powComplex, imaginary part

Percentage Accurate: 40.8% → 80.2%

Time: 10.7s

Alternatives: 22

Speedup: 1.6×

• 35.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{t\_1 \cdot y.re - t\_0}\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_5 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;t\_2 \cdot \sin \left(t\_1 \cdot y.im + t\_4\right) \leq -5 \cdot 10^{+193}:\\ \;\;\;\;t\_2 \cdot \left(\sin t\_3 + y.im \cdot \left(\cos t\_3 \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t\_5 \cdot y.re - t\_0} \cdot \sin \left(t\_5 \cdot y.im + t\_4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (exp (- (* t_1 y.re) t_0)))
        (t_3 (* y.re (atan2 x.im x.re)))
        (t_4 (* (atan2 x.im x.re) y.re))
        (t_5 (log (hypot x.re x.im))))
   (if (<= (* t_2 (sin (+ (* t_1 y.im) t_4))) -5e+193)
     (*
      t_2
      (+
       (sin t_3)
       (* y.im (* (cos t_3) (log (sqrt (fma x.im x.im (* x.re x.re))))))))
     (* (exp (- (* t_5 y.re) t_0)) (sin (+ (* t_5 y.im) t_4))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = exp(((t_1 * y_46_re) - t_0));
	double t_3 = y_46_re * atan2(x_46_im, x_46_re);
	double t_4 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_5 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if ((t_2 * sin(((t_1 * y_46_im) + t_4))) <= -5e+193) {
		tmp = t_2 * (sin(t_3) + (y_46_im * (cos(t_3) * log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))))));
	} else {
		tmp = exp(((t_5 * y_46_re) - t_0)) * sin(((t_5 * y_46_im) + t_4));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = exp(Float64(Float64(t_1 * y_46_re) - t_0))
	t_3 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_4 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_5 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (Float64(t_2 * sin(Float64(Float64(t_1 * y_46_im) + t_4))) <= -5e+193)
		tmp = Float64(t_2 * Float64(sin(t_3) + Float64(y_46_im * Float64(cos(t_3) * log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))))))));
	else
		tmp = Float64(exp(Float64(Float64(t_5 * y_46_re) - t_0)) * sin(Float64(Float64(t_5 * y_46_im) + t_4)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$5 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[Sin[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e+193], N[(t$95$2 * N[(N[Sin[t$95$3], $MachinePrecision] + N[(y$46$im * N[(N[Cos[t$95$3], $MachinePrecision] * N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t$95$5 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$5 * y$46$im), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -4.99999999999999972e193

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

   if -4.99999999999999972e193 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 40.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 40.8

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

   3. Applied rewrites 40.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 80.1

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

   5. Applied rewrites 80.1%

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

Alternative 2: 80.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

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.hypot(x_46_re, x_46_im));
	return Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 40.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 40.8

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

3. Applied rewrites 40.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 80.1

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

5. Applied rewrites 80.1%

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

Alternative 3: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 0.016:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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)))
          (sin (* y.re (atan2 x.im x.re))))))
   (if (<= y.re -3.1e-53)
     t_0
     (if (<= y.re 0.016)
       (*
        (exp (- (* y.im (atan2 x.im x.re))))
        (sin (+ (* (log (hypot x.re x.im)) y.im) (* (atan2 x.im x.re) y.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -3.1e-53) {
		tmp = t_0;
	} else if (y_46_re <= 0.016) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(((log(hypot(x_46_re, x_46_im)) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -3.1e-53) {
		tmp = t_0;
	} else if (y_46_re <= 0.016) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin(((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_re <= -3.1e-53:
		tmp = t_0
	elif y_46_re <= 0.016:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin(((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_re <= -3.1e-53)
		tmp = t_0;
	elseif (y_46_re <= 0.016)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_re <= -3.1e-53)
		tmp = t_0;
	elseif (y_46_re <= 0.016)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(((log(hypot(x_46_re, x_46_im)) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.1e-53], t$95$0, If[LessEqual[y$46$re, 0.016], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.10000000000000015e-53 or 0.016 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

