powComplex, real part

Percentage Accurate: 40.8% → 80.1%

Time: 8.6s

Alternatives: 7

Speedup: 3.8×

• 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: 40.8% 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: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{if}\;x.re \leq -1.8 \cdot 10^{-22}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(\log \left(-x.re\right), y.im, \mathsf{fma}\left(0.5, \pi, t\_1\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 4.7 \cdot 10^{+18}:\\ \;\;\;\;e^{\log \left(\left|x.im\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)) (t_1 (* (atan2 x.im x.re) y.re)))
   (if (<= x.re -1.8e-22)
     (*
      (exp (- (* (log (* -1.0 x.re)) y.re) t_0))
      (sin (fma (log (- x.re)) y.im (fma 0.5 PI t_1))))
     (if (<= x.re 4.7e+18)
       (* (exp (- (* (log (fabs x.im)) y.re) t_0)) 1.0)
       (* (exp (- (* (log x.re) y.re) t_0)) (sin (fma PI 0.5 t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	double tmp;
	if (x_46_re <= -1.8e-22) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(fma(log(-x_46_re), y_46_im, fma(0.5, ((double) M_PI), t_1)));
	} else if (x_46_re <= 4.7e+18) {
		tmp = exp(((log(fabs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp(((log(x_46_re) * y_46_re) - t_0)) * sin(fma(((double) M_PI), 0.5, t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	tmp = 0.0
	if (x_46_re <= -1.8e-22)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(fma(log(Float64(-x_46_re)), y_46_im, fma(0.5, pi, t_1))));
	elseif (x_46_re <= 4.7e+18)
		tmp = Float64(exp(Float64(Float64(log(abs(x_46_im)) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(log(x_46_re) * y_46_re) - t_0)) * sin(fma(pi, 0.5, t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, If[LessEqual[x$46$re, -1.8e-22], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[Log[(-x$46$re)], $MachinePrecision] * y$46$im + N[(0.5 * Pi + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.7e+18], N[(N[Exp[N[(N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(Pi * 0.5 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.7999999999999999e-22

   1. Initial program 40.8%

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

   2. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 18.7

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

   4. Applied rewrites 18.7%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 36.9

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

   7. Applied rewrites 36.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

   9. Applied rewrites 36.8%

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

   if -1.7999999999999999e-22 < x.re < 4.7e18

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sum-to-mult N/A

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

      11. sqrt-prod N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-fabs.f64 65.6

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

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in x.re around 0

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

       1. lower-fabs.f64 72.5

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

   12. Applied rewrites 72.5%

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

   if 4.7e18 < x.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 35.3

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

   7. Applied rewrites 35.3%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sum-to-mult N/A

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

      7. lift-PI.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. sum-to-mult N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lower-sin.f64 35.2

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 35.2

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

   9. Applied rewrites 35.2%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-log.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. log-rec N/A

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

       6. remove-double-neg N/A

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

       7. lower-log.f64 35.2

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

   11. Applied rewrites 35.2%

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

Alternative 2: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -1.9 \cdot 10^{-83}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{elif}\;x.re \leq 4.7 \cdot 10^{+18}:\\ \;\;\;\;e^{\log \left(\left|x.im\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -1.9e-83)
     (* (exp (- (* (log (- x.re)) y.re) t_0)) 1.0)
     (if (<= x.re 4.7e+18)
       (* (exp (- (* (log (fabs x.im)) y.re) t_0)) 1.0)
       (*
        (exp (- (* (log x.re) y.re) t_0))
        (sin (fma PI 0.5 (* (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 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -1.9e-83) {
		tmp = exp(((log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_re <= 4.7e+18) {
		tmp = exp(((log(fabs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp(((log(x_46_re) * y_46_re) - t_0)) * sin(fma(((double) M_PI), 0.5, (atan2(x_46_im, x_46_re) * y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -1.9e-83)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_re)) * y_46_re) - t_0)) * 1.0);
	elseif (x_46_re <= 4.7e+18)
		tmp = Float64(exp(Float64(Float64(log(abs(x_46_im)) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(log(x_46_re) * y_46_re) - t_0)) * sin(fma(pi, 0.5, Float64(atan(x_46_im, x_46_re) * y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -1.9e-83], N[(N[Exp[N[(N[(N[Log[(-x$46$re)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 4.7e+18], N[(N[Exp[N[(N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(Pi * 0.5 + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.89999999999999988e-83

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 37.5

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

   12. Applied rewrites 37.5%

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

   if -1.89999999999999988e-83 < x.re < 4.7e18

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sum-to-mult N/A

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

      11. sqrt-prod N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-fabs.f64 65.6

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

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in x.re around 0

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

       1. lower-fabs.f64 72.5

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

   12. Applied rewrites 72.5%

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

   if 4.7e18 < x.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 35.3

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

   7. Applied rewrites 35.3%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sum-to-mult N/A

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

      7. lift-PI.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. sum-to-mult N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lower-sin.f64 35.2

