powComplex, real part

Percentage Accurate: 40.4% → 75.3%

Time: 13.0s

Alternatives: 17

Speedup: 2.2×

• Could not converge on a ground truth

  1. x.re = 1.6103039053503067e-267
  2. x.im = -3.562469460080166e-243
  3. y.re = 2.1198192545759713e+248
  4. y.im = 3.58300642610764e+187

• 35.3% 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: 40.4% 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: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-14}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(t\_0, \frac{y.im}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.06 \cdot 10^{+60}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(t\_0 \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re))))
   (if (<= y.re -1.16e-14)
     (*
      (exp
       (-
        (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
        (* (atan2 x.im x.re) y.im)))
      (cos (* (fma t_0 (/ y.im y.re) (atan2 x.im x.re)) y.re)))
     (if (<= y.re 1.06e+60)
       (* (pow (exp (- y.im)) (atan2 x.im x.re)) (cos (* t_0 y.im)))
       (*
        (sin (fma (atan2 x.im x.re) y.re (/ (PI) 2.0)))
        (pow (hypot x.im x.re) y.re))))))

Derivation

1. Split input into 3 regimes

2. if y.re < -1.16000000000000007e-14

   1. Initial program 53.6%

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

   3. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-atan2.f64 84.3

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 86.0%

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

   if -1.16000000000000007e-14 < y.re < 1.0599999999999999e60

   1. Initial program 41.8%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-atan2.f64 82.1

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

   8. Applied rewrites 82.1%

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

   if 1.0599999999999999e60 < y.re

   1. Initial program 22.9%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 58.4

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

   8. Applied rewrites 58.4%

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

   10. Applied rewrites 75.1%

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

Alternative 2: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{-15}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.06 \cdot 10^{+60}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -9e-15)
   (*
    (exp
     (-
      (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
      (* (atan2 x.im x.re) y.im)))
    (cos (* (atan2 x.im x.re) y.re)))
   (if (<= y.re 1.06e+60)
     (*
      (pow (exp (- y.im)) (atan2 x.im x.re))
      (cos (* (log (hypot x.im x.re)) y.im)))
     (*
      (sin (fma (atan2 x.im x.re) y.re (/ (PI) 2.0)))
      (pow (hypot x.im x.re) y.re)))))

Derivation

1. Split input into 3 regimes

2. if y.re < -8.9999999999999995e-15

   1. Initial program 53.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 84.5

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

   5. Applied rewrites 84.5%

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

   if -8.9999999999999995e-15 < y.re < 1.0599999999999999e60

   1. Initial program 41.8%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-atan2.f64 82.1

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

   8. Applied rewrites 82.1%

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

   if 1.0599999999999999e60 < y.re

   1. Initial program 22.9%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 58.4

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

   8. Applied rewrites 58.4%

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

   10. Applied rewrites 75.1%

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

Alternative 3: 73.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{-8} \lor \neg \left(y.im \leq 3.2 \cdot 10^{+95}\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.3e-8) (not (<= y.im 3.2e+95)))
   (*
    (exp
     (-
      (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
      (* (atan2 x.im x.re) y.im)))
    (cos (* (atan2 x.im x.re) y.re)))
   (* (sin (* 0.5 (PI))) (pow (hypot x.im x.re) y.re))))

Derivation

1. Split input into 2 regimes

2. if y.im < -3.29999999999999977e-8 or 3.2000000000000001e95 < y.im

   1. Initial program 37.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 55.2

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

   5. Applied rewrites 55.2%

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

   if -3.29999999999999977e-8 < y.im < 3.2000000000000001e95

   1. Initial program 45.1%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 80.5

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

   8. Applied rewrites 80.5%

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

   10. Applied rewrites 85.7%

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

   11. Taylor expanded in y.re around 0

       \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(\color{blue}{x.im}, x.re\right)\right)}^{y.re} \]
   12. Step-by-step derivation