   if -3.10000000000000015e-53 < y.re < 0.016

   1. Initial program 40.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 40.8

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

   3. Applied rewrites 40.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lift-atan2.f64 N/A

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

      4. lift-*.f64 53.3

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

   8. Applied rewrites 53.3%

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

Alternative 4: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ t_2 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq -3.2 \cdot 10^{+192}:\\ \;\;\;\;t\_1 \cdot e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right)}\\ \mathbf{elif}\;x.im \leq -3.9 \cdot 10^{+41}:\\ \;\;\;\;e^{-t\_2} \cdot \sin \left(\log \left(-1 \cdot x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.im \leq 3.15 \cdot 10^{+40}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t\_2} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sin t_0))
        (t_2 (* y.im (atan2 x.im x.re))))
   (if (<= x.im -3.2e+192)
     (* t_1 (exp (* -1.0 (* y.re (log (/ -1.0 x.im))))))
     (if (<= x.im -3.9e+41)
       (*
        (exp (- t_2))
        (sin (+ (* (log (* -1.0 x.im)) y.im) (* (atan2 x.im x.re) y.re))))
       (if (<= x.im 3.15e+40)
         (*
          (exp
           (-
            (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          t_1)
         (*
          (exp (- (* y.re (log x.im)) t_2))
          (sin (fma y.im (log x.im) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = y_46_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= -3.2e+192) {
		tmp = t_1 * exp((-1.0 * (y_46_re * log((-1.0 / x_46_im)))));
	} else if (x_46_im <= -3.9e+41) {
		tmp = exp(-t_2) * sin(((log((-1.0 * x_46_im)) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_im <= 3.15e+40) {
		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))) * t_1;
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(fma(y_46_im, log(x_46_im), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = Float64(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_im <= -3.2e+192)
		tmp = Float64(t_1 * exp(Float64(-1.0 * Float64(y_46_re * log(Float64(-1.0 / x_46_im))))));
	elseif (x_46_im <= -3.9e+41)
		tmp = Float64(exp(Float64(-t_2)) * sin(Float64(Float64(log(Float64(-1.0 * x_46_im)) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_im <= 3.15e+40)
		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))) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin(fma(y_46_im, log(x_46_im), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -3.2e+192], N[(t$95$1 * N[Exp[N[(-1.0 * N[(y$46$re * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -3.9e+41], N[(N[Exp[(-t$95$2)], $MachinePrecision] * N[Sin[N[(N[(N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 3.15e+40], 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] * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -3.20000000000000023e192

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 19.3

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

   7. Applied rewrites 19.3%

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

   if -3.20000000000000023e192 < x.im < -3.8999999999999997e41

   1. Initial program 40.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 40.8

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

   3. Applied rewrites 40.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lift-atan2.f64 N/A

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

      4. lift-*.f64 53.3

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in x.im around -inf

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

       1. lower-*.f64 23.4

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

   11. Applied rewrites 23.4%

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

   if -3.8999999999999997e41 < x.im < 3.15000000000000003e40

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

   if 3.15000000000000003e40 < x.im

   1. Initial program 40.8%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 30.7

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

   4. Applied rewrites 30.7%

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

Alternative 5: 59.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_0\\ \mathbf{if}\;y.re \leq -9.6 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 0.046:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1
         (*
          (exp
           (-
            (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          t_0)))
   (if (<= y.re -9.6e-6)
     t_1
     (if (<= y.re 0.046) (* (exp (- (* y.im (atan2 x.im x.re)))) t_0) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = 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))) * t_0;
	double tmp;
	if (y_46_re <= -9.6e-6) {
		tmp = t_1;
	} else if (y_46_re <= 0.046) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * t_0
    if (y_46re <= (-9.6d-6)) then
        tmp = t_1
    else if (y_46re <= 0.046d0) then
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * t_0
    else
        tmp = t_1
    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.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = 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))) * t_0;
	double tmp;
	if (y_46_re <= -9.6e-6) {
		tmp = t_1;
	} else if (y_46_re <= 0.046) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = 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))) * t_0
	tmp = 0
	if y_46_re <= -9.6e-6:
		tmp = t_1
	elif y_46_re <= 0.046:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = 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))) * t_0)
	tmp = 0.0
	if (y_46_re <= -9.6e-6)
		tmp = t_1;
	elseif (y_46_re <= 0.046)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = 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))) * t_0;
	tmp = 0.0;
	if (y_46_re <= -9.6e-6)
		tmp = t_1;
	elseif (y_46_re <= 0.046)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -9.6e-6], t$95$1, If[LessEqual[y$46$re, 0.046], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -9.5999999999999996e-6 or 0.045999999999999999 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

   if -9.5999999999999996e-6 < y.re < 0.045999999999999999

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 39.6

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

   7. Applied rewrites 39.6%

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

Alternative 6: 58.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ t_2 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -0.31:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+68}:\\ \;\;\;\;e^{-t\_2} \cdot t\_0\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+177}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t\_2} \cdot \sin \left(y.im \cdot \log x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (* t_0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re)))
        (t_2 (* y.im (atan2 x.im x.re))))
   (if (<= y.re -0.31)
     t_1
     (if (<= y.re 1.35e+68)
       (* (exp (- t_2)) t_0)
       (if (<= y.re 4e+177)
         (* (exp (- (* y.re (log x.re)) t_2)) (sin (* y.im (log x.re))))
         t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double t_2 = y_46_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -0.31) {
		tmp = t_1;
	} else if (y_46_re <= 1.35e+68) {
		tmp = exp(-t_2) * t_0;
	} else if (y_46_re <= 4e+177) {
		tmp = exp(((y_46_re * log(x_46_re)) - t_2)) * sin((y_46_im * log(x_46_re)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	t_2 = Float64(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -0.31)
		tmp = t_1;
	elseif (y_46_re <= 1.35e+68)
		tmp = Float64(exp(Float64(-t_2)) * t_0);
	elseif (y_46_re <= 4e+177)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_2)) * sin(Float64(y_46_im * log(x_46_re))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$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]}, Block[{t$95$2 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -0.31], t$95$1, If[LessEqual[y$46$re, 1.35e+68], N[(N[Exp[(-t$95$2)], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 4e+177], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.309999999999999998 or 4e177 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   if -0.309999999999999998 < y.re < 1.34999999999999995e68

