powComplex, real part

Percentage Accurate: 39.8% → 78.9%

Time: 8.9s

Alternatives: 15

Speedup: 3.4×

• 35.5% 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: 39.8% 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: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left|x.im\right|\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_3 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;t\_2 \cdot \cos t\_1\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-85}:\\ \;\;\;\;e^{-t\_3} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, y.im \cdot t\_0\right)\right)\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+57}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y.im, t\_0, t\_1\right)\right) \cdot e^{y.re \cdot t\_0 - t\_3}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (fabs x.im)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im))))
        (t_3 (* y.im (atan2 x.im x.re))))
   (if (<= y.re -5.1e-8)
     (* t_2 (cos t_1))
     (if (<= y.re 2e-85)
       (* (exp (- t_3)) (sin (fma 0.5 PI (* y.im t_0))))
       (if (<= y.re 2.5e+57)
         (* (cos (fma y.im t_0 t_1)) (exp (- (* y.re t_0) t_3)))
         (* t_2 1.0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(fabs(x_46_im));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_3 = y_46_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -5.1e-8) {
		tmp = t_2 * cos(t_1);
	} else if (y_46_re <= 2e-85) {
		tmp = exp(-t_3) * sin(fma(0.5, ((double) M_PI), (y_46_im * t_0)));
	} else if (y_46_re <= 2.5e+57) {
		tmp = cos(fma(y_46_im, t_0, t_1)) * exp(((y_46_re * t_0) - t_3));
	} else {
		tmp = t_2 * 1.0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(abs(x_46_im))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_3 = Float64(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -5.1e-8)
		tmp = Float64(t_2 * cos(t_1));
	elseif (y_46_re <= 2e-85)
		tmp = Float64(exp(Float64(-t_3)) * sin(fma(0.5, pi, Float64(y_46_im * t_0))));
	elseif (y_46_re <= 2.5e+57)
		tmp = Float64(cos(fma(y_46_im, t_0, t_1)) * exp(Float64(Float64(y_46_re * t_0) - t_3)));
	else
		tmp = Float64(t_2 * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.1e-8], N[(t$95$2 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2e-85], N[(N[Exp[(-t$95$3)], $MachinePrecision] * N[Sin[N[(0.5 * Pi + N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+57], N[(N[Cos[N[(y$46$im * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * 1.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.10000000000000001e-8

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   if -5.10000000000000001e-8 < y.re < 2e-85

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 70.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-atan2.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift-atan2.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 70.8

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

   9. Applied rewrites 70.8%

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

   10. Taylor expanded in y.re around 0

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

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

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

       2. lower-*.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. lift-atan2.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-log.f64 N/A

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

       12. lift-fabs.f64 52.6

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

   12. Applied rewrites 52.6%

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

   if 2e-85 < y.re < 2.49999999999999986e57

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 70.5%

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

   if 2.49999999999999986e57 < y.re

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

Alternative 2: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left|x.im\right|\right)\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+173}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+52}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im, t\_0, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \frac{\pi}{2}\right) \cdot e^{y.re \cdot t\_0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (fabs x.im))))
   (if (<= y.re -9.5e+173)
     (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
     (if (<= y.re 3e+52)
       (*
        (sin (+ (fma y.im t_0 (* y.re (atan2 x.im x.re))) (/ PI 2.0)))
        (exp (- (* y.re t_0) (* y.im (atan2 x.im x.re)))))
       (*
        (exp
         (-
          (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
          (* (atan2 x.im x.re) y.im)))
        1.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(fabs(x_46_im));
	double tmp;
	if (y_46_re <= -9.5e+173) {
		tmp = 1.0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	} else if (y_46_re <= 3e+52) {
		tmp = sin((fma(y_46_im, t_0, (y_46_re * atan2(x_46_im, x_46_re))) + (((double) M_PI) / 2.0))) * exp(((y_46_re * t_0) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(abs(x_46_im))
	tmp = 0.0
	if (y_46_re <= -9.5e+173)
		tmp = Float64(1.0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re));
	elseif (y_46_re <= 3e+52)
		tmp = Float64(sin(Float64(fma(y_46_im, t_0, Float64(y_46_re * atan(x_46_im, x_46_re))) + Float64(pi / 2.0))) * exp(Float64(Float64(y_46_re * t_0) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	else
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -9.5e+173], N[(1.0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3e+52], N[(N[Sin[N[(N[(y$46$im * t$95$0 + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * t$95$0), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.5000000000000005e173