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 35.2

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

   9. Applied rewrites 35.2%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-log.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. log-rec N/A

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

       6. remove-double-neg N/A

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

       7. lower-log.f64 35.2

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

   11. Applied rewrites 35.2%

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

Alternative 3: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -1.9 \cdot 10^{-83}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{elif}\;x.re \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;e^{\log \left(\left|x.im\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t\_0} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -1.9e-83)
     (* (exp (- (* (log (- x.re)) y.re) t_0)) 1.0)
     (if (<= x.re 4.5e-33)
       (* (exp (- (* (log (fabs x.im)) y.re) t_0)) 1.0)
       (*
        (exp (- (* (log x.re) y.re) t_0))
        (cos (* y.re (atan2 x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -1.9e-83) {
		tmp = exp(((log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_re <= 4.5e-33) {
		tmp = exp(((log(fabs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp(((log(x_46_re) * y_46_re) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46re <= (-1.9d-83)) then
        tmp = exp(((log(-x_46re) * y_46re) - t_0)) * 1.0d0
    else if (x_46re <= 4.5d-33) then
        tmp = exp(((log(abs(x_46im)) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((log(x_46re) * y_46re) - t_0)) * cos((y_46re * atan2(x_46im, x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -1.9e-83) {
		tmp = Math.exp(((Math.log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_re <= 4.5e-33) {
		tmp = Math.exp(((Math.log(Math.abs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = Math.exp(((Math.log(x_46_re) * y_46_re) - t_0)) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_re <= -1.9e-83:
		tmp = math.exp(((math.log(-x_46_re) * y_46_re) - t_0)) * 1.0
	elif x_46_re <= 4.5e-33:
		tmp = math.exp(((math.log(math.fabs(x_46_im)) * y_46_re) - t_0)) * 1.0
	else:
		tmp = math.exp(((math.log(x_46_re) * y_46_re) - t_0)) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -1.9e-83)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_re)) * y_46_re) - t_0)) * 1.0);
	elseif (x_46_re <= 4.5e-33)
		tmp = Float64(exp(Float64(Float64(log(abs(x_46_im)) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(log(x_46_re) * y_46_re) - t_0)) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_re <= -1.9e-83)
		tmp = exp(((log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	elseif (x_46_re <= 4.5e-33)
		tmp = exp(((log(abs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp(((log(x_46_re) * y_46_re) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -1.9e-83], N[(N[Exp[N[(N[(N[Log[(-x$46$re)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 4.5e-33], N[(N[Exp[N[(N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.89999999999999988e-83

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 37.5

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

   12. Applied rewrites 37.5%

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

   if -1.89999999999999988e-83 < x.re < 4.49999999999999991e-33

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sum-to-mult N/A

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

      11. sqrt-prod N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-fabs.f64 65.6

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

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in x.re around 0

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

       1. lower-fabs.f64 72.5

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

   12. Applied rewrites 72.5%

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

   if 4.49999999999999991e-33 < x.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 35.3

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

   7. Applied rewrites 35.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-log.f64 35.3

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

   9. Applied rewrites 35.3%

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

Alternative 4: 79.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -1.9 \cdot 10^{-83}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{elif}\;x.re \leq 46000:\\ \;\;\;\;e^{\log \left(\left|x.im\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{1}{x.re}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -1.9e-83)
     (* (exp (- (* (log (- x.re)) y.re) t_0)) 1.0)
     (if (<= x.re 46000.0)
       (* (exp (- (* (log (fabs x.im)) y.re) t_0)) 1.0)
       (* (exp (- (* (* -1.0 (log (/ 1.0 x.re))) y.re) t_0)) 1.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -1.9e-83) {
		tmp = exp(((log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_re <= 46000.0) {
		tmp = exp(((log(fabs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp((((-1.0 * log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46re <= (-1.9d-83)) then
        tmp = exp(((log(-x_46re) * y_46re) - t_0)) * 1.0d0
    else if (x_46re <= 46000.0d0) then
        tmp = exp(((log(abs(x_46im)) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((((-1.0d0) * log((1.0d0 / x_46re))) * y_46re) - t_0)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -1.9e-83) {
		tmp = Math.exp(((Math.log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_re <= 46000.0) {
		tmp = Math.exp(((Math.log(Math.abs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = Math.exp((((-1.0 * Math.log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_re <= -1.9e-83:
		tmp = math.exp(((math.log(-x_46_re) * y_46_re) - t_0)) * 1.0
	elif x_46_re <= 46000.0:
		tmp = math.exp(((math.log(math.fabs(x_46_im)) * y_46_re) - t_0)) * 1.0
	else:
		tmp = math.exp((((-1.0 * math.log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -1.9e-83)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_re)) * y_46_re) - t_0)) * 1.0);
	elseif (x_46_re <= 46000.0)
		tmp = Float64(exp(Float64(Float64(log(abs(x_46_im)) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(Float64(-1.0 * log(Float64(1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_re <= -1.9e-83)
		tmp = exp(((log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	elseif (x_46_re <= 46000.0)
		tmp = exp(((log(abs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp((((-1.0 * log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -1.9e-83], N[(N[Exp[N[(N[(N[Log[(-x$46$re)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 46000.0], N[(N[Exp[N[(N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[(-1.0 * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.89999999999999988e-83