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 39.6

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

   7. Applied rewrites 39.6%

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

   if 1.34999999999999995e68 < y.re < 4e177

   1. Initial program 40.8%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 32.4

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

   4. Applied rewrites 32.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 28.2

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

   7. Applied rewrites 28.2%

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

Alternative 7: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -0.31:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 0.46:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (* t_0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -0.31)
     t_1
     (if (<= y.re 0.46) (* (exp (- (* y.im (atan2 x.im x.re)))) t_0) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_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 <= -0.31) {
		tmp = t_1;
	} else if (y_46_re <= 0.46) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_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 <= -0.31)
		tmp = t_1;
	elseif (y_46_re <= 0.46)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$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, -0.31], t$95$1, If[LessEqual[y$46$re, 0.46], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.309999999999999998 or 0.46000000000000002 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   if -0.309999999999999998 < y.re < 0.46000000000000002

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 39.6

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

   7. Applied rewrites 39.6%

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

Alternative 8: 49.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {t\_0}^{y.re}\\ \mathbf{if}\;y.re \leq -3.15 \cdot 10^{-186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{-205}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log t\_0\right)\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-34}:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sqrt (fma x.im x.im (* x.re x.re))))
        (t_1 (* (sin (* y.re (atan2 x.im x.re))) (pow t_0 y.re))))
   (if (<= y.re -3.15e-186)
     t_1
     (if (<= y.re 6.4e-205)
       (* (exp (* y.im (- (atan2 x.im x.re)))) (sin (* y.im (log t_0))))
       (if (<= y.re 1.3e-34)
         (*
          (*
           y.re
           (fma
            -0.16666666666666666
            (* (* y.re y.re) (pow (atan2 x.im x.re) 3.0))
            (atan2 x.im x.re)))
          1.0)
         t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(t_0, y_46_re);
	double tmp;
	if (y_46_re <= -3.15e-186) {
		tmp = t_1;
	} else if (y_46_re <= 6.4e-205) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re))) * sin((y_46_im * log(t_0)));
	} else if (y_46_re <= 1.3e-34) {
		tmp = (y_46_re * fma(-0.16666666666666666, ((y_46_re * y_46_re) * pow(atan2(x_46_im, x_46_re), 3.0)), atan2(x_46_im, x_46_re))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_1 = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (t_0 ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.15e-186)
		tmp = t_1;
	elseif (y_46_re <= 6.4e-205)
		tmp = Float64(exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))) * sin(Float64(y_46_im * log(t_0))));
	elseif (y_46_re <= 1.3e-34)
		tmp = Float64(Float64(y_46_re * fma(-0.16666666666666666, Float64(Float64(y_46_re * y_46_re) * (atan(x_46_im, x_46_re) ^ 3.0)), atan(x_46_im, x_46_re))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$0, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.15e-186], t$95$1, If[LessEqual[y$46$re, 6.4e-205], N[(N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.3e-34], N[(N[(y$46$re * N[(-0.16666666666666666 * N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.1499999999999999e-186 or 1.3e-34 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   if -3.1499999999999999e-186 < y.re < 6.40000000000000018e-205

   1. Initial program 40.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower-exp.f64 N/A

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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

   4. Applied rewrites 22.4%

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

   if 6.40000000000000018e-205 < y.re < 1.3e-34

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in y.re around 0

      \[\leadsto \left(y.re \cdot \left(\frac{-1}{6} \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1 \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lift-atan2.f64 20.2

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

   10. Applied rewrites 20.2%

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

Alternative 9: 47.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{-185}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-34}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (* t_0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -3.5e-239)
     t_1
     (if (<= y.re 2.8e-185)
       (* (exp (- (* y.im (atan2 x.im x.re)))) (sin (* y.im (log x.re))))
       (if (<= y.re 1.3e-34) (* t_0 1.0) t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_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 <= -3.5e-239) {
		tmp = t_1;
	} else if (y_46_re <= 2.8e-185) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(x_46_re)));
	} else if (y_46_re <= 1.3e-34) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_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 <= -3.5e-239)
		tmp = t_1;
	elseif (y_46_re <= 2.8e-185)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_im * log(x_46_re))));
	elseif (y_46_re <= 1.3e-34)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$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, -3.5e-239], t$95$1, If[LessEqual[y$46$re, 2.8e-185], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.3e-34], N[(t$95$0 * 1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.50000000000000005e-239 or 1.3e-34 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   if -3.50000000000000005e-239 < y.re < 2.79999999999999991e-185