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot {\color{blue}{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}}^{y.re} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

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

   if -9.5000000000000005e173 < y.re < 3e52

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 70.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-atan2.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift-atan2.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 3e52 < y.re

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

Alternative 3: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, y.im \cdot \log \left(\left|x.im\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -5.1e-8)
     (* t_0 (cos (* y.re (atan2 x.im x.re))))
     (if (<= y.re 2.5e+31)
       (*
        (exp (- (* y.im (atan2 x.im x.re))))
        (sin (fma 0.5 PI (* y.im (log (fabs x.im))))))
       (* 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 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -5.1e-8) {
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 2.5e+31) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(fma(0.5, ((double) M_PI), (y_46_im * log(fabs(x_46_im)))));
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -5.1e-8)
		tmp = Float64(t_0 * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 2.5e+31)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(0.5, pi, Float64(y_46_im * log(abs(x_46_im))))));
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -5.1e-8], N[(t$95$0 * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+31], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(0.5 * Pi + N[(y$46$im * N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.10000000000000001e-8

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   if -5.10000000000000001e-8 < y.re < 2.50000000000000013e31

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 70.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-atan2.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift-atan2.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 70.8

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

   9. Applied rewrites 70.8%

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

   10. Taylor expanded in y.re around 0

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

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

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

       2. lower-*.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. lift-atan2.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-log.f64 N/A

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

       12. lift-fabs.f64 52.6

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

   12. Applied rewrites 52.6%

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

   if 2.50000000000000013e31 < y.re

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

Alternative 4: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{if}\;y.re \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, y.im \cdot \log \left(\left|x.im\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (exp
           (-
            (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          1.0)))
   (if (<= y.re -5.1e-8)
     t_0
     (if (<= y.re 2.5e+31)
       (*
        (exp (- (* y.im (atan2 x.im x.re))))
        (sin (fma 0.5 PI (* y.im (log (fabs x.im))))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	double tmp;
	if (y_46_re <= -5.1e-8) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e+31) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(fma(0.5, ((double) M_PI), (y_46_im * log(fabs(x_46_im)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0)
	tmp = 0.0
	if (y_46_re <= -5.1e-8)
		tmp = t_0;
	elseif (y_46_re <= 2.5e+31)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(0.5, pi, Float64(y_46_im * log(abs(x_46_im))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -5.1e-8], t$95$0, If[LessEqual[y$46$re, 2.5e+31], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(0.5 * Pi + N[(y$46$im * N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.10000000000000001e-8 or 2.50000000000000013e31 < y.re

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

   if -5.10000000000000001e-8 < y.re < 2.50000000000000013e31

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 70.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-atan2.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift-atan2.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 70.8

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

   9. Applied rewrites 70.8%

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

   10. Taylor expanded in y.re around 0

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

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

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

       2. lower-*.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. lift-atan2.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-log.f64 N/A