   13. Applied rewrites 87.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{-8} \lor \neg \left(y.im \leq 3.2 \cdot 10^{+95}\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+144}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(-x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+50}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log x.im \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.95e+144)
   (*
    (exp (* (- y.im) (atan2 x.im x.re)))
    (cos (+ (* (log (- x.im)) y.im) (* (atan2 x.im x.re) y.re))))
   (if (<= y.im 4.2e+50)
     (* (pow (hypot x.im x.re) y.re) (cos (* (log (hypot x.im x.re)) y.im)))
     (* (pow (exp (- y.im)) (atan2 x.im x.re)) (cos (* (log x.im) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.95e+144) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos(((log(-x_46_im) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (y_46_im <= 4.2e+50) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) * cos((log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = pow(exp(-y_46_im), atan2(x_46_im, x_46_re)) * cos((log(x_46_im) * y_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.95e+144) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.cos(((Math.log(-x_46_im) * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (y_46_im <= 4.2e+50) {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re) * Math.cos((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = Math.pow(Math.exp(-y_46_im), Math.atan2(x_46_im, x_46_re)) * Math.cos((Math.log(x_46_im) * y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.95e+144:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.cos(((math.log(-x_46_im) * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))
	elif y_46_im <= 4.2e+50:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re) * math.cos((math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = math.pow(math.exp(-y_46_im), math.atan2(x_46_im, x_46_re)) * math.cos((math.log(x_46_im) * y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.95e+144)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(Float64(Float64(log(Float64(-x_46_im)) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (y_46_im <= 4.2e+50)
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) * cos(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = Float64((exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)) * cos(Float64(log(x_46_im) * y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.95e+144)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos(((log(-x_46_im) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	elseif (y_46_im <= 4.2e+50)
		tmp = (hypot(x_46_im, x_46_re) ^ y_46_re) * cos((log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = (exp(-y_46_im) ^ atan2(x_46_im, x_46_re)) * cos((log(x_46_im) * y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.95e+144], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.2e+50], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Cos[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.95000000000000009e144

   1. Initial program 37.1%

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

   3. Taylor expanded in x.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 19.5

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

   5. Applied rewrites 19.5%

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

   6. Taylor expanded in x.im around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 19.6

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

   8. Applied rewrites 19.6%

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

   9. Taylor expanded in y.re around 0

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-atan2.f64 47.0

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

   11. Applied rewrites 47.0%

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

   if -1.95000000000000009e144 < y.im < 4.1999999999999999e50

   1. Initial program 47.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 81.2

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

   8. Applied rewrites 81.2%

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

   if 4.1999999999999999e50 < y.im

   1. Initial program 25.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 26.8

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

   8. Applied rewrites 26.8%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 14.8%

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

   12. Taylor expanded in y.re around 0

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

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

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

       2. exp-prod N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lower-atan2.f64 44.8

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

   14. Applied rewrites 44.8%

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

Alternative 5: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+243}:\\ \;\;\;\;{\left(1 - y.im\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\mathsf{fma}\left(\log x.re, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+50}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log x.im \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.9e+243)
   (*
    (pow (- 1.0 y.im) (atan2 x.im x.re))
    (cos (fma (log x.re) y.im (* (atan2 x.im x.re) y.re))))
   (if (<= y.im -2.7e+144)
     (*
      (sin (fma (atan2 x.im x.re) y.re (/ (PI) 2.0)))
      (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re))
     (if (<= y.im 4.2e+50)
       (* (pow (hypot x.im x.re) y.re) (cos (* (log (hypot x.im x.re)) y.im)))
       (* (pow (exp (- y.im)) (atan2 x.im x.re)) (cos (* (log x.im) y.im)))))))

Derivation

1. Split input into 4 regimes

2. if y.im < -2.90000000000000006e243

   1. Initial program 36.8%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 32.0

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

   5. Applied rewrites 32.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-atan2.f64 42.4

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

   8. Applied rewrites 42.4%

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

   9. Taylor expanded in y.im around 0

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

   11. Applied rewrites 42.4%

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

   if -2.90000000000000006e243 < y.im < -2.70000000000000015e144

   1. Initial program 37.4%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 26.3

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

   8. Applied rewrites 26.3%

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

   10. Applied rewrites 26.3%

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

   11. Taylor expanded in x.im around 0

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

   13. Applied rewrites 43.5%

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

   if -2.70000000000000015e144 < y.im < 4.1999999999999999e50

   1. Initial program 47.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 81.2

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

   8. Applied rewrites 81.2%

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

   if 4.1999999999999999e50 < y.im

   1. Initial program 25.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 26.8

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

   8. Applied rewrites 26.8%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 14.8%

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

   12. Taylor expanded in y.re around 0

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

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

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

       2. exp-prod N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lower-atan2.f64 44.8