   1. Initial program 40.8%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 32.4

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

   4. Applied rewrites 32.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 17.7

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

   7. Applied rewrites 17.7%

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

   if 2.79999999999999991e-185 < y.re < 1.3e-34

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

Alternative 10: 43.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.re \leq -22500000000000:\\ \;\;\;\;t\_0 \cdot e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)}\\ \mathbf{elif}\;x.re \leq 2.7 \cdot 10^{-75}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 9.8 \cdot 10^{+122}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.re -22500000000000.0)
     (* t_0 (exp (* -1.0 (* y.re (log (/ -1.0 x.re))))))
     (if (<= x.re 2.7e-75)
       (* t_0 (pow (sqrt (* x.im x.im)) y.re))
       (if (<= x.re 9.8e+122)
         (* (exp (- (* y.im (atan2 x.im x.re)))) (sin (* y.im (log x.re))))
         (* t_0 (pow x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -22500000000000.0) {
		tmp = t_0 * exp((-1.0 * (y_46_re * log((-1.0 / x_46_re)))));
	} else if (x_46_re <= 2.7e-75) {
		tmp = t_0 * pow(sqrt((x_46_im * x_46_im)), y_46_re);
	} else if (x_46_re <= 9.8e+122) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(x_46_re)));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46re <= (-22500000000000.0d0)) then
        tmp = t_0 * exp(((-1.0d0) * (y_46re * log(((-1.0d0) / x_46re)))))
    else if (x_46re <= 2.7d-75) then
        tmp = t_0 * (sqrt((x_46im * x_46im)) ** y_46re)
    else if (x_46re <= 9.8d+122) then
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * sin((y_46im * log(x_46re)))
    else
        tmp = t_0 * (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 t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -22500000000000.0) {
		tmp = t_0 * Math.exp((-1.0 * (y_46_re * Math.log((-1.0 / x_46_re)))));
	} else if (x_46_re <= 2.7e-75) {
		tmp = t_0 * Math.pow(Math.sqrt((x_46_im * x_46_im)), y_46_re);
	} else if (x_46_re <= 9.8e+122) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((y_46_im * Math.log(x_46_re)));
	} else {
		tmp = t_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):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_re <= -22500000000000.0:
		tmp = t_0 * math.exp((-1.0 * (y_46_re * math.log((-1.0 / x_46_re)))))
	elif x_46_re <= 2.7e-75:
		tmp = t_0 * math.pow(math.sqrt((x_46_im * x_46_im)), y_46_re)
	elif x_46_re <= 9.8e+122:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((y_46_im * math.log(x_46_re)))
	else:
		tmp = t_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)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_re <= -22500000000000.0)
		tmp = Float64(t_0 * exp(Float64(-1.0 * Float64(y_46_re * log(Float64(-1.0 / x_46_re))))));
	elseif (x_46_re <= 2.7e-75)
		tmp = Float64(t_0 * (sqrt(Float64(x_46_im * x_46_im)) ^ y_46_re));
	elseif (x_46_re <= 9.8e+122)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_im * log(x_46_re))));
	else
		tmp = Float64(t_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)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_re <= -22500000000000.0)
		tmp = t_0 * exp((-1.0 * (y_46_re * log((-1.0 / x_46_re)))));
	elseif (x_46_re <= 2.7e-75)
		tmp = t_0 * (sqrt((x_46_im * x_46_im)) ^ y_46_re);
	elseif (x_46_re <= 9.8e+122)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(x_46_re)));
	else
		tmp = t_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_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -22500000000000.0], N[(t$95$0 * N[Exp[N[(-1.0 * N[(y$46$re * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.7e-75], N[(t$95$0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 9.8e+122], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -2.25e13

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 20.2

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

   7. Applied rewrites 20.2%

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

   if -2.25e13 < x.re < 2.6999999999999998e-75

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 36.8

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

   7. Applied rewrites 36.8%

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

   if 2.6999999999999998e-75 < x.re < 9.7999999999999995e122

   1. Initial program 40.8%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 32.4

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

   4. Applied rewrites 32.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 17.7

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

   7. Applied rewrites 17.7%

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

   if 9.7999999999999995e122 < x.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   7. Applied rewrites 31.5%

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

Alternative 11: 41.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq -0.68:\\ \;\;\;\;t\_0 \cdot e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right)}\\ \mathbf{elif}\;x.im \leq 1.16 \cdot 10^{-76}:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot e^{y.re \cdot \log x.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.im -0.68)
     (* t_0 (exp (* -1.0 (* y.re (log (/ -1.0 x.im))))))
     (if (<= x.im 1.16e-76)
       (* t_0 (pow x.re y.re))
       (* t_0 (exp (* y.re (log x.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -0.68) {
		tmp = t_0 * exp((-1.0 * (y_46_re * log((-1.0 / x_46_im)))));
	} else if (x_46_im <= 1.16e-76) {
		tmp = t_0 * pow(x_46_re, y_46_re);
	} else {
		tmp = t_0 * exp((y_46_re * log(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 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46im <= (-0.68d0)) then
        tmp = t_0 * exp(((-1.0d0) * (y_46re * log(((-1.0d0) / x_46im)))))
    else if (x_46im <= 1.16d-76) then
        tmp = t_0 * (x_46re ** y_46re)
    else
        tmp = t_0 * exp((y_46re * log(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.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -0.68) {
		tmp = t_0 * Math.exp((-1.0 * (y_46_re * Math.log((-1.0 / x_46_im)))));
	} else if (x_46_im <= 1.16e-76) {
		tmp = t_0 * Math.pow(x_46_re, y_46_re);
	} else {
		tmp = t_0 * Math.exp((y_46_re * Math.log(x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_im <= -0.68:
		tmp = t_0 * math.exp((-1.0 * (y_46_re * math.log((-1.0 / x_46_im)))))
	elif x_46_im <= 1.16e-76:
		tmp = t_0 * math.pow(x_46_re, y_46_re)
	else:
		tmp = t_0 * math.exp((y_46_re * math.log(x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -0.68)
		tmp = Float64(t_0 * exp(Float64(-1.0 * Float64(y_46_re * log(Float64(-1.0 / x_46_im))))));
	elseif (x_46_im <= 1.16e-76)
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	else
		tmp = Float64(t_0 * exp(Float64(y_46_re * log(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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -0.68)
		tmp = t_0 * exp((-1.0 * (y_46_re * log((-1.0 / x_46_im)))));
	elseif (x_46_im <= 1.16e-76)
		tmp = t_0 * (x_46_re ^ y_46_re);
	else
		tmp = t_0 * exp((y_46_re * log(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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -0.68], N[(t$95$0 * N[Exp[N[(-1.0 * N[(y$46$re * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.16e-76], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -0.680000000000000049