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

       12. lift-fabs.f64 52.6

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

   12. Applied rewrites 52.6%

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

Alternative 5: 77.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{if}\;y.re \leq -0.0118:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(1 + -0.5 \cdot {\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (exp
           (-
            (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          1.0)))
   (if (<= y.re -0.0118)
     t_0
     (if (<= y.re 2.5e+31)
       (*
        (exp (- (* y.im (atan2 x.im x.re))))
        (+ 1.0 (* -0.5 (pow (* y.re (atan2 x.im x.re)) 2.0))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	double tmp;
	if (y_46_re <= -0.0118) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e+31) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * (1.0 + (-0.5 * pow((y_46_re * atan2(x_46_im, x_46_re)), 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * 1.0d0
    if (y_46re <= (-0.0118d0)) then
        tmp = t_0
    else if (y_46re <= 2.5d+31) then
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * (1.0d0 + ((-0.5d0) * ((y_46re * atan2(x_46im, x_46re)) ** 2.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	double tmp;
	if (y_46_re <= -0.0118) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e+31) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * (1.0 + (-0.5 * Math.pow((y_46_re * Math.atan2(x_46_im, x_46_re)), 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0
	tmp = 0
	if y_46_re <= -0.0118:
		tmp = t_0
	elif y_46_re <= 2.5e+31:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * (1.0 + (-0.5 * math.pow((y_46_re * math.atan2(x_46_im, x_46_re)), 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0)
	tmp = 0.0
	if (y_46_re <= -0.0118)
		tmp = t_0;
	elseif (y_46_re <= 2.5e+31)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * Float64(1.0 + Float64(-0.5 * (Float64(y_46_re * atan(x_46_im, x_46_re)) ^ 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	tmp = 0.0;
	if (y_46_re <= -0.0118)
		tmp = t_0;
	elseif (y_46_re <= 2.5e+31)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * (1.0 + (-0.5 * ((y_46_re * atan2(x_46_im, x_46_re)) ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -0.0118], t$95$0, If[LessEqual[y$46$re, 2.5e+31], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.0117999999999999997 or 2.50000000000000013e31 < y.re

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

   if -0.0117999999999999997 < y.re < 2.50000000000000013e31

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 51.6

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-atan2.f64 N/A

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

      6. lift-*.f64 50.1

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

   10. Applied rewrites 50.1%

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

Alternative 6: 67.0% accurate, 0.7× speedup

Math

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

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)))))
        (t_1 (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))))
   (if (<= (* t_1 (cos (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re)))) INFINITY)
     (* t_1 1.0)
     (* (cos (* y.im (log (fabs x.im)))) (pow (fabs x.im) 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))));
	double t_1 = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if ((t_1 * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)))) <= ((double) INFINITY)) {
		tmp = t_1 * 1.0;
	} else {
		tmp = cos((y_46_im * log(fabs(x_46_im)))) * pow(fabs(x_46_im), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_1 = Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if ((t_1 * Math.cos(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * 1.0;
	} else {
		tmp = Math.cos((y_46_im * Math.log(Math.abs(x_46_im)))) * Math.pow(Math.abs(x_46_im), y_46_re);
	}
	return tmp;
}

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))))
	t_1 = math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	tmp = 0
	if (t_1 * math.cos(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))) <= math.inf:
		tmp = t_1 * 1.0
	else:
		tmp = math.cos((y_46_im * math.log(math.fabs(x_46_im)))) * math.pow(math.fabs(x_46_im), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_1 = exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re)))) <= Inf)
		tmp = Float64(t_1 * 1.0);
	else
		tmp = Float64(cos(Float64(y_46_im * log(abs(x_46_im)))) * (abs(x_46_im) ^ y_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 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	t_1 = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	tmp = 0.0;
	if ((t_1 * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)))) <= Inf)
		tmp = t_1 * 1.0;
	else
		tmp = cos((y_46_im * log(abs(x_46_im)))) * (abs(x_46_im) ^ y_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[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[N[(t$95$1 * 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], Infinity], N[(t$95$1 * 1.0), $MachinePrecision], N[(N[Cos[N[(y$46$im * N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in y.im around 0

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

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

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

      2. pow2 N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

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

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

      6. lift-fabs.f64 51.1

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

   10. Applied rewrites 51.1%

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

   11. Taylor expanded in y.re around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \cos \left(y.im \cdot \log \left(\left|x.im\right|\right)\right) \cdot {\left(\left|x.im\right|\right)}^{y.re} \]

       2. lift-log.f64 N/A

          \[\leadsto \cos \left(y.im \cdot \log \left(\left|x.im\right|\right)\right) \cdot {\left(\left|x.im\right|\right)}^{y.re} \]

       3. lift-fabs.f64 53.1

          \[\leadsto \cos \left(y.im \cdot \log \left(\left|x.im\right|\right)\right) \cdot {\left(\left|x.im\right|\right)}^{y.re} \]