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

   14. Applied rewrites 44.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+243}:\\ \;\;\;\;{\left(1 - y.im\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\mathsf{fma}\left(\log x.re, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+50}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log x.im \cdot y.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+154}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\mathsf{fma}\left(\log x.re, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+50}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log x.im \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.1e+154)
   (*
    (exp (* (- y.im) (atan2 x.im x.re)))
    (cos (fma (log x.re) y.im (* (atan2 x.im x.re) y.re))))
   (if (<= y.im 4.2e+50)
     (* (pow (hypot x.im x.re) y.re) (cos (* (log (hypot x.im x.re)) y.im)))
     (* (pow (exp (- y.im)) (atan2 x.im x.re)) (cos (* (log x.im) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.1e+154) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos(fma(log(x_46_re), y_46_im, (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (y_46_im <= 4.2e+50) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) * cos((log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = pow(exp(-y_46_im), atan2(x_46_im, x_46_re)) * cos((log(x_46_im) * y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.1e+154)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(fma(log(x_46_re), y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (y_46_im <= 4.2e+50)
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) * cos(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = Float64((exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)) * cos(Float64(log(x_46_im) * y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.1e+154], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Log[x$46$re], $MachinePrecision] * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.2e+50], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Cos[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.1e154

   1. Initial program 37.4%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-atan2.f64 40.3

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

   8. Applied rewrites 40.3%

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

   10. Applied rewrites 40.2%

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

   if -4.1e154 < y.im < 4.1999999999999999e50

   1. Initial program 47.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 40.9

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 80.4

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

   8. Applied rewrites 80.4%

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

   if 4.1999999999999999e50 < y.im

   1. Initial program 25.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 26.8

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

   8. Applied rewrites 26.8%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 14.8%

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

   12. Taylor expanded in y.re around 0

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

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

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

       2. exp-prod N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lower-atan2.f64 44.8

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

   14. Applied rewrites 44.8%

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

Alternative 7: 65.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+50}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log x.im \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.7e+144)
   (*
    (sin (fma (atan2 x.im x.re) y.re (/ (PI) 2.0)))
    (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re))
   (if (<= y.im 4.2e+50)
     (* (pow (hypot x.im x.re) y.re) (cos (* (log (hypot x.im x.re)) y.im)))
     (* (pow (exp (- y.im)) (atan2 x.im x.re)) (cos (* (log x.im) y.im))))))

Derivation

1. Split input into 3 regimes

2. if y.im < -2.70000000000000015e144

   1. Initial program 37.1%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 17.8

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

   8. Applied rewrites 17.8%

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

   10. Applied rewrites 17.8%

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

   11. Taylor expanded in x.im around 0

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

   13. Applied rewrites 31.6%

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

   if -2.70000000000000015e144 < y.im < 4.1999999999999999e50

   1. Initial program 47.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 81.2

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

   8. Applied rewrites 81.2%

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

   if 4.1999999999999999e50 < y.im

   1. Initial program 25.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 26.8

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

   8. Applied rewrites 26.8%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 14.8%

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

   12. Taylor expanded in y.re around 0

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

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

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

       2. exp-prod N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lower-atan2.f64 44.8

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

   14. Applied rewrites 44.8%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+50}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log x.im \cdot y.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-183}:\\ \;\;\;\;t\_1 \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (fma (atan2 x.im x.re) y.re (/ (PI) 2.0))))
        (t_1 (pow (hypot x.im x.re) y.re)))
   (if (<= y.im -2.7e+144)
     (* t_0 (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re))
     (if (<= y.im 3.6e-183)
       (* t_1 (cos (* (log (hypot x.im x.re)) y.im)))
       (* t_0 t_1)))))

Derivation

1. Split input into 3 regimes

2. if y.im < -2.70000000000000015e144

   1. Initial program 37.1%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 17.8

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

   8. Applied rewrites 17.8%

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

   10. Applied rewrites 17.8%

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

   11. Taylor expanded in x.im around 0

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

   13. Applied rewrites 31.6%

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

   if -2.70000000000000015e144 < y.im < 3.6000000000000001e-183

   1. Initial program 49.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if 3.6000000000000001e-183 < y.im

   1. Initial program 32.7%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 49.7

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

   8. Applied rewrites 49.7%

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

   10. Applied rewrites 55.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-183}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+145}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.5e+145)
   (*
    (sin (fma (atan2 x.im x.re) y.re (/ (PI) 2.0)))
    (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re))
   (* (sin (* 0.5 (PI))) (pow (hypot x.im x.re) y.re))))