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 19.3

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

   7. Applied rewrites 19.3%

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

   if -0.680000000000000049 < x.im < 1.1599999999999999e-76

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   7. Applied rewrites 31.5%

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

   if 1.1599999999999999e-76 < x.im

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around inf

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 18.8

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

   7. Applied rewrites 18.8%

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

   8. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 18.8

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

   10. Applied rewrites 18.8%

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

Alternative 12: 41.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq -0.68:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 1.16 \cdot 10^{-76}:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot e^{y.re \cdot \log x.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.im -0.68)
     (* t_0 (pow (sqrt (* x.im x.im)) y.re))
     (if (<= x.im 1.16e-76)
       (* t_0 (pow x.re y.re))
       (* t_0 (exp (* y.re (log x.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -0.68) {
		tmp = t_0 * pow(sqrt((x_46_im * x_46_im)), y_46_re);
	} else if (x_46_im <= 1.16e-76) {
		tmp = t_0 * pow(x_46_re, y_46_re);
	} else {
		tmp = t_0 * exp((y_46_re * log(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 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46im <= (-0.68d0)) then
        tmp = t_0 * (sqrt((x_46im * x_46im)) ** y_46re)
    else if (x_46im <= 1.16d-76) then
        tmp = t_0 * (x_46re ** y_46re)
    else
        tmp = t_0 * exp((y_46re * log(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.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -0.68) {
		tmp = t_0 * Math.pow(Math.sqrt((x_46_im * x_46_im)), y_46_re);
	} else if (x_46_im <= 1.16e-76) {
		tmp = t_0 * Math.pow(x_46_re, y_46_re);
	} else {
		tmp = t_0 * Math.exp((y_46_re * Math.log(x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_im <= -0.68:
		tmp = t_0 * math.pow(math.sqrt((x_46_im * x_46_im)), y_46_re)
	elif x_46_im <= 1.16e-76:
		tmp = t_0 * math.pow(x_46_re, y_46_re)
	else:
		tmp = t_0 * math.exp((y_46_re * math.log(x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -0.68)
		tmp = Float64(t_0 * (sqrt(Float64(x_46_im * x_46_im)) ^ y_46_re));
	elseif (x_46_im <= 1.16e-76)
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	else
		tmp = Float64(t_0 * exp(Float64(y_46_re * log(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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -0.68)
		tmp = t_0 * (sqrt((x_46_im * x_46_im)) ^ y_46_re);
	elseif (x_46_im <= 1.16e-76)
		tmp = t_0 * (x_46_re ^ y_46_re);
	else
		tmp = t_0 * exp((y_46_re * log(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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -0.68], N[(t$95$0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.16e-76], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -0.680000000000000049

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 36.8

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

   7. Applied rewrites 36.8%

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

   if -0.680000000000000049 < x.im < 1.1599999999999999e-76

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   7. Applied rewrites 31.5%

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

   if 1.1599999999999999e-76 < x.im

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around inf

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 18.8

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

   7. Applied rewrites 18.8%

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

   8. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 18.8

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

   10. Applied rewrites 18.8%

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

Alternative 13: 35.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -530000:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{-185}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+191}:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -530000.0)
     (* t_0 (pow x.re y.re))
     (if (<= y.re 2.8e-185)
       (* (exp (- (* y.im (atan2 x.im x.re)))) (sin (* y.im (log x.re))))
       (if (<= y.re 7.5e+191)
         (*
          (*
           y.re
           (fma
            -0.16666666666666666
            (* (* y.re y.re) (pow (atan2 x.im x.re) 3.0))
            (atan2 x.im x.re)))
          1.0)
         (* t_0 (pow 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -530000.0) {
		tmp = t_0 * pow(x_46_re, y_46_re);
	} else if (y_46_re <= 2.8e-185) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(x_46_re)));
	} else if (y_46_re <= 7.5e+191) {
		tmp = (y_46_re * fma(-0.16666666666666666, ((y_46_re * y_46_re) * pow(atan2(x_46_im, x_46_re), 3.0)), atan2(x_46_im, x_46_re))) * 1.0;
	} else {
		tmp = t_0 * pow(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 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -530000.0)
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	elseif (y_46_re <= 2.8e-185)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_im * log(x_46_re))));
	elseif (y_46_re <= 7.5e+191)
		tmp = Float64(Float64(y_46_re * fma(-0.16666666666666666, Float64(Float64(y_46_re * y_46_re) * (atan(x_46_im, x_46_re) ^ 3.0)), atan(x_46_im, x_46_re))) * 1.0);
	else
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -530000.0], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.8e-185], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.5e+191], N[(N[(y$46$re * N[(-0.16666666666666666 * N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.3e5

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   7. Applied rewrites 31.5%

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

   if -5.3e5 < y.re < 2.79999999999999991e-185

   1. Initial program 40.8%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 32.4

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

   4. Applied rewrites 32.4%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 17.7

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

   7. Applied rewrites 17.7%

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

   if 2.79999999999999991e-185 < y.re < 7.5e191

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in y.re around 0

      \[\leadsto \left(y.re \cdot \left(\frac{-1}{6} \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1 \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lift-atan2.f64 20.2