   13. Applied rewrites 53.1%

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

Alternative 7: 65.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-184}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (exp
           (-
            (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          1.0)))
   (if (<= y.re -3.2e-27) t_0 (if (<= y.re 1.85e-184) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	double tmp;
	if (y_46_re <= -3.2e-27) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-184) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * 1.0d0
    if (y_46re <= (-3.2d-27)) then
        tmp = t_0
    else if (y_46re <= 1.85d-184) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	double tmp;
	if (y_46_re <= -3.2e-27) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-184) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0
	tmp = 0
	if y_46_re <= -3.2e-27:
		tmp = t_0
	elif y_46_re <= 1.85e-184:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0)
	tmp = 0.0
	if (y_46_re <= -3.2e-27)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-184)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	tmp = 0.0;
	if (y_46_re <= -3.2e-27)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-184)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -3.2e-27], t$95$0, If[LessEqual[y$46$re, 1.85e-184], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.19999999999999991e-27 or 1.8499999999999999e-184 < y.re

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

   if -3.19999999999999991e-27 < y.re < 1.8499999999999999e-184

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.7%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.7 \cdot 10^{+187}:\\ \;\;\;\;\left(1 + -0.5 \cdot {\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}\right) \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -2.7e+187)
   (*
    (+ 1.0 (* -0.5 (pow (* y.re (atan2 x.im x.re)) 2.0)))
    (pow (fabs x.im) y.re))
   (*
    (exp
     (-
      (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
      (* (atan2 x.im x.re) y.im)))
    1.0)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.7e+187) {
		tmp = (1.0 + (-0.5 * pow((y_46_re * atan2(x_46_im, x_46_re)), 2.0))) * pow(fabs(x_46_im), y_46_re);
	} else {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 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_46im <= (-2.7d+187)) then
        tmp = (1.0d0 + ((-0.5d0) * ((y_46re * atan2(x_46im, x_46re)) ** 2.0d0))) * (abs(x_46im) ** y_46re)
    else
        tmp = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * 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_im <= -2.7e+187) {
		tmp = (1.0 + (-0.5 * Math.pow((y_46_re * Math.atan2(x_46_im, x_46_re)), 2.0))) * Math.pow(Math.abs(x_46_im), y_46_re);
	} else {
		tmp = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -2.7e+187:
		tmp = (1.0 + (-0.5 * math.pow((y_46_re * math.atan2(x_46_im, x_46_re)), 2.0))) * math.pow(math.fabs(x_46_im), y_46_re)
	else:
		tmp = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * 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_im <= -2.7e+187)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * (Float64(y_46_re * atan(x_46_im, x_46_re)) ^ 2.0))) * (abs(x_46_im) ^ y_46_re));
	else
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 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_im <= -2.7e+187)
		tmp = (1.0 + (-0.5 * ((y_46_re * atan2(x_46_im, x_46_re)) ^ 2.0))) * (abs(x_46_im) ^ y_46_re);
	else
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -2.7e+187], N[(N[(1.0 + N[(-0.5 * N[Power[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.70000000000000008e187

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-atan2.f64 N/A

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

      6. lift-*.f64 36.4

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

   10. Applied rewrites 36.4%

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

   11. Taylor expanded in x.re around 0

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

       1. pow2 N/A

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

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

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

       3. lift-pow.f64 N/A

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

       4. lift-fabs.f64 40.7

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

   13. Applied rewrites 40.7%

       \[\leadsto \left(1 + -0.5 \cdot {\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}\right) \cdot {\left(\left|x.im\right|\right)}^{\color{blue}{y.re}} \]

   if -2.70000000000000008e187 < x.im

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

Alternative 9: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 3.9 \cdot 10^{+161}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot {\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}\right) \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 3.9e+161)
   (*
    (exp
     (-
      (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
      (* (atan2 x.im x.re) y.im)))
    1.0)
   (* (+ 1.0 (* -0.5 (pow (* y.re (atan2 x.im x.re)) 2.0))) (pow x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 3.9e+161) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	} else {
		tmp = (1.0 + (-0.5 * pow((y_46_re * atan2(x_46_im, x_46_re)), 2.0))) * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 3.9d+161) then
        tmp = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * 1.0d0
    else
        tmp = (1.0d0 + ((-0.5d0) * ((y_46re * atan2(x_46im, x_46re)) ** 2.0d0))) * (x_46re ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 3.9e+161) {
		tmp = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	} else {
		tmp = (1.0 + (-0.5 * Math.pow((y_46_re * Math.atan2(x_46_im, x_46_re)), 2.0))) * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 3.9e+161:
		tmp = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0
	else:
		tmp = (1.0 + (-0.5 * math.pow((y_46_re * math.atan2(x_46_im, x_46_re)), 2.0))) * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 3.9e+161)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * (Float64(y_46_re * atan(x_46_im, x_46_re)) ^ 2.0))) * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 3.9e+161)
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	else
		tmp = (1.0 + (-0.5 * ((y_46_re * atan2(x_46_im, x_46_re)) ^ 2.0))) * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 3.9e+161], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[Power[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.9000000000000002e161