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 37.5

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

   12. Applied rewrites 37.5%

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

   if -1.89999999999999988e-83 < x.re < 46000

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sum-to-mult N/A

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

      11. sqrt-prod N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-fabs.f64 65.6

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

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in x.re around 0

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

       1. lower-fabs.f64 72.5

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

   12. Applied rewrites 72.5%

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

   if 46000 < x.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 35.4

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

   10. Applied rewrites 35.4%

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

Alternative 5: 75.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -1.9 \cdot 10^{-83}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left|x.im\right|\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -1.9e-83)
     (* (exp (- (* (log (- x.re)) y.re) t_0)) 1.0)
     (* (exp (- (* (log (fabs x.im)) y.re) t_0)) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -1.9e-83) {
		tmp = exp(((log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp(((log(fabs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46re <= (-1.9d-83)) then
        tmp = exp(((log(-x_46re) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((log(abs(x_46im)) * y_46re) - t_0)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -1.9e-83) {
		tmp = Math.exp(((Math.log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = Math.exp(((Math.log(Math.abs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_re <= -1.9e-83:
		tmp = math.exp(((math.log(-x_46_re) * y_46_re) - t_0)) * 1.0
	else:
		tmp = math.exp(((math.log(math.fabs(x_46_im)) * y_46_re) - t_0)) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -1.9e-83)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_re)) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(log(abs(x_46_im)) * y_46_re) - t_0)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_re <= -1.9e-83)
		tmp = exp(((log(-x_46_re) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp(((log(abs(x_46_im)) * y_46_re) - t_0)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -1.9e-83], N[(N[Exp[N[(N[(N[Log[(-x$46$re)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.89999999999999988e-83

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 37.5

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

   12. Applied rewrites 37.5%

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

   if -1.89999999999999988e-83 < x.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sum-to-mult N/A

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

      11. sqrt-prod N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-fabs.f64 65.6

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

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in x.re around 0

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

       1. lower-fabs.f64 72.5

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

   12. Applied rewrites 72.5%

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

Alternative 6: 60.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-216}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -5e-216)
   (* (exp (- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im))) 1.0)
   (* (exp (- (* y.im (atan2 x.im x.re)))) 1.0)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -5e-216) {
		tmp = exp(((log(-x_46_re) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	} else {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-5d-216)) then
        tmp = exp(((log(-x_46re) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * 1.0d0
    else
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -5e-216) {
		tmp = Math.exp(((Math.log(-x_46_re) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	} else {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -5e-216:
		tmp = math.exp(((math.log(-x_46_re) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0
	else:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -5e-216)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_re)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0);
	else
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -5e-216)
		tmp = exp(((log(-x_46_re) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	else
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -5e-216], N[(N[Exp[N[(N[(N[Log[(-x$46$re)], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -5.00000000000000021e-216

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 37.5

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

   12. Applied rewrites 37.5%

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

   if -5.00000000000000021e-216 < x.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 62.2

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 63.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sum-to-mult N/A

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

      11. sqrt-prod N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-fabs.f64 65.6

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

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in y.re around 0

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

       1. lower-exp.f64 N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 53.3

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

   12. Applied rewrites 53.3%

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

Alternative 7: 53.3% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (exp (- (* y.im (atan2 x.im x.re)))) 1.0))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp(-(y_46_im * atan2(x_46_im, x_46_re))) * 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 = exp(-(y_46im * atan2(x_46im, x_46re))) * 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 Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * 1.0)
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * 1.0;
end

Wolfram

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

Derivation

1. Initial program 40.8%

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

2. Taylor expanded in y.im around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-atan2.f64 62.2

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

4. Applied rewrites 62.2%

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

5. Taylor expanded in y.re around 0

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

7. Applied rewrites 63.8%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. pow1/2 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sum-to-mult N/A

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

   11. sqrt-prod N/A

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

   12. lift-*.f64 N/A

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

   13. rem-sqrt-square-rev N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. associate-/l* N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. lower-fabs.f64 65.6

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

9. Applied rewrites 65.6%

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

10. Taylor expanded in y.re around 0

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

    1. lower-exp.f64 N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-atan2.f64 53.3

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

12. Applied rewrites 53.3%

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

Reproduce

herbie shell --seed 2025156 
(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)))))