Derivation

1. Split input into 2 regimes

2. if y.im < -7.50000000000000006e145

   1. Initial program 37.1%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 17.8

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

   8. Applied rewrites 17.8%

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

   10. Applied rewrites 17.8%

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

   11. Taylor expanded in x.im around 0

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

   13. Applied rewrites 31.6%

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

   if -7.50000000000000006e145 < y.im

   1. Initial program 42.2%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 64.7

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

   8. Applied rewrites 64.7%

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

   10. Applied rewrites 67.5%

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

   11. Taylor expanded in y.re around 0

       \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(\color{blue}{x.im}, x.re\right)\right)}^{y.re} \]
   12. Step-by-step derivation

   13. Applied rewrites 68.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+145}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+142}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.65e+142)
   (*
    (cos (* (atan2 x.im x.re) y.re))
    (pow (fma (/ (* x.im x.im) x.re) 0.5 x.re) y.re))
   (* (sin (* 0.5 (PI))) (pow (hypot x.im x.re) y.re))))

Derivation

1. Split input into 2 regimes

2. if y.im < -1.6500000000000001e142

   1. Initial program 37.1%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 17.8

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

   8. Applied rewrites 17.8%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 29.2%

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

   if -1.6500000000000001e142 < y.im

   1. Initial program 42.2%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 64.7

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

   8. Applied rewrites 64.7%

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

   10. Applied rewrites 67.5%

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

   11. Taylor expanded in y.re around 0

       \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(\color{blue}{x.im}, x.re\right)\right)}^{y.re} \]
   12. Step-by-step derivation

   13. Applied rewrites 68.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+142}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -480000000 \lor \neg \left(y.re \leq 1.46 \cdot 10^{+16}\right):\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -480000000.0) (not (<= y.re 1.46e+16)))
   (* (cos (* (atan2 x.im x.re) y.re)) (pow x.im y.re))
   (fma y.re (log (hypot x.re x.im)) 1.0)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -480000000.0) || !(y_46_re <= 1.46e+16)) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * pow(x_46_im, y_46_re);
	} else {
		tmp = fma(y_46_re, log(hypot(x_46_re, x_46_im)), 1.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -480000000.0) || !(y_46_re <= 1.46e+16))
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * (x_46_im ^ y_46_re));
	else
		tmp = fma(y_46_re, log(hypot(x_46_re, x_46_im)), 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -480000000.0], N[Not[LessEqual[y$46$re, 1.46e+16]], $MachinePrecision]], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.8e8 or 1.46e16 < y.re

   1. Initial program 39.8%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 66.3

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

   8. Applied rewrites 66.3%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 55.5%

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

   if -4.8e8 < y.re < 1.46e16

   1. Initial program 42.7%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 49.4

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 45.5%

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

   12. Taylor expanded in y.re around 0

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

   14. Applied rewrites 45.8%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -480000000 \lor \neg \left(y.re \leq 1.46 \cdot 10^{+16}\right):\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -480000000:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -480000000.0)
     (* t_0 (pow x.im y.re))
     (if (<= y.re 5e+21)
       (fma y.re (log (hypot x.re x.im)) 1.0)
       (* t_0 (pow 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 = cos((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -480000000.0) {
		tmp = t_0 * pow(x_46_im, y_46_re);
	} else if (y_46_re <= 5e+21) {
		tmp = fma(y_46_re, log(hypot(x_46_re, x_46_im)), 1.0);
	} else {
		tmp = t_0 * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -480000000.0)
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	elseif (y_46_re <= 5e+21)
		tmp = fma(y_46_re, log(hypot(x_46_re, x_46_im)), 1.0);
	else
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -480000000.0], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5e+21], N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.8e8

   1. Initial program 55.7%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 75.7

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 64.4%

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

   if -4.8e8 < y.re < 5e21

   1. Initial program 43.1%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 49.1

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 45.2%

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

   12. Taylor expanded in y.re around 0

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

   14. Applied rewrites 45.5%

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

   if 5e21 < y.re

   1. Initial program 21.4%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 57.3

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in x.im around 0

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

   11. Applied rewrites 46.9%

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

Alternative 13: 41.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(y.re \cdot y.re\right), {\tan^{-1}_* \frac{x.im}{x.re}}^{2}, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \cos \left(\log x.im \cdot y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -5e-310)
   (* (fma (* -0.5 (* y.re y.re)) (pow (atan2 x.im x.re) 2.0) 1.0) 1.0)
   (* (pow x.im y.re) (cos (* (log x.im) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -5e-310) {
		tmp = fma((-0.5 * (y_46_re * y_46_re)), pow(atan2(x_46_im, x_46_re), 2.0), 1.0) * 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re) * cos((log(x_46_im) * y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -5e-310)
		tmp = Float64(fma(Float64(-0.5 * Float64(y_46_re * y_46_re)), (atan(x_46_im, x_46_re) ^ 2.0), 1.0) * 1.0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * cos(Float64(log(x_46_im) * y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -5e-310], N[(N[(N[(-0.5 * N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * N[Cos[N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.5%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 52.5