   1. Initial program 39.8%

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

   2. 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. lift-atan2.f64 60.9

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

   4. Applied rewrites 60.9%

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

   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 63.6%

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

   if 3.9000000000000002e161 < x.re

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-atan2.f64 N/A

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

      6. lift-*.f64 36.4

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

   10. Applied rewrites 36.4%

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

   11. Taylor expanded in x.re around 0

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

       1. lower-pow.f64 31.1

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

   13. Applied rewrites 31.1%

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

Alternative 10: 61.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -6e-25) t_0 (if (<= y.re 2.1e-33) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -6e-25) {
		tmp = t_0;
	} else if (y_46_re <= 2.1e-33) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -6e-25)
		tmp = t_0;
	elseif (y_46_re <= 2.1e-33)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6e-25], t$95$0, If[LessEqual[y$46$re, 2.1e-33], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.9999999999999995e-25 or 2.1e-33 < y.re

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot {\color{blue}{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}}^{y.re} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

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

   if -5.9999999999999995e-25 < y.re < 2.1e-33

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.7%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 33.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left|x.im\right|\right)\\ \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+34}:\\ \;\;\;\;1 + y.re \cdot \log \left(\sqrt{x.im \cdot x.im}\right)\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-37}:\\ \;\;\;\;1 + y.re \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 + y.re \cdot \left(t\_0 + 0.5 \cdot \frac{x.re \cdot x.re}{x.im \cdot x.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (fabs x.im))))
   (if (<= y.re -1.05e+34)
     (+ 1.0 (* y.re (log (sqrt (* x.im x.im)))))
     (if (<= y.re 3.2e-37)
       (+ 1.0 (* y.re t_0))
       (+ 1.0 (* y.re (+ t_0 (* 0.5 (/ (* x.re x.re) (* x.im x.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(fabs(x_46_im));
	double tmp;
	if (y_46_re <= -1.05e+34) {
		tmp = 1.0 + (y_46_re * log(sqrt((x_46_im * x_46_im))));
	} else if (y_46_re <= 3.2e-37) {
		tmp = 1.0 + (y_46_re * t_0);
	} else {
		tmp = 1.0 + (y_46_re * (t_0 + (0.5 * ((x_46_re * x_46_re) / (x_46_im * x_46_im)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(abs(x_46im))
    if (y_46re <= (-1.05d+34)) then
        tmp = 1.0d0 + (y_46re * log(sqrt((x_46im * x_46im))))
    else if (y_46re <= 3.2d-37) then
        tmp = 1.0d0 + (y_46re * t_0)
    else
        tmp = 1.0d0 + (y_46re * (t_0 + (0.5d0 * ((x_46re * x_46re) / (x_46im * x_46im)))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.abs(x_46_im));
	double tmp;
	if (y_46_re <= -1.05e+34) {
		tmp = 1.0 + (y_46_re * Math.log(Math.sqrt((x_46_im * x_46_im))));
	} else if (y_46_re <= 3.2e-37) {
		tmp = 1.0 + (y_46_re * t_0);
	} else {
		tmp = 1.0 + (y_46_re * (t_0 + (0.5 * ((x_46_re * x_46_re) / (x_46_im * x_46_im)))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.fabs(x_46_im))
	tmp = 0
	if y_46_re <= -1.05e+34:
		tmp = 1.0 + (y_46_re * math.log(math.sqrt((x_46_im * x_46_im))))
	elif y_46_re <= 3.2e-37:
		tmp = 1.0 + (y_46_re * t_0)
	else:
		tmp = 1.0 + (y_46_re * (t_0 + (0.5 * ((x_46_re * x_46_re) / (x_46_im * x_46_im)))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(abs(x_46_im))
	tmp = 0.0
	if (y_46_re <= -1.05e+34)
		tmp = Float64(1.0 + Float64(y_46_re * log(sqrt(Float64(x_46_im * x_46_im)))));
	elseif (y_46_re <= 3.2e-37)
		tmp = Float64(1.0 + Float64(y_46_re * t_0));
	else
		tmp = Float64(1.0 + Float64(y_46_re * Float64(t_0 + Float64(0.5 * Float64(Float64(x_46_re * x_46_re) / Float64(x_46_im * x_46_im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(abs(x_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.05e+34)
		tmp = 1.0 + (y_46_re * log(sqrt((x_46_im * x_46_im))));
	elseif (y_46_re <= 3.2e-37)
		tmp = 1.0 + (y_46_re * t_0);
	else
		tmp = 1.0 + (y_46_re * (t_0 + (0.5 * ((x_46_re * x_46_re) / (x_46_im * x_46_im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.05e+34], N[(1.0 + N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.2e-37], N[(1.0 + N[(y$46$re * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y$46$re * N[(t$95$0 + N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.05000000000000009e34