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

   10. Applied rewrites 20.2%

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

   if 7.5e191 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 31.6

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

   7. Applied rewrites 31.6%

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

Alternative 14: 32.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -210000000000:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+191}:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -210000000000.0)
     (* t_0 (pow x.re y.re))
     (if (<= y.re 7.5e+191)
       (*
        (*
         y.re
         (fma
          -0.16666666666666666
          (* (* y.re y.re) (pow (atan2 x.im x.re) 3.0))
          (atan2 x.im x.re)))
        1.0)
       (* t_0 (pow 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -210000000000.0) {
		tmp = t_0 * pow(x_46_re, y_46_re);
	} else if (y_46_re <= 7.5e+191) {
		tmp = (y_46_re * fma(-0.16666666666666666, ((y_46_re * y_46_re) * pow(atan2(x_46_im, x_46_re), 3.0)), atan2(x_46_im, x_46_re))) * 1.0;
	} else {
		tmp = t_0 * pow(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 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -210000000000.0)
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	elseif (y_46_re <= 7.5e+191)
		tmp = Float64(Float64(y_46_re * fma(-0.16666666666666666, Float64(Float64(y_46_re * y_46_re) * (atan(x_46_im, x_46_re) ^ 3.0)), atan(x_46_im, x_46_re))) * 1.0);
	else
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -210000000000.0], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.5e+191], N[(N[(y$46$re * N[(-0.16666666666666666 * N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.1e11

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   7. Applied rewrites 31.5%

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

   if -2.1e11 < y.re < 7.5e191

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in y.re around 0

      \[\leadsto \left(y.re \cdot \left(\frac{-1}{6} \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1 \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lift-atan2.f64 20.2

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

   10. Applied rewrites 20.2%

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

   if 7.5e191 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 31.6

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

   7. Applied rewrites 31.6%

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

Alternative 15: 31.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+191}:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (sin (* y.re (atan2 x.im x.re))) (pow x.im y.re))))
   (if (<= y.re -2e+20)
     t_0
     (if (<= y.re 7.5e+191)
       (*
        (*
         y.re
         (fma
          -0.16666666666666666
          (* (* y.re y.re) (pow (atan2 x.im x.re) 3.0))
          (atan2 x.im x.re)))
        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 = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(x_46_im, y_46_re);
	double tmp;
	if (y_46_re <= -2e+20) {
		tmp = t_0;
	} else if (y_46_re <= 7.5e+191) {
		tmp = (y_46_re * fma(-0.16666666666666666, ((y_46_re * y_46_re) * pow(atan2(x_46_im, x_46_re), 3.0)), atan2(x_46_im, x_46_re))) * 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(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (x_46_im ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2e+20)
		tmp = t_0;
	elseif (y_46_re <= 7.5e+191)
		tmp = Float64(Float64(y_46_re * fma(-0.16666666666666666, Float64(Float64(y_46_re * y_46_re) * (atan(x_46_im, x_46_re) ^ 3.0)), atan(x_46_im, x_46_re))) * 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[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2e+20], t$95$0, If[LessEqual[y$46$re, 7.5e+191], N[(N[(y$46$re * N[(-0.16666666666666666 * N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2e20 or 7.5e191 < y.re

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 31.6

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

   7. Applied rewrites 31.6%

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

   if -2e20 < y.re < 7.5e191

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in y.re around 0

      \[\leadsto \left(y.re \cdot \left(\frac{-1}{6} \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1 \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lift-atan2.f64 20.2

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

   10. Applied rewrites 20.2%

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

Alternative 16: 23.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\\ \mathbf{if}\;y.im \leq -0.122:\\ \;\;\;\;y.re \cdot \left(y.re \cdot \left(t\_0 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-171}:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(t\_0, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (fma x.im x.im (* x.re x.re))))))
   (if (<= y.im -0.122)
     (* y.re (* y.re (* t_0 (atan2 x.im x.re))))
     (if (<= y.im 3.4e-171)
       (*
        (*
         y.re
         (fma
          -0.16666666666666666
          (* (* y.re y.re) (pow (atan2 x.im x.re) 3.0))
          (atan2 x.im x.re)))
        1.0)
       (*
        (* y.re y.re)
        (fma t_0 (atan2 x.im x.re) (/ (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(fma(x_46_im, x_46_im, (x_46_re * x_46_re))));
	double tmp;
	if (y_46_im <= -0.122) {
		tmp = y_46_re * (y_46_re * (t_0 * atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 3.4e-171) {
		tmp = (y_46_re * fma(-0.16666666666666666, ((y_46_re * y_46_re) * pow(atan2(x_46_im, x_46_re), 3.0)), atan2(x_46_im, x_46_re))) * 1.0;
	} else {
		tmp = (y_46_re * y_46_re) * fma(t_0, atan2(x_46_im, x_46_re), (atan2(x_46_im, x_46_re) / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))))
	tmp = 0.0
	if (y_46_im <= -0.122)
		tmp = Float64(y_46_re * Float64(y_46_re * Float64(t_0 * atan(x_46_im, x_46_re))));
	elseif (y_46_im <= 3.4e-171)
		tmp = Float64(Float64(y_46_re * fma(-0.16666666666666666, Float64(Float64(y_46_re * y_46_re) * (atan(x_46_im, x_46_re) ^ 3.0)), atan(x_46_im, x_46_re))) * 1.0);
	else
		tmp = Float64(Float64(y_46_re * y_46_re) * fma(t_0, atan(x_46_im, x_46_re), Float64(atan(x_46_im, x_46_re) / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -0.122], N[(y$46$re * N[(y$46$re * N[(t$95$0 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.4e-171], N[(N[(y$46$re * N[(-0.16666666666666666 * N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[(t$95$0 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -0.122