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 23.3%

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

   12. Taylor expanded in y.re around 0

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

   14. Applied rewrites 27.6%

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

   if -4.999999999999985e-310 < x.im

   1. Initial program 43.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 34.5

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 62.6

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 61.9%

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

   12. Taylor expanded in x.re around 0

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

   14. Applied rewrites 56.3%

       \[\leadsto {x.im}^{\color{blue}{y.re}} \cdot \cos \left(\log x.im \cdot y.im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (sin (* 0.5 (PI))) (pow (hypot x.im x.re) y.re)))

Derivation

1. Initial program 41.4%

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

3. Taylor expanded in y.re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 63.8

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

5. Applied rewrites 63.8%

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

6. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-atan2.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. lower-hypot.f64 57.2

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

8. Applied rewrites 57.2%

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

10. Applied rewrites 59.5%

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

11. Taylor expanded in y.re around 0

    \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(\color{blue}{x.im}, x.re\right)\right)}^{y.re} \]
12. Step-by-step derivation

13. Applied rewrites 59.2%

    \[\leadsto \sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(\mathsf{hypot}\left(\color{blue}{x.im}, x.re\right)\right)}^{y.re} \]
14. Add Preprocessing

Alternative 15: 30.0% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (fma (* -0.5 (* y.re y.re)) (pow (atan2 x.im x.re) 2.0) 1.0) 1.0))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma((-0.5 * (y_46_re * y_46_re)), pow(atan2(x_46_im, x_46_re), 2.0), 1.0) * 1.0;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(fma(Float64(-0.5 * Float64(y_46_re * y_46_re)), (atan(x_46_im, x_46_re) ^ 2.0), 1.0) * 1.0)
end

Wolfram

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

Derivation

1. Initial program 41.4%

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

3. Taylor expanded in y.re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 63.8

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

5. Applied rewrites 63.8%

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

6. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-atan2.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. lower-hypot.f64 57.2

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

8. Applied rewrites 57.2%

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

9. Taylor expanded in y.re around 0

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

11. Applied rewrites 25.7%

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

12. Taylor expanded in y.re around 0

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

14. Applied rewrites 31.3%

    \[\leadsto \mathsf{fma}\left(-0.5 \cdot \left(y.re \cdot y.re\right), {\tan^{-1}_* \frac{x.im}{x.re}}^{2}, 1\right) \cdot 1 \]
15. Add Preprocessing

Alternative 16: 26.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y.re, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.re (log (hypot x.re x.im)) 1.0))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma(y_46_re, log(hypot(x_46_re, x_46_im)), 1.0);
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return fma(y_46_re, log(hypot(x_46_re, x_46_im)), 1.0)
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 41.4%

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

3. Taylor expanded in y.re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 63.8

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

5. Applied rewrites 63.8%

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

6. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-atan2.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. lower-hypot.f64 57.2

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

8. Applied rewrites 57.2%

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

9. Taylor expanded in y.re around 0

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

11. Applied rewrites 25.7%

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

12. Taylor expanded in y.re around 0

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

14. Applied rewrites 25.9%

    \[\leadsto \mathsf{fma}\left(y.re, \color{blue}{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, 1\right) \]
15. Add Preprocessing

Alternative 17: 26.0% accurate, 680.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 41.4%

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

3. Taylor expanded in y.re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 63.8

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

5. Applied rewrites 63.8%

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

6. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-atan2.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. lower-hypot.f64 57.2

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

8. Applied rewrites 57.2%

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

9. Taylor expanded in y.re around 0

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

11. Applied rewrites 25.7%

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

12. Taylor expanded in y.re around 0

    \[\leadsto 1 \]
13. Step-by-step derivation

14. Applied rewrites 25.7%

    \[\leadsto 1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024347 
(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)))))