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 22.4

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

   10. Applied rewrites 22.4%

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

   if -1.05000000000000009e34 < y.re < 3.1999999999999999e-37

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

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

         \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]

      3. lift-fabs.f64 25.9

         \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]

   10. Applied rewrites 25.9%

       \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]

   if 3.1999999999999999e-37 < y.re

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

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

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

      3. unpow2 N/A

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

      4. pow2 N/A

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

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

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

      6. pow2 N/A

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

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

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

      8. sqr-abs-rev N/A

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

      9. pow2 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 20.0

          \[\leadsto 1 + y.re \cdot \left(\log \left(\left|x.im\right|\right) + 0.5 \cdot \frac{x.re \cdot x.re}{x.im \cdot x.im}\right) \]

   10. Applied rewrites 20.0%

       \[\leadsto 1 + y.re \cdot \left(\log \left(\left|x.im\right|\right) + 0.5 \cdot \color{blue}{\frac{x.re \cdot x.re}{x.im \cdot x.im}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 31.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+34}:\\ \;\;\;\;1 + y.re \cdot \log \left(\sqrt{x.im \cdot x.im}\right)\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-33}:\\ \;\;\;\;1 + y.re \cdot \log \left(\left|x.im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.05e+34)
   (+ 1.0 (* y.re (log (sqrt (* x.im x.im)))))
   (if (<= y.re 2.1e-33)
     (+ 1.0 (* y.re (log (fabs x.im))))
     (+ 1.0 (* y.re (log (sqrt (fma x.im x.im (* x.re x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.05e+34) {
		tmp = 1.0 + (y_46_re * log(sqrt((x_46_im * x_46_im))));
	} else if (y_46_re <= 2.1e-33) {
		tmp = 1.0 + (y_46_re * log(fabs(x_46_im)));
	} else {
		tmp = 1.0 + (y_46_re * log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.05e+34)
		tmp = Float64(1.0 + Float64(y_46_re * log(sqrt(Float64(x_46_im * x_46_im)))));
	elseif (y_46_re <= 2.1e-33)
		tmp = Float64(1.0 + Float64(y_46_re * log(abs(x_46_im))));
	else
		tmp = Float64(1.0 + Float64(y_46_re * log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.05e+34], N[(1.0 + N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-33], N[(1.0 + N[(y$46$re * N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.05000000000000009e34

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 22.4

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

   10. Applied rewrites 22.4%

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

   if -1.05000000000000009e34 < y.re < 2.1e-33

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

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

         \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]

      3. lift-fabs.f64 25.9

         \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]

