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

   if 6.8000000000000005e30 < y.im

   1. Initial program 34.5%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. lower--.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-atan2.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 46.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-hypot.f64 52.5

         \[\leadsto e^{\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

   7. Applied rewrites 52.5%

      \[\leadsto e^{\color{blue}{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow1/2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-neg.f64 74.0

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

   9. Applied rewrites 74.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(-1 \cdot x.im, x.re\right)\right), \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - y.im \cdot \left(\log \left(\mathsf{hypot}\left(-1 \cdot x.im, -1 \cdot x.re\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(-1 \cdot x.im, x.re\right)\right), \frac{\pi}{2}\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (exp (- (* (log (hypot x.re x.im)) y.re) (* (atan2 x.im x.re) y.im)))
  (sin (fma y.im (log (hypot (* -1.0 x.im) x.re)) (/ PI 2.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(fma(y_46_im, log(hypot((-1.0 * x_46_im), x_46_re)), (((double) M_PI) / 2.0)));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(exp(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(fma(y_46_im, log(hypot(Float64(-1.0 * x_46_im), x_46_re)), Float64(pi / 2.0))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Exp[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[N[Sqrt[N[(-1.0 * x$46$im), $MachinePrecision] ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.8%

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

3. Taylor expanded in y.im around 0

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

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

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

   2. lower--.f64 N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-atan2.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

5. Applied rewrites 50.6%

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-hypot.f64 52.5

      \[\leadsto e^{\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

7. Applied rewrites 52.5%

   \[\leadsto e^{\color{blue}{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow1/2 N/A

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

   5. sqr-neg-rev N/A

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

   6. sqr-neg-rev N/A

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

   7. lower-hypot.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-neg.f64 82.9

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

9. Applied rewrites 82.9%

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

10. Taylor expanded in y.re around 0

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

    1. sin-+PI/2-rev N/A

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

    2. lower-sin.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. lower-fma.f64 N/A

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

12. Applied rewrites 85.4%

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

13. Final simplification 85.4%

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

Alternative 3: 66.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.55 \cdot 10^{+162}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(0.5 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(0.16666666666666666, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, 0.5 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im 1.55e+162)
   (*
    (exp
     (-
      (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
      (* (atan2 x.im x.re) y.im)))
    (sin (* 0.5 PI)))
   (*
    (cos (fma y.im (log x.im) (* y.re (atan2 x.im x.re))))
    (/
     (exp (* y.re (log x.im)))
     (+
      1.0
      (*
       y.im
       (fma
        y.im
        (fma
         0.16666666666666666
         (* y.im (pow (atan2 x.im x.re) 3.0))
         (* 0.5 (pow (atan2 x.im x.re) 2.0)))
        (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_im <= 1.55e+162) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((0.5 * ((double) M_PI)));
	} else {
		tmp = cos(fma(y_46_im, log(x_46_im), (y_46_re * atan2(x_46_im, x_46_re)))) * (exp((y_46_re * log(x_46_im))) / (1.0 + (y_46_im * fma(y_46_im, fma(0.16666666666666666, (y_46_im * pow(atan2(x_46_im, x_46_re), 3.0)), (0.5 * pow(atan2(x_46_im, x_46_re), 2.0))), 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_im <= 1.55e+162)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(0.5 * pi)));
	else
		tmp = Float64(cos(fma(y_46_im, log(x_46_im), Float64(y_46_re * atan(x_46_im, x_46_re)))) * Float64(exp(Float64(y_46_re * log(x_46_im))) / Float64(1.0 + Float64(y_46_im * fma(y_46_im, fma(0.16666666666666666, Float64(y_46_im * (atan(x_46_im, x_46_re) ^ 3.0)), Float64(0.5 * (atan(x_46_im, x_46_re) ^ 2.0))), 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$46$im, 1.55e+162], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(y$46$im * N[(y$46$im * N[(0.16666666666666666 * N[(y$46$im * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.55e162

   1. Initial program 48.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. lower--.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-atan2.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 54.4%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) - 1 \cdot \left(y.im \cdot \left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)} \]

   6. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 69.7

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

   8. Applied rewrites 69.7%

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

   if 1.55e162 < x.im

   1. Initial program 0.0%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-atan2.f64 N/A

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

      7. exp-diff N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-exp.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + \color{blue}{y.im \cdot \left(y.im \cdot \left(\frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \color{blue}{\left(y.im \cdot \left(\frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \left(y.im \cdot \left(\frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \color{blue}{\frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot \color{blue}{{\tan^{-1}_* \frac{x.im}{x.re}}^{3}}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{\color{blue}{3}}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      7. lift-atan2.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      10. lift-atan2.f64 N/A

          \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      11. lift-atan2.f64 79.5

          \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(0.16666666666666666, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, 0.5 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Applied rewrites 79.5%

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + \color{blue}{y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(0.16666666666666666, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, 0.5 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 48.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq 2.8 \cdot 10^{-308}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y.im, \log x.re, t\_1\right)\right) \cdot \frac{e^{y.re \cdot \log x.re}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y.im, \log x.im, t\_1\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{t\_0}\\ \end{array} \end{array} \]