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-atan2.f64 N/A

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

      11. lift-atan2.f64 17.3

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in y.re around inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. lift-*.f64 14.9

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

   10. Applied rewrites 14.9%

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

   if -0.122 < y.im < 3.39999999999999985e-171

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in y.re around 0

      \[\leadsto \left(y.re \cdot \left(\frac{-1}{6} \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot 1 \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lift-atan2.f64 20.2

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

   10. Applied rewrites 20.2%

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

   if 3.39999999999999985e-171 < y.im

   1. Initial program 40.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in y.re around inf

      \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{\tan^{-1}_* \frac{x.im}{x.re}}{\color{blue}{y.re}}\right) \]

      2. pow2 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re} + \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      5. pow2 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      6. pow2 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      10. lift-log.f64 N/A

          \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      11. lift-atan2.f64 N/A

          \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

      13. lift-atan2.f64 17.6

          \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right), \tan^{-1}_* \frac{x.im}{x.re}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]

   10. Applied rewrites 17.6%

       \[\leadsto \left(y.re \cdot y.re\right) \cdot \mathsf{fma}\left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.re}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 19.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-67}:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 1.52 \cdot 10^{+49}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.5e-67)
   (*
    y.re
    (fma
     y.re
     (* (log (sqrt (* x.im x.im))) (atan2 x.im x.re))
     (atan2 x.im x.re)))
   (if (<= y.re 1.52e+49)
     (* (sin (* y.re (atan2 x.im x.re))) 1.0)
     (*
      (* y.re y.re)
      (* (log (sqrt (fma x.im x.im (* x.re x.re)))) (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.5e-67) {
		tmp = y_46_re * fma(y_46_re, (log(sqrt((x_46_im * x_46_im))) * atan2(x_46_im, x_46_re)), atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 1.52e+49) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * 1.0;
	} else {
		tmp = (y_46_re * y_46_re) * (log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))) * atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.5e-67)
		tmp = Float64(y_46_re * fma(y_46_re, Float64(log(sqrt(Float64(x_46_im * x_46_im))) * atan(x_46_im, x_46_re)), atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 1.52e+49)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * 1.0);
	else
		tmp = Float64(Float64(y_46_re * y_46_re) * Float64(log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))) * atan(x_46_im, x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.5e-67], N[(y$46$re * N[(y$46$re * N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.52e+49], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.4999999999999999e-67

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in x.re around 0

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lower-*.f64 17.5

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 17.5%

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -2.4999999999999999e-67 < y.re < 1.52e49

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 13.9%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot 1 \]

   if 1.52e49 < y.re

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in y.re around inf

      \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. pow2 N/A

         \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}\right) \]

      4. pow2 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-log.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-*.f64 14.0

          \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 14.0%

       \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 18.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(y.re, -1 \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -5e-310)
   (*
    y.re
    (fma
     y.re
     (* -1.0 (* (log (/ -1.0 x.re)) (atan2 x.im x.re)))
     (atan2 x.im x.re)))
   (* y.re (fma y.re (* (log x.re) (atan2 x.im x.re)) (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -5e-310) {
		tmp = y_46_re * fma(y_46_re, (-1.0 * (log((-1.0 / x_46_re)) * atan2(x_46_im, x_46_re))), atan2(x_46_im, x_46_re));
	} else {
		tmp = y_46_re * fma(y_46_re, (log(x_46_re) * atan2(x_46_im, x_46_re)), atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -5e-310)
		tmp = Float64(y_46_re * fma(y_46_re, Float64(-1.0 * Float64(log(Float64(-1.0 / x_46_re)) * atan(x_46_im, x_46_re))), atan(x_46_im, x_46_re)));
	else
		tmp = Float64(y_46_re * fma(y_46_re, Float64(log(x_46_re) * atan(x_46_im, x_46_re)), atan(x_46_im, x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -5e-310], N[(y$46$re * N[(y$46$re * N[(-1.0 * N[(N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(y$46$re * N[(N[Log[x$46$re], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.999999999999985e-310

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in x.re around -inf

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, -1 \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, -1 \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, -1 \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, -1 \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, -1 \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. lift-atan2.f64 9.4

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, -1 \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 9.4%

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, -1 \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -4.999999999999985e-310 < x.re

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in x.im around 0

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lift-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lift-atan2.f64 8.9

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 8.9%

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 18.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 0.072:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 0.072)
   (*
    y.re
    (fma
     y.re
     (* (log (sqrt (* x.im x.im))) (atan2 x.im x.re))
     (atan2 x.im x.re)))
   (* y.re (fma y.re (* (log x.re) (atan2 x.im x.re)) (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 0.072) {
		tmp = y_46_re * fma(y_46_re, (log(sqrt((x_46_im * x_46_im))) * atan2(x_46_im, x_46_re)), atan2(x_46_im, x_46_re));
	} else {
		tmp = y_46_re * fma(y_46_re, (log(x_46_re) * atan2(x_46_im, x_46_re)), atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 0.072)
		tmp = Float64(y_46_re * fma(y_46_re, Float64(log(sqrt(Float64(x_46_im * x_46_im))) * atan(x_46_im, x_46_re)), atan(x_46_im, x_46_re)));
	else
		tmp = Float64(y_46_re * fma(y_46_re, Float64(log(x_46_re) * atan(x_46_im, x_46_re)), atan(x_46_im, x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 0.072], N[(y$46$re * N[(y$46$re * N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(y$46$re * N[(N[Log[x$46$re], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 0.0719999999999999946