   10. Applied rewrites 25.9%

       \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]

   if 2.1e-33 < y.re

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

      \[\leadsto 1 + \color{blue}{y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 31.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y.re \cdot \log \left(\sqrt{x.im \cdot x.im}\right)\\ \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 400000000000:\\ \;\;\;\;1 + y.re \cdot \log \left(\left|x.im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y.re (log (sqrt (* x.im x.im)))))))
   (if (<= y.re -1.05e+34)
     t_0
     (if (<= y.re 400000000000.0) (+ 1.0 (* y.re (log (fabs x.im)))) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 + (y_46_re * log(sqrt((x_46_im * x_46_im))));
	double tmp;
	if (y_46_re <= -1.05e+34) {
		tmp = t_0;
	} else if (y_46_re <= 400000000000.0) {
		tmp = 1.0 + (y_46_re * log(fabs(x_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y_46re * log(sqrt((x_46im * x_46im))))
    if (y_46re <= (-1.05d+34)) then
        tmp = t_0
    else if (y_46re <= 400000000000.0d0) then
        tmp = 1.0d0 + (y_46re * log(abs(x_46im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 + (y_46_re * Math.log(Math.sqrt((x_46_im * x_46_im))));
	double tmp;
	if (y_46_re <= -1.05e+34) {
		tmp = t_0;
	} else if (y_46_re <= 400000000000.0) {
		tmp = 1.0 + (y_46_re * Math.log(Math.abs(x_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 + (y_46_re * math.log(math.sqrt((x_46_im * x_46_im))))
	tmp = 0
	if y_46_re <= -1.05e+34:
		tmp = t_0
	elif y_46_re <= 400000000000.0:
		tmp = 1.0 + (y_46_re * math.log(math.fabs(x_46_im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 + Float64(y_46_re * log(sqrt(Float64(x_46_im * x_46_im)))))
	tmp = 0.0
	if (y_46_re <= -1.05e+34)
		tmp = t_0;
	elseif (y_46_re <= 400000000000.0)
		tmp = Float64(1.0 + Float64(y_46_re * log(abs(x_46_im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 + (y_46_re * log(sqrt((x_46_im * x_46_im))));
	tmp = 0.0;
	if (y_46_re <= -1.05e+34)
		tmp = t_0;
	elseif (y_46_re <= 400000000000.0)
		tmp = 1.0 + (y_46_re * log(abs(x_46_im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 + N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.05e+34], t$95$0, If[LessEqual[y$46$re, 400000000000.0], N[(1.0 + N[(y$46$re * N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.05000000000000009e34 or 4e11 < y.re

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 22.4

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

   10. Applied rewrites 22.4%

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

   if -1.05000000000000009e34 < y.re < 4e11

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

      8. lift-log.f64 24.0

         \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

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

         \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]

      3. lift-fabs.f64 25.9

         \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]

   10. Applied rewrites 25.9%

       \[\leadsto 1 + y.re \cdot \log \left(\left|x.im\right|\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 25.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ 1 + y.re \cdot \log \left(\left|x.re\right|\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ 1.0 (* y.re (log (fabs x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 + (y_46_re * log(fabs(x_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
    code = 1.0d0 + (y_46re * log(abs(x_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 + (y_46_re * Math.log(Math.abs(x_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 + (y_46_re * math.log(math.fabs(x_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 + Float64(y_46_re * log(abs(x_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 + (y_46_re * log(abs(x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 + N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.8%

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

2. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. pow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 51.4

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

4. Applied rewrites 51.4%

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

5. Taylor expanded in y.re around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. pow2 N/A

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

   5. lift-fma.f64 N/A

      \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   8. lift-log.f64 24.0

      \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

7. Applied rewrites 24.0%

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

8. Taylor expanded in x.im around 0

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

   1. pow2 N/A

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

   2. rem-sqrt-square N/A

      \[\leadsto 1 + y.re \cdot \log \left(\left|x.re\right|\right) \]

   3. lower-fabs.f64 25.8

      \[\leadsto 1 + y.re \cdot \log \left(\left|x.re\right|\right) \]

10. Applied rewrites 25.8%

    \[\leadsto 1 + y.re \cdot \log \left(\left|x.re\right|\right) \]
11. Add Preprocessing

Alternative 15: 25.7% accurate, 126.4× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 39.8%

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

2. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. pow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 51.4

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

4. Applied rewrites 51.4%

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

5. Taylor expanded in y.re around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. pow2 N/A

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

   5. lift-fma.f64 N/A

      \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

   8. lift-log.f64 24.0

      \[\leadsto 1 + y.re \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \]

7. Applied rewrites 24.0%

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

8. Taylor expanded in y.re around 0

   \[\leadsto 1 \]
9. Step-by-step derivation

10. Applied rewrites 25.7%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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)))))