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 (* y.re (atan2 x.im x.re))))
   (if (<= x.im 2.8e-308)
     (* (cos (fma y.im (log x.re) t_1)) (/ (exp (* y.re (log x.re))) t_0))
     (* (cos (fma y.im (log x.im) t_1)) (/ (exp (* y.re (log x.im))) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= 2.8e-308) {
		tmp = cos(fma(y_46_im, log(x_46_re), t_1)) * (exp((y_46_re * log(x_46_re))) / t_0);
	} else {
		tmp = cos(fma(y_46_im, log(x_46_im), t_1)) * (exp((y_46_re * log(x_46_im))) / t_0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_im <= 2.8e-308)
		tmp = Float64(cos(fma(y_46_im, log(x_46_re), t_1)) * Float64(exp(Float64(y_46_re * log(x_46_re))) / t_0));
	else
		tmp = Float64(cos(fma(y_46_im, log(x_46_im), t_1)) * Float64(exp(Float64(y_46_re * log(x_46_im))) / t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, 2.8e-308], N[(N[Cos[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.79999999999999984e-308

   1. Initial program 41.6%

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

   3. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-atan2.f64 N/A

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

      7. exp-diff N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-exp.f64 N/A

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

   5. Applied rewrites 30.3%

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

   if 2.79999999999999984e-308 < x.im

   1. Initial program 43.9%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-atan2.f64 N/A

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

      7. exp-diff N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-exp.f64 N/A

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

   5. Applied rewrites 68.6%

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

Alternative 5: 31.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (cos (fma y.im (log x.im) (* y.re (atan2 x.im x.re))))
  (/ (exp (* y.re (log x.im))) (exp (* y.im (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return cos(fma(y_46_im, log(x_46_im), (y_46_re * atan2(x_46_im, x_46_re)))) * (exp((y_46_re * log(x_46_im))) / exp((y_46_im * atan2(x_46_im, x_46_re))));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(cos(fma(y_46_im, log(x_46_im), Float64(y_46_re * atan(x_46_im, x_46_re)))) * Float64(exp(Float64(y_46_re * log(x_46_im))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re)))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Cos[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.8%

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

3. Taylor expanded in x.re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-atan2.f64 N/A

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

   7. exp-diff N/A

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

   8. lower-/.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-exp.f64 N/A

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

5. Applied rewrites 34.3%

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

Alternative 6: 28.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(0.16666666666666666, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, 0.5 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (cos (fma y.im (log x.im) (* y.re (atan2 x.im x.re))))
  (/
   (exp (* y.re (log x.im)))
   (+
    1.0
    (*
     y.im
     (fma
      y.im
      (fma
       0.16666666666666666
       (* y.im (pow (atan2 x.im x.re) 3.0))
       (* 0.5 (pow (atan2 x.im x.re) 2.0)))
      (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return cos(fma(y_46_im, log(x_46_im), (y_46_re * atan2(x_46_im, x_46_re)))) * (exp((y_46_re * log(x_46_im))) / (1.0 + (y_46_im * fma(y_46_im, fma(0.16666666666666666, (y_46_im * pow(atan2(x_46_im, x_46_re), 3.0)), (0.5 * pow(atan2(x_46_im, x_46_re), 2.0))), atan2(x_46_im, x_46_re)))));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(cos(fma(y_46_im, log(x_46_im), Float64(y_46_re * atan(x_46_im, x_46_re)))) * Float64(exp(Float64(y_46_re * log(x_46_im))) / Float64(1.0 + Float64(y_46_im * fma(y_46_im, fma(0.16666666666666666, Float64(y_46_im * (atan(x_46_im, x_46_re) ^ 3.0)), Float64(0.5 * (atan(x_46_im, x_46_re) ^ 2.0))), atan(x_46_im, x_46_re))))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Cos[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(y$46$im * N[(y$46$im * N[(0.16666666666666666 * N[(y$46$im * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.8%

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

3. Taylor expanded in x.re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-atan2.f64 N/A

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

   7. exp-diff N/A

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

   8. lower-/.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-exp.f64 N/A

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

5. Applied rewrites 34.3%

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

6. Taylor expanded in y.im around 0

   \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + \color{blue}{y.im \cdot \left(y.im \cdot \left(\frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \color{blue}{\left(y.im \cdot \left(\frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \left(y.im \cdot \left(\frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \color{blue}{\frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot \color{blue}{{\tan^{-1}_* \frac{x.im}{x.re}}^{3}}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{\color{blue}{3}}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   7. lift-atan2.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. lower-pow.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   10. lift-atan2.f64 N/A

       \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(\frac{1}{6}, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   11. lift-atan2.f64 32.9

       \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(0.16666666666666666, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, 0.5 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

8. Applied rewrites 32.9%

   \[\leadsto \cos \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{e^{y.re \cdot \log x.im}}{1 + \color{blue}{y.im \cdot \mathsf{fma}\left(y.im, \mathsf{fma}\left(0.16666666666666666, y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, 0.5 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))