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in x.re around 0

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lower-*.f64 17.5

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 17.5%

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if 0.0719999999999999946 < x.re

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in x.im around 0

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lift-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lift-atan2.f64 8.9

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 8.9%

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 17.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7.5 \cdot 10^{-266}:\\ \;\;\;\;y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 7.5e-266)
   (*
    y.re
    (* y.re (* (log (sqrt (fma x.im x.im (* x.re x.re)))) (atan2 x.im x.re))))
   (* y.re (fma y.re (* (log x.re) (atan2 x.im x.re)) (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 7.5e-266) {
		tmp = y_46_re * (y_46_re * (log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))) * atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_re * fma(y_46_re, (log(x_46_re) * atan2(x_46_im, x_46_re)), atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 7.5e-266)
		tmp = Float64(y_46_re * Float64(y_46_re * Float64(log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))) * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(y_46_re * fma(y_46_re, Float64(log(x_46_re) * atan(x_46_im, x_46_re)), atan(x_46_im, x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 7.5e-266], N[(y$46$re * N[(y$46$re * N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(y$46$re * N[(N[Log[x$46$re], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.4999999999999995e-266

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in y.re around inf

      \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. pow2 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}\right)\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. lift-log.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lift-atan2.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. lift-*.f64 14.9

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   10. Applied rewrites 14.9%

       \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   if 7.4999999999999995e-266 < x.re

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in x.im around 0

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lift-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lift-atan2.f64 8.9

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 8.9%

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.re \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 16.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.35 \cdot 10^{-255}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(y.re, \log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im 2.35e-255)
   (*
    (* y.re y.re)
    (* (log (sqrt (fma x.im x.im (* x.re x.re)))) (atan2 x.im x.re)))
   (* y.re (fma y.re (* (log x.im) (atan2 x.im x.re)) (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= 2.35e-255) {
		tmp = (y_46_re * y_46_re) * (log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))) * atan2(x_46_im, x_46_re));
	} else {
		tmp = y_46_re * fma(y_46_re, (log(x_46_im) * atan2(x_46_im, x_46_re)), atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= 2.35e-255)
		tmp = Float64(Float64(y_46_re * y_46_re) * Float64(log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))) * atan(x_46_im, x_46_re)));
	else
		tmp = Float64(y_46_re * fma(y_46_re, Float64(log(x_46_im) * atan(x_46_im, x_46_re)), atan(x_46_im, x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, 2.35e-255], N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(y$46$re * N[(N[Log[x$46$im], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.34999999999999985e-255

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in y.re around inf

      \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. pow2 N/A

         \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}\right) \]

      4. pow2 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-log.f64 N/A

         \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-*.f64 14.0

          \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 14.0%

       \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   if 2.34999999999999985e-255 < x.im

   1. Initial program 40.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

      4. lift-atan2.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

      7. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

      9. pow2 N/A

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

      10. lift-*.f64 44.2

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   4. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. pow2 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      10. lift-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      11. lift-atan2.f64 17.3

          \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 17.3%

      \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in x.re around 0

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.im \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lift-log.f64 N/A

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lift-atan2.f64 9.3

         \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied rewrites 9.3%

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log x.im \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 14.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (* y.re y.re)
  (* (log (sqrt (fma x.im x.im (* x.re x.re)))) (atan2 x.im x.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (y_46_re * y_46_re) * (log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))) * atan2(x_46_im, x_46_re));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(y_46_re * y_46_re) * Float64(log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))) * atan(x_46_im, x_46_re)))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.8%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

2. Taylor expanded in y.im around 0

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \]

   3. lower-*.f64 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re} \]

   4. lift-atan2.f64 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re} \]

   5. lower-pow.f64 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{\color{blue}{y.re}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \]

   7. pow2 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)}^{y.re} \]

   8. lower-fma.f64 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}\right)}^{y.re} \]

   9. pow2 N/A

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

   10. lift-*.f64 44.2

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]

4. Applied rewrites 44.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

5. Taylor expanded in y.re around 0

   \[\leadsto y.re \cdot \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   4. lower-log.f64 N/A

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   5. pow2 N/A

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   6. pow2 N/A

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   9. lift-sqrt.f64 N/A

      \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. lift-atan2.f64 N/A

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. lift-atan2.f64 17.3

       \[\leadsto y.re \cdot \mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]

7. Applied rewrites 17.3%

   \[\leadsto y.re \cdot \color{blue}{\mathsf{fma}\left(y.re, \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

8. Taylor expanded in y.re around inf

   \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   2. pow2 N/A

      \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto {y.re}^{2} \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}\right) \]

   4. pow2 N/A

      \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   9. lift-log.f64 N/A

      \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. lift-atan2.f64 N/A

       \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. lift-*.f64 14.0

       \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

10. Applied rewrites 14.0%

    \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))